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1922 lines
43 KiB
1922 lines
43 KiB
// math-inlining.
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const { abs, cos, sin, acos, atan2, sqrt, pow } = Math;
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// cube root function yielding real roots
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function crt(v) {
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return v < 0 ? -pow(-v, 1 / 3) : pow(v, 1 / 3);
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}
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// trig constants
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const pi = Math.PI,
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tau = 2 * pi,
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quart = pi / 2,
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// float precision significant decimal
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epsilon = 0.000001,
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// extremas used in bbox calculation and similar algorithms
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nMax = Number.MAX_SAFE_INTEGER || 9007199254740991,
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nMin = Number.MIN_SAFE_INTEGER || -9007199254740991,
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// a zero coordinate, which is surprisingly useful
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ZERO = { x: 0, y: 0, z: 0 };
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// Bezier utility functions
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const utils = {
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// Legendre-Gauss abscissae with n=24 (x_i values, defined at i=n as the roots of the nth order Legendre polynomial Pn(x))
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Tvalues: [
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-0.0640568928626056260850430826247450385909,
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0.0640568928626056260850430826247450385909,
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-0.1911188674736163091586398207570696318404,
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0.1911188674736163091586398207570696318404,
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-0.3150426796961633743867932913198102407864,
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0.3150426796961633743867932913198102407864,
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-0.4337935076260451384870842319133497124524,
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0.4337935076260451384870842319133497124524,
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-0.5454214713888395356583756172183723700107,
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0.5454214713888395356583756172183723700107,
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-0.6480936519369755692524957869107476266696,
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0.6480936519369755692524957869107476266696,
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-0.7401241915785543642438281030999784255232,
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0.7401241915785543642438281030999784255232,
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-0.8200019859739029219539498726697452080761,
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0.8200019859739029219539498726697452080761,
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-0.8864155270044010342131543419821967550873,
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0.8864155270044010342131543419821967550873,
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-0.9382745520027327585236490017087214496548,
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0.9382745520027327585236490017087214496548,
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-0.9747285559713094981983919930081690617411,
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0.9747285559713094981983919930081690617411,
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-0.9951872199970213601799974097007368118745,
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0.9951872199970213601799974097007368118745,
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],
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// Legendre-Gauss weights with n=24 (w_i values, defined by a function linked to in the Bezier primer article)
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Cvalues: [
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0.1279381953467521569740561652246953718517,
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0.1279381953467521569740561652246953718517,
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0.1258374563468282961213753825111836887264,
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0.1258374563468282961213753825111836887264,
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0.121670472927803391204463153476262425607,
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0.121670472927803391204463153476262425607,
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0.1155056680537256013533444839067835598622,
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0.1155056680537256013533444839067835598622,
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0.1074442701159656347825773424466062227946,
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0.1074442701159656347825773424466062227946,
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0.0976186521041138882698806644642471544279,
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0.0976186521041138882698806644642471544279,
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0.086190161531953275917185202983742667185,
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0.086190161531953275917185202983742667185,
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0.0733464814110803057340336152531165181193,
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0.0733464814110803057340336152531165181193,
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0.0592985849154367807463677585001085845412,
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0.0592985849154367807463677585001085845412,
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0.0442774388174198061686027482113382288593,
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0.0442774388174198061686027482113382288593,
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0.0285313886289336631813078159518782864491,
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0.0285313886289336631813078159518782864491,
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0.0123412297999871995468056670700372915759,
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0.0123412297999871995468056670700372915759,
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],
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arcfn: function (t, derivativeFn) {
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const d = derivativeFn(t);
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let l = d.x * d.x + d.y * d.y;
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if (typeof d.z !== "undefined") {
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l += d.z * d.z;
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}
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return sqrt(l);
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},
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compute: function (t, points, _3d) {
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// shortcuts
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if (t === 0) {
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points[0].t = 0;
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return points[0];
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}
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const order = points.length - 1;
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if (t === 1) {
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points[order].t = 1;
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return points[order];
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}
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const mt = 1 - t;
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let p = points;
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// constant?
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if (order === 0) {
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points[0].t = t;
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return points[0];
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}
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// linear?
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if (order === 1) {
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const ret = {
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x: mt * p[0].x + t * p[1].x,
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y: mt * p[0].y + t * p[1].y,
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t: t,
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};
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if (_3d) {
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ret.z = mt * p[0].z + t * p[1].z;
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}
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return ret;
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}
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// quadratic/cubic curve?
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if (order < 4) {
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let mt2 = mt * mt,
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t2 = t * t,
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a,
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b,
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c,
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d = 0;
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if (order === 2) {
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p = [p[0], p[1], p[2], ZERO];
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a = mt2;
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b = mt * t * 2;
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c = t2;
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} else if (order === 3) {
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a = mt2 * mt;
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b = mt2 * t * 3;
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c = mt * t2 * 3;
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d = t * t2;
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}
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const ret = {
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x: a * p[0].x + b * p[1].x + c * p[2].x + d * p[3].x,
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y: a * p[0].y + b * p[1].y + c * p[2].y + d * p[3].y,
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t: t,
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};
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if (_3d) {
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ret.z = a * p[0].z + b * p[1].z + c * p[2].z + d * p[3].z;
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}
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return ret;
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}
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// higher order curves: use de Casteljau's computation
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const dCpts = JSON.parse(JSON.stringify(points));
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while (dCpts.length > 1) {
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for (let i = 0; i < dCpts.length - 1; i++) {
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dCpts[i] = {
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x: dCpts[i].x + (dCpts[i + 1].x - dCpts[i].x) * t,
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y: dCpts[i].y + (dCpts[i + 1].y - dCpts[i].y) * t,
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};
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if (typeof dCpts[i].z !== "undefined") {
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dCpts[i] = dCpts[i].z + (dCpts[i + 1].z - dCpts[i].z) * t;
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}
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}
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dCpts.splice(dCpts.length - 1, 1);
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}
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dCpts[0].t = t;
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return dCpts[0];
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},
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computeWithRatios: function (t, points, ratios, _3d) {
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const mt = 1 - t,
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r = ratios,
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p = points;
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let f1 = r[0],
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f2 = r[1],
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f3 = r[2],
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f4 = r[3],
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d;
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// spec for linear
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f1 *= mt;
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f2 *= t;
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if (p.length === 2) {
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d = f1 + f2;
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return {
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x: (f1 * p[0].x + f2 * p[1].x) / d,
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y: (f1 * p[0].y + f2 * p[1].y) / d,
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z: !_3d ? false : (f1 * p[0].z + f2 * p[1].z) / d,
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t: t,
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};
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}
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// upgrade to quadratic
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f1 *= mt;
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f2 *= 2 * mt;
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f3 *= t * t;
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if (p.length === 3) {
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d = f1 + f2 + f3;
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return {
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x: (f1 * p[0].x + f2 * p[1].x + f3 * p[2].x) / d,
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y: (f1 * p[0].y + f2 * p[1].y + f3 * p[2].y) / d,
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z: !_3d ? false : (f1 * p[0].z + f2 * p[1].z + f3 * p[2].z) / d,
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t: t,
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};
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}
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// upgrade to cubic
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f1 *= mt;
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f2 *= 1.5 * mt;
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f3 *= 3 * mt;
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f4 *= t * t * t;
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if (p.length === 4) {
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d = f1 + f2 + f3 + f4;
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return {
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x: (f1 * p[0].x + f2 * p[1].x + f3 * p[2].x + f4 * p[3].x) / d,
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y: (f1 * p[0].y + f2 * p[1].y + f3 * p[2].y + f4 * p[3].y) / d,
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z: !_3d
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? false
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: (f1 * p[0].z + f2 * p[1].z + f3 * p[2].z + f4 * p[3].z) / d,
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t: t,
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};
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}
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},
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derive: function (points, _3d) {
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const dpoints = [];
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for (let p = points, d = p.length, c = d - 1; d > 1; d--, c--) {
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const list = [];
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for (let j = 0, dpt; j < c; j++) {
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dpt = {
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x: c * (p[j + 1].x - p[j].x),
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y: c * (p[j + 1].y - p[j].y),
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};
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if (_3d) {
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dpt.z = c * (p[j + 1].z - p[j].z);
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}
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list.push(dpt);
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}
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dpoints.push(list);
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p = list;
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}
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return dpoints;
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},
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between: function (v, m, M) {
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return (
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(m <= v && v <= M) ||
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utils.approximately(v, m) ||
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utils.approximately(v, M)
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);
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},
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approximately: function (a, b, precision) {
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return abs(a - b) <= (precision || epsilon);
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},
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length: function (derivativeFn) {
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const z = 0.5,
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len = utils.Tvalues.length;
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let sum = 0;
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for (let i = 0, t; i < len; i++) {
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t = z * utils.Tvalues[i] + z;
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sum += utils.Cvalues[i] * utils.arcfn(t, derivativeFn);
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}
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return z * sum;
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},
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map: function (v, ds, de, ts, te) {
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const d1 = de - ds,
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d2 = te - ts,
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v2 = v - ds,
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r = v2 / d1;
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return ts + d2 * r;
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},
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lerp: function (r, v1, v2) {
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const ret = {
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x: v1.x + r * (v2.x - v1.x),
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y: v1.y + r * (v2.y - v1.y),
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};
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if (!!v1.z && !!v2.z) {
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ret.z = v1.z + r * (v2.z - v1.z);
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}
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return ret;
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},
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pointToString: function (p) {
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let s = p.x + "/" + p.y;
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if (typeof p.z !== "undefined") {
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s += "/" + p.z;
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}
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return s;
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},
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pointsToString: function (points) {
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return "[" + points.map(utils.pointToString).join(", ") + "]";
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},
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copy: function (obj) {
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return JSON.parse(JSON.stringify(obj));
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},
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angle: function (o, v1, v2) {
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const dx1 = v1.x - o.x,
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dy1 = v1.y - o.y,
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dx2 = v2.x - o.x,
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dy2 = v2.y - o.y,
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cross = dx1 * dy2 - dy1 * dx2,
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dot = dx1 * dx2 + dy1 * dy2;
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return atan2(cross, dot);
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},
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// round as string, to avoid rounding errors
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round: function (v, d) {
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const s = "" + v;
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const pos = s.indexOf(".");
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return parseFloat(s.substring(0, pos + 1 + d));
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},
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dist: function (p1, p2) {
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const dx = p1.x - p2.x,
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dy = p1.y - p2.y;
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return sqrt(dx * dx + dy * dy);
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},
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closest: function (LUT, point) {
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let mdist = pow(2, 63),
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mpos,
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d;
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LUT.forEach(function (p, idx) {
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d = utils.dist(point, p);
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if (d < mdist) {
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mdist = d;
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mpos = idx;
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}
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});
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return { mdist: mdist, mpos: mpos };
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},
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abcratio: function (t, n) {
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// see ratio(t) note on http://pomax.github.io/bezierinfo/#abc
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if (n !== 2 && n !== 3) {
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return false;
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}
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if (typeof t === "undefined") {
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t = 0.5;
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} else if (t === 0 || t === 1) {
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return t;
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}
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const bottom = pow(t, n) + pow(1 - t, n),
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top = bottom - 1;
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return abs(top / bottom);
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},
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projectionratio: function (t, n) {
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// see u(t) note on http://pomax.github.io/bezierinfo/#abc
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if (n !== 2 && n !== 3) {
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return false;
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}
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if (typeof t === "undefined") {
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t = 0.5;
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} else if (t === 0 || t === 1) {
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return t;
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}
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const top = pow(1 - t, n),
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bottom = pow(t, n) + top;
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return top / bottom;
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},
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lli8: function (x1, y1, x2, y2, x3, y3, x4, y4) {
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const nx =
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(x1 * y2 - y1 * x2) * (x3 - x4) - (x1 - x2) * (x3 * y4 - y3 * x4),
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ny = (x1 * y2 - y1 * x2) * (y3 - y4) - (y1 - y2) * (x3 * y4 - y3 * x4),
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d = (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4);
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if (d == 0) {
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return false;
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}
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return { x: nx / d, y: ny / d };
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},
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lli4: function (p1, p2, p3, p4) {
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const x1 = p1.x,
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y1 = p1.y,
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x2 = p2.x,
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y2 = p2.y,
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x3 = p3.x,
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y3 = p3.y,
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x4 = p4.x,
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y4 = p4.y;
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return utils.lli8(x1, y1, x2, y2, x3, y3, x4, y4);
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},
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lli: function (v1, v2) {
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return utils.lli4(v1, v1.c, v2, v2.c);
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},
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makeline: function (p1, p2) {
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const x1 = p1.x,
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y1 = p1.y,
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x2 = p2.x,
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y2 = p2.y,
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dx = (x2 - x1) / 3,
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dy = (y2 - y1) / 3;
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return new Bezier(
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x1,
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y1,
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x1 + dx,
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y1 + dy,
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x1 + 2 * dx,
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y1 + 2 * dy,
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x2,
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y2
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);
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},
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findbbox: function (sections) {
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let mx = nMax,
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my = nMax,
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MX = nMin,
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MY = nMin;
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sections.forEach(function (s) {
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const bbox = s.bbox();
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if (mx > bbox.x.min) mx = bbox.x.min;
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if (my > bbox.y.min) my = bbox.y.min;
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if (MX < bbox.x.max) MX = bbox.x.max;
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if (MY < bbox.y.max) MY = bbox.y.max;
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});
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return {
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x: { min: mx, mid: (mx + MX) / 2, max: MX, size: MX - mx },
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y: { min: my, mid: (my + MY) / 2, max: MY, size: MY - my },
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};
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},
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shapeintersections: function (
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s1,
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bbox1,
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s2,
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bbox2,
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curveIntersectionThreshold
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) {
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if (!utils.bboxoverlap(bbox1, bbox2)) return [];
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const intersections = [];
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const a1 = [s1.startcap, s1.forward, s1.back, s1.endcap];
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const a2 = [s2.startcap, s2.forward, s2.back, s2.endcap];
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a1.forEach(function (l1) {
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if (l1.virtual) return;
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a2.forEach(function (l2) {
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if (l2.virtual) return;
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const iss = l1.intersects(l2, curveIntersectionThreshold);
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if (iss.length > 0) {
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iss.c1 = l1;
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iss.c2 = l2;
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iss.s1 = s1;
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iss.s2 = s2;
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intersections.push(iss);
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}
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});
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});
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return intersections;
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},
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makeshape: function (forward, back, curveIntersectionThreshold) {
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const bpl = back.points.length;
|
|
const fpl = forward.points.length;
|
|
const start = utils.makeline(back.points[bpl - 1], forward.points[0]);
|
|
const end = utils.makeline(forward.points[fpl - 1], back.points[0]);
|
|
const shape = {
|
|
startcap: start,
|
|
forward: forward,
|
|
back: back,
|
|
endcap: end,
|
|
bbox: utils.findbbox([start, forward, back, end]),
|
|
};
|
|
shape.intersections = function (s2) {
|
|
return utils.shapeintersections(
|
|
shape,
|
|
shape.bbox,
|
|
s2,
|
|
s2.bbox,
|
|
curveIntersectionThreshold
|
|
);
|
|
};
|
|
return shape;
|
|
},
|
|
|
|
getminmax: function (curve, d, list) {
|
|
if (!list) return { min: 0, max: 0 };
|
|
let min = nMax,
|
|
max = nMin,
|
|
t,
|
|
c;
|
|
if (list.indexOf(0) === -1) {
|
|
list = [0].concat(list);
|
|
}
|
|
if (list.indexOf(1) === -1) {
|
|
list.push(1);
|
|
}
|
|
for (let i = 0, len = list.length; i < len; i++) {
|
|
t = list[i];
|
|
c = curve.get(t);
|
|
if (c[d] < min) {
|
|
min = c[d];
|
|
}
|
|
if (c[d] > max) {
|
|
max = c[d];
|
|
}
|
|
}
|
|
return { min: min, mid: (min + max) / 2, max: max, size: max - min };
|
|
},
|
|
|
|
align: function (points, line) {
|
|
const tx = line.p1.x,
|
|
ty = line.p1.y,
|
|
a = -atan2(line.p2.y - ty, line.p2.x - tx),
|
|
d = function (v) {
|
|
return {
|
|
x: (v.x - tx) * cos(a) - (v.y - ty) * sin(a),
|
|
y: (v.x - tx) * sin(a) + (v.y - ty) * cos(a),
|
|
};
|
|
};
|
|
return points.map(d);
|
|
},
|
|
|
|
roots: function (points, line) {
|
|
line = line || { p1: { x: 0, y: 0 }, p2: { x: 1, y: 0 } };
|
|
|
|
const order = points.length - 1;
|
|
const aligned = utils.align(points, line);
|
|
const reduce = function (t) {
|
|
return 0 <= t && t <= 1;
|
|
};
|
|
|
|
if (order === 2) {
|
|
const a = aligned[0].y,
|
|
b = aligned[1].y,
|
|
c = aligned[2].y,
|
|
d = a - 2 * b + c;
|
|
if (d !== 0) {
|
|
const m1 = -sqrt(b * b - a * c),
|
|
m2 = -a + b,
|
|
v1 = -(m1 + m2) / d,
|
|
v2 = -(-m1 + m2) / d;
|
|
return [v1, v2].filter(reduce);
|
|
} else if (b !== c && d === 0) {
|
|
return [(2 * b - c) / (2 * b - 2 * c)].filter(reduce);
|
|
}
|
|
return [];
|
|
}
|
|
|
|
// see http://www.trans4mind.com/personal_development/mathematics/polynomials/cubicAlgebra.htm
|
|
const pa = aligned[0].y,
|
|
pb = aligned[1].y,
|
|
pc = aligned[2].y,
|
|
pd = aligned[3].y;
|
|
|
|
let d = -pa + 3 * pb - 3 * pc + pd,
|
|
a = 3 * pa - 6 * pb + 3 * pc,
|
|
b = -3 * pa + 3 * pb,
|
|
c = pa;
|
|
|
|
if (utils.approximately(d, 0)) {
|
|
// this is not a cubic curve.
|
|
if (utils.approximately(a, 0)) {
|
|
// in fact, this is not a quadratic curve either.
|
|
if (utils.approximately(b, 0)) {
|
|
// in fact in fact, there are no solutions.
|
|
return [];
|
|
}
|
|
// linear solution:
|
|
return [-c / b].filter(reduce);
|
|
}
|
|
// quadratic solution:
|
|
const q = sqrt(b * b - 4 * a * c),
|
|
a2 = 2 * a;
|
|
return [(q - b) / a2, (-b - q) / a2].filter(reduce);
|
|
}
|
|
|
|
// at this point, we know we need a cubic solution:
|
|
|
|
a /= d;
|
|
b /= d;
|
|
c /= d;
|
|
|
|
const p = (3 * b - a * a) / 3,
|
|
p3 = p / 3,
|
|
q = (2 * a * a * a - 9 * a * b + 27 * c) / 27,
|
|
q2 = q / 2,
|
|
discriminant = q2 * q2 + p3 * p3 * p3;
|
|
|
|
let u1, v1, x1, x2, x3;
|
|
if (discriminant < 0) {
|
|
const mp3 = -p / 3,
|
|
mp33 = mp3 * mp3 * mp3,
|
|
r = sqrt(mp33),
|
|
t = -q / (2 * r),
|
|
cosphi = t < -1 ? -1 : t > 1 ? 1 : t,
|
|
phi = acos(cosphi),
|
|
crtr = crt(r),
|
|
t1 = 2 * crtr;
|
|
x1 = t1 * cos(phi / 3) - a / 3;
|
|
x2 = t1 * cos((phi + tau) / 3) - a / 3;
|
|
x3 = t1 * cos((phi + 2 * tau) / 3) - a / 3;
|
|
return [x1, x2, x3].filter(reduce);
|
|
} else if (discriminant === 0) {
|
|
u1 = q2 < 0 ? crt(-q2) : -crt(q2);
|
|
x1 = 2 * u1 - a / 3;
|
|
x2 = -u1 - a / 3;
|
|
return [x1, x2].filter(reduce);
|
|
} else {
|
|
const sd = sqrt(discriminant);
|
|
u1 = crt(-q2 + sd);
|
|
v1 = crt(q2 + sd);
|
|
return [u1 - v1 - a / 3].filter(reduce);
|
|
}
|
|
},
|
|
|
|
droots: function (p) {
|
|
// quadratic roots are easy
|
|
if (p.length === 3) {
|
|
const a = p[0],
|
|
b = p[1],
|
|
c = p[2],
|
|
d = a - 2 * b + c;
|
|
if (d !== 0) {
|
|
const m1 = -sqrt(b * b - a * c),
|
|
m2 = -a + b,
|
|
v1 = -(m1 + m2) / d,
|
|
v2 = -(-m1 + m2) / d;
|
|
return [v1, v2];
|
|
} else if (b !== c && d === 0) {
|
|
return [(2 * b - c) / (2 * (b - c))];
|
|
}
|
|
return [];
|
|
}
|
|
|
|
// linear roots are even easier
|
|
if (p.length === 2) {
|
|
const a = p[0],
|
|
b = p[1];
|
|
if (a !== b) {
|
|
return [a / (a - b)];
|
|
}
|
|
return [];
|
|
}
|
|
|
|
return [];
|
|
},
|
|
|
|
curvature: function (t, d1, d2, _3d, kOnly) {
|
|
let num,
|
|
dnm,
|
|
adk,
|
|
dk,
|
|
k = 0,
|
|
r = 0;
|
|
|
|
//
|
|
// We're using the following formula for curvature:
|
|
//
|
|
// x'y" - y'x"
|
|
// k(t) = ------------------
|
|
// (x'² + y'²)^(3/2)
|
|
//
|
|
// from https://en.wikipedia.org/wiki/Radius_of_curvature#Definition
|
|
//
|
|
// With it corresponding 3D counterpart:
|
|
//
|
|
// sqrt( (y'z" - y"z')² + (z'x" - z"x')² + (x'y" - x"y')²)
|
|
// k(t) = -------------------------------------------------------
|
|
// (x'² + y'² + z'²)^(3/2)
|
|
//
|
|
|
|
const d = utils.compute(t, d1);
|
|
const dd = utils.compute(t, d2);
|
|
const qdsum = d.x * d.x + d.y * d.y;
|
|
|
|
if (_3d) {
|
|
num = sqrt(
|
|
pow(d.y * dd.z - dd.y * d.z, 2) +
|
|
pow(d.z * dd.x - dd.z * d.x, 2) +
|
|
pow(d.x * dd.y - dd.x * d.y, 2)
|
|
);
|
|
dnm = pow(qdsum + d.z * d.z, 3 / 2);
|
|
} else {
|
|
num = d.x * dd.y - d.y * dd.x;
|
|
dnm = pow(qdsum, 3 / 2);
|
|
}
|
|
|
|
if (num === 0 || dnm === 0) {
|
|
return { k: 0, r: 0 };
|
|
}
|
|
|
|
k = num / dnm;
|
|
r = dnm / num;
|
|
|
|
// We're also computing the derivative of kappa, because
|
|
// there is value in knowing the rate of change for the
|
|
// curvature along the curve. And we're just going to
|
|
// ballpark it based on an epsilon.
|
|
if (!kOnly) {
|
|
// compute k'(t) based on the interval before, and after it,
|
|
// to at least try to not introduce forward/backward pass bias.
|
|
const pk = utils.curvature(t - 0.001, d1, d2, _3d, true).k;
|
|
const nk = utils.curvature(t + 0.001, d1, d2, _3d, true).k;
|
|
dk = (nk - k + (k - pk)) / 2;
|
|
adk = (abs(nk - k) + abs(k - pk)) / 2;
|
|
}
|
|
|
|
return { k: k, r: r, dk: dk, adk: adk };
|
|
},
|
|
|
|
inflections: function (points) {
|
|
if (points.length < 4) return [];
|
|
|
|
// FIXME: TODO: add in inflection abstraction for quartic+ curves?
|
|
|
|
const p = utils.align(points, { p1: points[0], p2: points.slice(-1)[0] }),
|
|
a = p[2].x * p[1].y,
|
|
b = p[3].x * p[1].y,
|
|
c = p[1].x * p[2].y,
|
|
d = p[3].x * p[2].y,
|
|
v1 = 18 * (-3 * a + 2 * b + 3 * c - d),
|
|
v2 = 18 * (3 * a - b - 3 * c),
|
|
v3 = 18 * (c - a);
|
|
|
|
if (utils.approximately(v1, 0)) {
|
|
if (!utils.approximately(v2, 0)) {
|
|
let t = -v3 / v2;
|
|
if (0 <= t && t <= 1) return [t];
|
|
}
|
|
return [];
|
|
}
|
|
|
|
const trm = v2 * v2 - 4 * v1 * v3,
|
|
sq = Math.sqrt(trm),
|
|
d2 = 2 * v1;
|
|
|
|
if (utils.approximately(d2, 0)) return [];
|
|
|
|
return [(sq - v2) / d2, -(v2 + sq) / d2].filter(function (r) {
|
|
return 0 <= r && r <= 1;
|
|
});
|
|
},
|
|
|
|
bboxoverlap: function (b1, b2) {
|
|
const dims = ["x", "y"],
|
|
len = dims.length;
|
|
|
|
for (let i = 0, dim, l, t, d; i < len; i++) {
|
|
dim = dims[i];
|
|
l = b1[dim].mid;
|
|
t = b2[dim].mid;
|
|
d = (b1[dim].size + b2[dim].size) / 2;
|
|
if (abs(l - t) >= d) return false;
|
|
}
|
|
return true;
|
|
},
|
|
|
|
expandbox: function (bbox, _bbox) {
|
|
if (_bbox.x.min < bbox.x.min) {
|
|
bbox.x.min = _bbox.x.min;
|
|
}
|
|
if (_bbox.y.min < bbox.y.min) {
|
|
bbox.y.min = _bbox.y.min;
|
|
}
|
|
if (_bbox.z && _bbox.z.min < bbox.z.min) {
|
|
bbox.z.min = _bbox.z.min;
|
|
}
|
|
if (_bbox.x.max > bbox.x.max) {
|
|
bbox.x.max = _bbox.x.max;
|
|
}
|
|
if (_bbox.y.max > bbox.y.max) {
|
|
bbox.y.max = _bbox.y.max;
|
|
}
|
|
if (_bbox.z && _bbox.z.max > bbox.z.max) {
|
|
bbox.z.max = _bbox.z.max;
|
|
}
|
|
bbox.x.mid = (bbox.x.min + bbox.x.max) / 2;
|
|
bbox.y.mid = (bbox.y.min + bbox.y.max) / 2;
|
|
if (bbox.z) {
|
|
bbox.z.mid = (bbox.z.min + bbox.z.max) / 2;
|
|
}
|
|
bbox.x.size = bbox.x.max - bbox.x.min;
|
|
bbox.y.size = bbox.y.max - bbox.y.min;
|
|
if (bbox.z) {
|
|
bbox.z.size = bbox.z.max - bbox.z.min;
|
|
}
|
|
},
|
|
|
|
pairiteration: function (c1, c2, curveIntersectionThreshold) {
|
|
const c1b = c1.bbox(),
|
|
c2b = c2.bbox(),
|
|
r = 100000,
|
|
threshold = curveIntersectionThreshold || 0.5;
|
|
|
|
if (
|
|
c1b.x.size + c1b.y.size < threshold &&
|
|
c2b.x.size + c2b.y.size < threshold
|
|
) {
|
|
return [
|
|
(((r * (c1._t1 + c1._t2)) / 2) | 0) / r +
|
|
"/" +
|
|
(((r * (c2._t1 + c2._t2)) / 2) | 0) / r,
|
|
];
|
|
}
|
|
|
|
let cc1 = c1.split(0.5),
|
|
cc2 = c2.split(0.5),
|
|
pairs = [
|
|
{ left: cc1.left, right: cc2.left },
|
|
{ left: cc1.left, right: cc2.right },
|
|
{ left: cc1.right, right: cc2.right },
|
|
{ left: cc1.right, right: cc2.left },
|
|
];
|
|
|
|
pairs = pairs.filter(function (pair) {
|
|
return utils.bboxoverlap(pair.left.bbox(), pair.right.bbox());
|
|
});
|
|
|
|
let results = [];
|
|
|
|
if (pairs.length === 0) return results;
|
|
|
|
pairs.forEach(function (pair) {
|
|
results = results.concat(
|
|
utils.pairiteration(pair.left, pair.right, threshold)
|
|
);
|
|
});
|
|
|
|
results = results.filter(function (v, i) {
|
|
return results.indexOf(v) === i;
|
|
});
|
|
|
|
return results;
|
|
},
|
|
|
|
getccenter: function (p1, p2, p3) {
|
|
const dx1 = p2.x - p1.x,
|
|
dy1 = p2.y - p1.y,
|
|
dx2 = p3.x - p2.x,
|
|
dy2 = p3.y - p2.y,
|
|
dx1p = dx1 * cos(quart) - dy1 * sin(quart),
|
|
dy1p = dx1 * sin(quart) + dy1 * cos(quart),
|
|
dx2p = dx2 * cos(quart) - dy2 * sin(quart),
|
|
dy2p = dx2 * sin(quart) + dy2 * cos(quart),
|
|
// chord midpoints
|
|
mx1 = (p1.x + p2.x) / 2,
|
|
my1 = (p1.y + p2.y) / 2,
|
|
mx2 = (p2.x + p3.x) / 2,
|
|
my2 = (p2.y + p3.y) / 2,
|
|
// midpoint offsets
|
|
mx1n = mx1 + dx1p,
|
|
my1n = my1 + dy1p,
|
|
mx2n = mx2 + dx2p,
|
|
my2n = my2 + dy2p,
|
|
// intersection of these lines:
|
|
arc = utils.lli8(mx1, my1, mx1n, my1n, mx2, my2, mx2n, my2n),
|
|
r = utils.dist(arc, p1);
|
|
|
|
// arc start/end values, over mid point:
|
|
let s = atan2(p1.y - arc.y, p1.x - arc.x),
|
|
m = atan2(p2.y - arc.y, p2.x - arc.x),
|
|
e = atan2(p3.y - arc.y, p3.x - arc.x),
|
|
_;
|
|
|
|
// determine arc direction (cw/ccw correction)
|
|
if (s < e) {
|
|
// if s<m<e, arc(s, e)
|
|
// if m<s<e, arc(e, s + tau)
|
|
// if s<e<m, arc(e, s + tau)
|
|
if (s > m || m > e) {
|
|
s += tau;
|
|
}
|
|
if (s > e) {
|
|
_ = e;
|
|
e = s;
|
|
s = _;
|
|
}
|
|
} else {
|
|
// if e<m<s, arc(e, s)
|
|
// if m<e<s, arc(s, e + tau)
|
|
// if e<s<m, arc(s, e + tau)
|
|
if (e < m && m < s) {
|
|
_ = e;
|
|
e = s;
|
|
s = _;
|
|
} else {
|
|
e += tau;
|
|
}
|
|
}
|
|
// assign and done.
|
|
arc.s = s;
|
|
arc.e = e;
|
|
arc.r = r;
|
|
return arc;
|
|
},
|
|
|
|
numberSort: function (a, b) {
|
|
return a - b;
|
|
},
|
|
};
|
|
|
|
/**
|
|
* Poly Bezier
|
|
* @param {[type]} curves [description]
|
|
*/
|
|
class PolyBezier {
|
|
constructor(curves) {
|
|
this.curves = [];
|
|
this._3d = false;
|
|
if (!!curves) {
|
|
this.curves = curves;
|
|
this._3d = this.curves[0]._3d;
|
|
}
|
|
}
|
|
|
|
valueOf() {
|
|
return this.toString();
|
|
}
|
|
|
|
toString() {
|
|
return (
|
|
"[" +
|
|
this.curves
|
|
.map(function (curve) {
|
|
return utils.pointsToString(curve.points);
|
|
})
|
|
.join(", ") +
|
|
"]"
|
|
);
|
|
}
|
|
|
|
addCurve(curve) {
|
|
this.curves.push(curve);
|
|
this._3d = this._3d || curve._3d;
|
|
}
|
|
|
|
length() {
|
|
return this.curves
|
|
.map(function (v) {
|
|
return v.length();
|
|
})
|
|
.reduce(function (a, b) {
|
|
return a + b;
|
|
});
|
|
}
|
|
|
|
curve(idx) {
|
|
return this.curves[idx];
|
|
}
|
|
|
|
bbox() {
|
|
const c = this.curves;
|
|
var bbox = c[0].bbox();
|
|
for (var i = 1; i < c.length; i++) {
|
|
utils.expandbox(bbox, c[i].bbox());
|
|
}
|
|
return bbox;
|
|
}
|
|
|
|
offset(d) {
|
|
const offset = [];
|
|
this.curves.forEach(function (v) {
|
|
offset.push(...v.offset(d));
|
|
});
|
|
return new PolyBezier(offset);
|
|
}
|
|
}
|
|
|
|
/**
|
|
A javascript Bezier curve library by Pomax.
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Based on http://pomax.github.io/bezierinfo
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This code is MIT licensed.
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**/
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// math-inlining.
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const { abs: abs$1, min, max, cos: cos$1, sin: sin$1, acos: acos$1, sqrt: sqrt$1 } = Math;
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const pi$1 = Math.PI;
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/**
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* Bezier curve constructor.
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*
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* ...docs pending...
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*/
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class Bezier {
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constructor(coords) {
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let args =
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coords && coords.forEach ? coords : Array.from(arguments).slice();
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let coordlen = false;
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if (typeof args[0] === "object") {
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coordlen = args.length;
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const newargs = [];
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args.forEach(function (point) {
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["x", "y", "z"].forEach(function (d) {
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if (typeof point[d] !== "undefined") {
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newargs.push(point[d]);
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}
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});
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});
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args = newargs;
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}
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let higher = false;
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const len = args.length;
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if (coordlen) {
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if (coordlen > 4) {
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if (arguments.length !== 1) {
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throw new Error(
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"Only new Bezier(point[]) is accepted for 4th and higher order curves"
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);
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}
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higher = true;
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}
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} else {
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if (len !== 6 && len !== 8 && len !== 9 && len !== 12) {
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if (arguments.length !== 1) {
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throw new Error(
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"Only new Bezier(point[]) is accepted for 4th and higher order curves"
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);
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}
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}
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}
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const _3d = (this._3d =
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(!higher && (len === 9 || len === 12)) ||
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(coords && coords[0] && typeof coords[0].z !== "undefined"));
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const points = (this.points = []);
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for (let idx = 0, step = _3d ? 3 : 2; idx < len; idx += step) {
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var point = {
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x: args[idx],
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y: args[idx + 1],
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};
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if (_3d) {
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point.z = args[idx + 2];
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}
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points.push(point);
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}
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const order = (this.order = points.length - 1);
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const dims = (this.dims = ["x", "y"]);
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if (_3d) dims.push("z");
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this.dimlen = dims.length;
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const aligned = utils.align(points, { p1: points[0], p2: points[order] });
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this._linear = !aligned.some((p) => abs$1(p.y) > 0.0001);
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this._lut = [];
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this._t1 = 0;
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this._t2 = 1;
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this.update();
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}
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static quadraticFromPoints(p1, p2, p3, t) {
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if (typeof t === "undefined") {
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t = 0.5;
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}
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// shortcuts, although they're really dumb
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if (t === 0) {
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return new Bezier(p2, p2, p3);
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}
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if (t === 1) {
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return new Bezier(p1, p2, p2);
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}
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// real fitting.
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const abc = Bezier.getABC(2, p1, p2, p3, t);
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return new Bezier(p1, abc.A, p3);
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}
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static cubicFromPoints(S, B, E, t, d1) {
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if (typeof t === "undefined") {
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t = 0.5;
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}
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const abc = Bezier.getABC(3, S, B, E, t);
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if (typeof d1 === "undefined") {
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d1 = utils.dist(B, abc.C);
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}
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const d2 = (d1 * (1 - t)) / t;
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const selen = utils.dist(S, E),
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lx = (E.x - S.x) / selen,
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ly = (E.y - S.y) / selen,
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bx1 = d1 * lx,
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by1 = d1 * ly,
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bx2 = d2 * lx,
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by2 = d2 * ly;
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// derivation of new hull coordinates
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const e1 = { x: B.x - bx1, y: B.y - by1 },
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e2 = { x: B.x + bx2, y: B.y + by2 },
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A = abc.A,
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v1 = { x: A.x + (e1.x - A.x) / (1 - t), y: A.y + (e1.y - A.y) / (1 - t) },
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v2 = { x: A.x + (e2.x - A.x) / t, y: A.y + (e2.y - A.y) / t },
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nc1 = { x: S.x + (v1.x - S.x) / t, y: S.y + (v1.y - S.y) / t },
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nc2 = {
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x: E.x + (v2.x - E.x) / (1 - t),
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y: E.y + (v2.y - E.y) / (1 - t),
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};
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// ...done
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return new Bezier(S, nc1, nc2, E);
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}
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static getUtils() {
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return utils;
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}
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getUtils() {
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return Bezier.getUtils();
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}
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static get PolyBezier() {
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return PolyBezier;
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}
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valueOf() {
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return this.toString();
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}
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toString() {
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return utils.pointsToString(this.points);
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}
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toSVG() {
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if (this._3d) return false;
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const p = this.points,
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x = p[0].x,
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y = p[0].y,
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s = ["M", x, y, this.order === 2 ? "Q" : "C"];
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for (let i = 1, last = p.length; i < last; i++) {
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s.push(p[i].x);
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s.push(p[i].y);
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}
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return s.join(" ");
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}
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setRatios(ratios) {
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if (ratios.length !== this.points.length) {
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throw new Error("incorrect number of ratio values");
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}
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this.ratios = ratios;
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this._lut = []; // invalidate any precomputed LUT
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}
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verify() {
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const print = this.coordDigest();
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if (print !== this._print) {
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this._print = print;
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this.update();
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}
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}
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coordDigest() {
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return this.points
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.map(function (c, pos) {
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return "" + pos + c.x + c.y + (c.z ? c.z : 0);
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})
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.join("");
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}
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update() {
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// invalidate any precomputed LUT
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this._lut = [];
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this.dpoints = utils.derive(this.points, this._3d);
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this.computedirection();
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}
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computedirection() {
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const points = this.points;
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const angle = utils.angle(points[0], points[this.order], points[1]);
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this.clockwise = angle > 0;
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}
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length() {
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return utils.length(this.derivative.bind(this));
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}
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static getABC(order = 2, S, B, E, t = 0.5) {
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const u = utils.projectionratio(t, order),
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um = 1 - u,
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C = {
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x: u * S.x + um * E.x,
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y: u * S.y + um * E.y,
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},
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s = utils.abcratio(t, order),
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A = {
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x: B.x + (B.x - C.x) / s,
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y: B.y + (B.y - C.y) / s,
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};
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return { A, B, C, S, E };
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}
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getABC(t, B) {
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B = B || this.get(t);
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let S = this.points[0];
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let E = this.points[this.order];
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return Bezier.getABC(this.order, S, B, E, t);
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}
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getLUT(steps) {
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this.verify();
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steps = steps || 100;
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if (this._lut.length === steps) {
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return this._lut;
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}
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this._lut = [];
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// We want a range from 0 to 1 inclusive, so
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// we decrement and then use <= rather than <:
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steps--;
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for (let i = 0, p, t; i < steps; i++) {
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t = i / (steps - 1);
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p = this.compute(t);
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p.t = t;
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this._lut.push(p);
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}
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return this._lut;
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}
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on(point, error) {
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error = error || 5;
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const lut = this.getLUT(),
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hits = [];
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for (let i = 0, c, t = 0; i < lut.length; i++) {
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c = lut[i];
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if (utils.dist(c, point) < error) {
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hits.push(c);
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t += i / lut.length;
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}
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}
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if (!hits.length) return false;
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return (t /= hits.length);
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}
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project(point) {
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// step 1: coarse check
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const LUT = this.getLUT(),
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l = LUT.length - 1,
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closest = utils.closest(LUT, point),
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mpos = closest.mpos,
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t1 = (mpos - 1) / l,
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t2 = (mpos + 1) / l,
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step = 0.1 / l;
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// step 2: fine check
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let mdist = closest.mdist,
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t = t1,
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ft = t,
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p;
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mdist += 1;
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for (let d; t < t2 + step; t += step) {
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p = this.compute(t);
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d = utils.dist(point, p);
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if (d < mdist) {
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mdist = d;
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ft = t;
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}
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}
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ft = ft < 0 ? 0 : ft > 1 ? 1 : ft;
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p = this.compute(ft);
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p.t = ft;
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p.d = mdist;
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return p;
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}
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get(t) {
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return this.compute(t);
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}
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point(idx) {
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return this.points[idx];
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}
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compute(t) {
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if (this.ratios) {
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return utils.computeWithRatios(t, this.points, this.ratios, this._3d);
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}
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return utils.compute(t, this.points, this._3d, this.ratios);
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}
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raise() {
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const p = this.points,
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np = [p[0]],
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k = p.length;
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for (let i = 1, pi, pim; i < k; i++) {
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pi = p[i];
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pim = p[i - 1];
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np[i] = {
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x: ((k - i) / k) * pi.x + (i / k) * pim.x,
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y: ((k - i) / k) * pi.y + (i / k) * pim.y,
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};
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}
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np[k] = p[k - 1];
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return new Bezier(np);
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}
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derivative(t) {
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return utils.compute(t, this.dpoints[0]);
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}
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dderivative(t) {
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return utils.compute(t, this.dpoints[1]);
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}
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align() {
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let p = this.points;
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return new Bezier(utils.align(p, { p1: p[0], p2: p[p.length - 1] }));
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}
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curvature(t) {
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return utils.curvature(t, this.dpoints[0], this.dpoints[1], this._3d);
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}
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inflections() {
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return utils.inflections(this.points);
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}
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normal(t) {
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return this._3d ? this.__normal3(t) : this.__normal2(t);
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}
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__normal2(t) {
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const d = this.derivative(t);
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const q = sqrt$1(d.x * d.x + d.y * d.y);
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return { x: -d.y / q, y: d.x / q };
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}
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__normal3(t) {
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// see http://stackoverflow.com/questions/25453159
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const r1 = this.derivative(t),
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r2 = this.derivative(t + 0.01),
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q1 = sqrt$1(r1.x * r1.x + r1.y * r1.y + r1.z * r1.z),
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q2 = sqrt$1(r2.x * r2.x + r2.y * r2.y + r2.z * r2.z);
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r1.x /= q1;
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r1.y /= q1;
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r1.z /= q1;
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r2.x /= q2;
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r2.y /= q2;
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r2.z /= q2;
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// cross product
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const c = {
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x: r2.y * r1.z - r2.z * r1.y,
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y: r2.z * r1.x - r2.x * r1.z,
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z: r2.x * r1.y - r2.y * r1.x,
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};
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const m = sqrt$1(c.x * c.x + c.y * c.y + c.z * c.z);
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c.x /= m;
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c.y /= m;
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c.z /= m;
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// rotation matrix
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const R = [
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c.x * c.x,
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c.x * c.y - c.z,
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c.x * c.z + c.y,
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c.x * c.y + c.z,
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c.y * c.y,
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c.y * c.z - c.x,
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c.x * c.z - c.y,
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c.y * c.z + c.x,
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c.z * c.z,
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];
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// normal vector:
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const n = {
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x: R[0] * r1.x + R[1] * r1.y + R[2] * r1.z,
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y: R[3] * r1.x + R[4] * r1.y + R[5] * r1.z,
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z: R[6] * r1.x + R[7] * r1.y + R[8] * r1.z,
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};
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return n;
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}
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hull(t) {
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let p = this.points,
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_p = [],
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q = [],
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idx = 0;
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q[idx++] = p[0];
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q[idx++] = p[1];
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q[idx++] = p[2];
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if (this.order === 3) {
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q[idx++] = p[3];
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}
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// we lerp between all points at each iteration, until we have 1 point left.
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while (p.length > 1) {
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_p = [];
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for (let i = 0, pt, l = p.length - 1; i < l; i++) {
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pt = utils.lerp(t, p[i], p[i + 1]);
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q[idx++] = pt;
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_p.push(pt);
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}
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p = _p;
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}
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return q;
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}
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split(t1, t2) {
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// shortcuts
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if (t1 === 0 && !!t2) {
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return this.split(t2).left;
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}
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if (t2 === 1) {
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return this.split(t1).right;
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}
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// no shortcut: use "de Casteljau" iteration.
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const q = this.hull(t1);
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const result = {
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left:
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this.order === 2
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? new Bezier([q[0], q[3], q[5]])
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: new Bezier([q[0], q[4], q[7], q[9]]),
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right:
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this.order === 2
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? new Bezier([q[5], q[4], q[2]])
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: new Bezier([q[9], q[8], q[6], q[3]]),
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span: q,
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};
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// make sure we bind _t1/_t2 information!
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result.left._t1 = utils.map(0, 0, 1, this._t1, this._t2);
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result.left._t2 = utils.map(t1, 0, 1, this._t1, this._t2);
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result.right._t1 = utils.map(t1, 0, 1, this._t1, this._t2);
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result.right._t2 = utils.map(1, 0, 1, this._t1, this._t2);
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// if we have no t2, we're done
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if (!t2) {
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return result;
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}
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// if we have a t2, split again:
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t2 = utils.map(t2, t1, 1, 0, 1);
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return result.right.split(t2).left;
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}
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extrema() {
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const result = {};
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let roots = [];
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this.dims.forEach(
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function (dim) {
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let mfn = function (v) {
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return v[dim];
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};
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let p = this.dpoints[0].map(mfn);
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result[dim] = utils.droots(p);
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if (this.order === 3) {
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p = this.dpoints[1].map(mfn);
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result[dim] = result[dim].concat(utils.droots(p));
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}
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result[dim] = result[dim].filter(function (t) {
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return t >= 0 && t <= 1;
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});
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roots = roots.concat(result[dim].sort(utils.numberSort));
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}.bind(this)
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);
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result.values = roots.sort(utils.numberSort).filter(function (v, idx) {
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return roots.indexOf(v) === idx;
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});
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return result;
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}
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bbox() {
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const extrema = this.extrema(),
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result = {};
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this.dims.forEach(
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function (d) {
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result[d] = utils.getminmax(this, d, extrema[d]);
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}.bind(this)
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);
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return result;
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}
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overlaps(curve) {
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const lbbox = this.bbox(),
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tbbox = curve.bbox();
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return utils.bboxoverlap(lbbox, tbbox);
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}
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offset(t, d) {
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if (typeof d !== "undefined") {
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const c = this.get(t),
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n = this.normal(t);
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const ret = {
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c: c,
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n: n,
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x: c.x + n.x * d,
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y: c.y + n.y * d,
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};
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if (this._3d) {
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ret.z = c.z + n.z * d;
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}
|
|
return ret;
|
|
}
|
|
if (this._linear) {
|
|
const nv = this.normal(0),
|
|
coords = this.points.map(function (p) {
|
|
const ret = {
|
|
x: p.x + t * nv.x,
|
|
y: p.y + t * nv.y,
|
|
};
|
|
if (p.z && nv.z) {
|
|
ret.z = p.z + t * nv.z;
|
|
}
|
|
return ret;
|
|
});
|
|
return [new Bezier(coords)];
|
|
}
|
|
return this.reduce().map(function (s) {
|
|
if (s._linear) {
|
|
return s.offset(t)[0];
|
|
}
|
|
return s.scale(t);
|
|
});
|
|
}
|
|
|
|
simple() {
|
|
if (this.order === 3) {
|
|
const a1 = utils.angle(this.points[0], this.points[3], this.points[1]);
|
|
const a2 = utils.angle(this.points[0], this.points[3], this.points[2]);
|
|
if ((a1 > 0 && a2 < 0) || (a1 < 0 && a2 > 0)) return false;
|
|
}
|
|
const n1 = this.normal(0);
|
|
const n2 = this.normal(1);
|
|
let s = n1.x * n2.x + n1.y * n2.y;
|
|
if (this._3d) {
|
|
s += n1.z * n2.z;
|
|
}
|
|
return abs$1(acos$1(s)) < pi$1 / 3;
|
|
}
|
|
|
|
reduce() {
|
|
// TODO: examine these var types in more detail...
|
|
let i,
|
|
t1 = 0,
|
|
t2 = 0,
|
|
step = 0.01,
|
|
segment,
|
|
pass1 = [],
|
|
pass2 = [];
|
|
// first pass: split on extrema
|
|
let extrema = this.extrema().values;
|
|
if (extrema.indexOf(0) === -1) {
|
|
extrema = [0].concat(extrema);
|
|
}
|
|
if (extrema.indexOf(1) === -1) {
|
|
extrema.push(1);
|
|
}
|
|
|
|
for (t1 = extrema[0], i = 1; i < extrema.length; i++) {
|
|
t2 = extrema[i];
|
|
segment = this.split(t1, t2);
|
|
segment._t1 = t1;
|
|
segment._t2 = t2;
|
|
pass1.push(segment);
|
|
t1 = t2;
|
|
}
|
|
|
|
// second pass: further reduce these segments to simple segments
|
|
pass1.forEach(function (p1) {
|
|
t1 = 0;
|
|
t2 = 0;
|
|
while (t2 <= 1) {
|
|
for (t2 = t1 + step; t2 <= 1 + step; t2 += step) {
|
|
segment = p1.split(t1, t2);
|
|
if (!segment.simple()) {
|
|
t2 -= step;
|
|
if (abs$1(t1 - t2) < step) {
|
|
// we can never form a reduction
|
|
return [];
|
|
}
|
|
segment = p1.split(t1, t2);
|
|
segment._t1 = utils.map(t1, 0, 1, p1._t1, p1._t2);
|
|
segment._t2 = utils.map(t2, 0, 1, p1._t1, p1._t2);
|
|
pass2.push(segment);
|
|
t1 = t2;
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
if (t1 < 1) {
|
|
segment = p1.split(t1, 1);
|
|
segment._t1 = utils.map(t1, 0, 1, p1._t1, p1._t2);
|
|
segment._t2 = p1._t2;
|
|
pass2.push(segment);
|
|
}
|
|
});
|
|
return pass2;
|
|
}
|
|
|
|
scale(d) {
|
|
const order = this.order;
|
|
let distanceFn = false;
|
|
if (typeof d === "function") {
|
|
distanceFn = d;
|
|
}
|
|
if (distanceFn && order === 2) {
|
|
return this.raise().scale(distanceFn);
|
|
}
|
|
|
|
// TODO: add special handling for degenerate (=linear) curves.
|
|
const clockwise = this.clockwise;
|
|
const r1 = distanceFn ? distanceFn(0) : d;
|
|
const r2 = distanceFn ? distanceFn(1) : d;
|
|
const v = [this.offset(0, 10), this.offset(1, 10)];
|
|
const points = this.points;
|
|
const np = [];
|
|
const o = utils.lli4(v[0], v[0].c, v[1], v[1].c);
|
|
|
|
if (!o) {
|
|
throw new Error("cannot scale this curve. Try reducing it first.");
|
|
}
|
|
// move all points by distance 'd' wrt the origin 'o'
|
|
|
|
// move end points by fixed distance along normal.
|
|
[0, 1].forEach(function (t) {
|
|
const p = (np[t * order] = utils.copy(points[t * order]));
|
|
p.x += (t ? r2 : r1) * v[t].n.x;
|
|
p.y += (t ? r2 : r1) * v[t].n.y;
|
|
});
|
|
|
|
if (!distanceFn) {
|
|
// move control points to lie on the intersection of the offset
|
|
// derivative vector, and the origin-through-control vector
|
|
[0, 1].forEach((t) => {
|
|
if (order === 2 && !!t) return;
|
|
const p = np[t * order];
|
|
const d = this.derivative(t);
|
|
const p2 = { x: p.x + d.x, y: p.y + d.y };
|
|
np[t + 1] = utils.lli4(p, p2, o, points[t + 1]);
|
|
});
|
|
return new Bezier(np);
|
|
}
|
|
|
|
// move control points by "however much necessary to
|
|
// ensure the correct tangent to endpoint".
|
|
[0, 1].forEach(function (t) {
|
|
if (order === 2 && !!t) return;
|
|
var p = points[t + 1];
|
|
var ov = {
|
|
x: p.x - o.x,
|
|
y: p.y - o.y,
|
|
};
|
|
var rc = distanceFn ? distanceFn((t + 1) / order) : d;
|
|
if (distanceFn && !clockwise) rc = -rc;
|
|
var m = sqrt$1(ov.x * ov.x + ov.y * ov.y);
|
|
ov.x /= m;
|
|
ov.y /= m;
|
|
np[t + 1] = {
|
|
x: p.x + rc * ov.x,
|
|
y: p.y + rc * ov.y,
|
|
};
|
|
});
|
|
return new Bezier(np);
|
|
}
|
|
|
|
outline(d1, d2, d3, d4) {
|
|
d2 = typeof d2 === "undefined" ? d1 : d2;
|
|
const reduced = this.reduce(),
|
|
len = reduced.length,
|
|
fcurves = [];
|
|
|
|
let bcurves = [],
|
|
p,
|
|
alen = 0,
|
|
tlen = this.length();
|
|
|
|
const graduated = typeof d3 !== "undefined" && typeof d4 !== "undefined";
|
|
|
|
function linearDistanceFunction(s, e, tlen, alen, slen) {
|
|
return function (v) {
|
|
const f1 = alen / tlen,
|
|
f2 = (alen + slen) / tlen,
|
|
d = e - s;
|
|
return utils.map(v, 0, 1, s + f1 * d, s + f2 * d);
|
|
};
|
|
}
|
|
|
|
// form curve oulines
|
|
reduced.forEach(function (segment) {
|
|
const slen = segment.length();
|
|
if (graduated) {
|
|
fcurves.push(
|
|
segment.scale(linearDistanceFunction(d1, d3, tlen, alen, slen))
|
|
);
|
|
bcurves.push(
|
|
segment.scale(linearDistanceFunction(-d2, -d4, tlen, alen, slen))
|
|
);
|
|
} else {
|
|
fcurves.push(segment.scale(d1));
|
|
bcurves.push(segment.scale(-d2));
|
|
}
|
|
alen += slen;
|
|
});
|
|
|
|
// reverse the "return" outline
|
|
bcurves = bcurves
|
|
.map(function (s) {
|
|
p = s.points;
|
|
if (p[3]) {
|
|
s.points = [p[3], p[2], p[1], p[0]];
|
|
} else {
|
|
s.points = [p[2], p[1], p[0]];
|
|
}
|
|
return s;
|
|
})
|
|
.reverse();
|
|
|
|
// form the endcaps as lines
|
|
const fs = fcurves[0].points[0],
|
|
fe = fcurves[len - 1].points[fcurves[len - 1].points.length - 1],
|
|
bs = bcurves[len - 1].points[bcurves[len - 1].points.length - 1],
|
|
be = bcurves[0].points[0],
|
|
ls = utils.makeline(bs, fs),
|
|
le = utils.makeline(fe, be),
|
|
segments = [ls].concat(fcurves).concat([le]).concat(bcurves);
|
|
|
|
return new PolyBezier(segments);
|
|
}
|
|
|
|
outlineshapes(d1, d2, curveIntersectionThreshold) {
|
|
d2 = d2 || d1;
|
|
const outline = this.outline(d1, d2).curves;
|
|
const shapes = [];
|
|
for (let i = 1, len = outline.length; i < len / 2; i++) {
|
|
const shape = utils.makeshape(
|
|
outline[i],
|
|
outline[len - i],
|
|
curveIntersectionThreshold
|
|
);
|
|
shape.startcap.virtual = i > 1;
|
|
shape.endcap.virtual = i < len / 2 - 1;
|
|
shapes.push(shape);
|
|
}
|
|
return shapes;
|
|
}
|
|
|
|
intersects(curve, curveIntersectionThreshold) {
|
|
if (!curve) return this.selfintersects(curveIntersectionThreshold);
|
|
if (curve.p1 && curve.p2) {
|
|
return this.lineIntersects(curve);
|
|
}
|
|
if (curve instanceof Bezier) {
|
|
curve = curve.reduce();
|
|
}
|
|
return this.curveintersects(
|
|
this.reduce(),
|
|
curve,
|
|
curveIntersectionThreshold
|
|
);
|
|
}
|
|
|
|
lineIntersects(line) {
|
|
const mx = min(line.p1.x, line.p2.x),
|
|
my = min(line.p1.y, line.p2.y),
|
|
MX = max(line.p1.x, line.p2.x),
|
|
MY = max(line.p1.y, line.p2.y);
|
|
return utils.roots(this.points, line).filter((t) => {
|
|
var p = this.get(t);
|
|
return utils.between(p.x, mx, MX) && utils.between(p.y, my, MY);
|
|
});
|
|
}
|
|
|
|
selfintersects(curveIntersectionThreshold) {
|
|
// "simple" curves cannot intersect with their direct
|
|
// neighbour, so for each segment X we check whether
|
|
// it intersects [0:x-2][x+2:last].
|
|
|
|
const reduced = this.reduce(),
|
|
len = reduced.length - 2,
|
|
results = [];
|
|
|
|
for (let i = 0, result, left, right; i < len; i++) {
|
|
left = reduced.slice(i, i + 1);
|
|
right = reduced.slice(i + 2);
|
|
result = this.curveintersects(left, right, curveIntersectionThreshold);
|
|
results.push(...result);
|
|
}
|
|
return results;
|
|
}
|
|
|
|
curveintersects(c1, c2, curveIntersectionThreshold) {
|
|
const pairs = [];
|
|
// step 1: pair off any overlapping segments
|
|
c1.forEach(function (l) {
|
|
c2.forEach(function (r) {
|
|
if (l.overlaps(r)) {
|
|
pairs.push({ left: l, right: r });
|
|
}
|
|
});
|
|
});
|
|
// step 2: for each pairing, run through the convergence algorithm.
|
|
let intersections = [];
|
|
pairs.forEach(function (pair) {
|
|
const result = utils.pairiteration(
|
|
pair.left,
|
|
pair.right,
|
|
curveIntersectionThreshold
|
|
);
|
|
if (result.length > 0) {
|
|
intersections = intersections.concat(result);
|
|
}
|
|
});
|
|
return intersections;
|
|
}
|
|
|
|
arcs(errorThreshold) {
|
|
errorThreshold = errorThreshold || 0.5;
|
|
return this._iterate(errorThreshold, []);
|
|
}
|
|
|
|
_error(pc, np1, s, e) {
|
|
const q = (e - s) / 4,
|
|
c1 = this.get(s + q),
|
|
c2 = this.get(e - q),
|
|
ref = utils.dist(pc, np1),
|
|
d1 = utils.dist(pc, c1),
|
|
d2 = utils.dist(pc, c2);
|
|
return abs$1(d1 - ref) + abs$1(d2 - ref);
|
|
}
|
|
|
|
_iterate(errorThreshold, circles) {
|
|
let t_s = 0,
|
|
t_e = 1,
|
|
safety;
|
|
// we do a binary search to find the "good `t` closest to no-longer-good"
|
|
do {
|
|
safety = 0;
|
|
|
|
// step 1: start with the maximum possible arc
|
|
t_e = 1;
|
|
|
|
// points:
|
|
let np1 = this.get(t_s),
|
|
np2,
|
|
np3,
|
|
arc,
|
|
prev_arc;
|
|
|
|
// booleans:
|
|
let curr_good = false,
|
|
prev_good = false,
|
|
done;
|
|
|
|
// numbers:
|
|
let t_m = t_e,
|
|
prev_e = 1;
|
|
|
|
// step 2: find the best possible arc
|
|
do {
|
|
prev_good = curr_good;
|
|
prev_arc = arc;
|
|
t_m = (t_s + t_e) / 2;
|
|
|
|
np2 = this.get(t_m);
|
|
np3 = this.get(t_e);
|
|
|
|
arc = utils.getccenter(np1, np2, np3);
|
|
|
|
//also save the t values
|
|
arc.interval = {
|
|
start: t_s,
|
|
end: t_e,
|
|
};
|
|
|
|
let error = this._error(arc, np1, t_s, t_e);
|
|
curr_good = error <= errorThreshold;
|
|
|
|
done = prev_good && !curr_good;
|
|
if (!done) prev_e = t_e;
|
|
|
|
// this arc is fine: we can move 'e' up to see if we can find a wider arc
|
|
if (curr_good) {
|
|
// if e is already at max, then we're done for this arc.
|
|
if (t_e >= 1) {
|
|
// make sure we cap at t=1
|
|
arc.interval.end = prev_e = 1;
|
|
prev_arc = arc;
|
|
// if we capped the arc segment to t=1 we also need to make sure that
|
|
// the arc's end angle is correct with respect to the bezier end point.
|
|
if (t_e > 1) {
|
|
let d = {
|
|
x: arc.x + arc.r * cos$1(arc.e),
|
|
y: arc.y + arc.r * sin$1(arc.e),
|
|
};
|
|
arc.e += utils.angle({ x: arc.x, y: arc.y }, d, this.get(1));
|
|
}
|
|
break;
|
|
}
|
|
// if not, move it up by half the iteration distance
|
|
t_e = t_e + (t_e - t_s) / 2;
|
|
} else {
|
|
// this is a bad arc: we need to move 'e' down to find a good arc
|
|
t_e = t_m;
|
|
}
|
|
} while (!done && safety++ < 100);
|
|
|
|
if (safety >= 100) {
|
|
break;
|
|
}
|
|
|
|
// console.log("L835: [F] arc found", t_s, prev_e, prev_arc.x, prev_arc.y, prev_arc.s, prev_arc.e);
|
|
|
|
prev_arc = prev_arc ? prev_arc : arc;
|
|
circles.push(prev_arc);
|
|
t_s = prev_e;
|
|
} while (t_e < 1);
|
|
return circles;
|
|
}
|
|
}
|
|
|
|
export { Bezier };
|