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https://github.com/Pomax/BezierInfo-2.git
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polybezier
This commit is contained in:
Pomax
21
article.js
21
article.js
File diff suppressed because one or more lines are too long
@@ -24,7 +24,7 @@ var Graphic = React.createClass({
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my:0,
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cx:0,
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cy:0,
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mp: { x: 0, y: 0},
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mp: false,
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offset: { x: 0, y: 0},
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lpts: [],
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colorSeed: 0,
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@@ -96,10 +96,11 @@ var Graphic = React.createClass({
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this.dragging = false;
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this.down = true;
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this.lpts.forEach(p => {
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this.lpts.forEach((p,idx) => {
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if(Math.abs(this.mx - p.x)<10 && Math.abs(this.my - p.y)<10) {
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this.moving = true;
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this.mp = p;
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this.mp_idx = idx;
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this.cx = p.x;
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this.cy = p.y;
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}
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@@ -39,10 +39,7 @@ module.exports = {
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catmullconv: require("./catmullconv"),
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catmullmoulding: require("./catmullmoulding"),
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/*
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// This requires bezier.js to have a proper poly implementation
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polybezier: require("./polybezier"),
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*/
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polybezier: require("./polybezier"),
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/*
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// This section is way too much work to port, and not worth implementing given paper.js etc.
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@@ -2,6 +2,8 @@ var React = require("react");
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var Graphic = require("../../Graphic.jsx");
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var SectionHeader = require("../../SectionHeader.jsx");
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var atan2 = Math.atan2, sqrt = Math.sqrt, sin = Math.sin, cos = Math.cos;
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var PolyBezier = React.createClass({
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getDefaultProps: function() {
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return {
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@@ -27,27 +29,213 @@ var PolyBezier = React.createClass({
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},
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setupCubic: function(api) {
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var w = api.getPanelWidth(),
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h = api.getPanelHeight(),
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cx = w/2, cy = h/2, pad = 40,
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r = (w - 2*pad)/2,
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k = 0.55228,
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kr = k*r,
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pts = [
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// first curve:
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{x:cx,y:pad}, {x:cx+kr,y:pad}, {x:w-pad,y:cy-kr}, {x:w-pad,y:cy},
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// subsequent curve
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{x:w-pad,y:cy+kr}, {x:cx+kr,y:h-pad}, {x:cx,y:h-pad},
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// subsequent curve
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{x:cx-kr,y:h-pad}, {x:pad,y:cy+kr}, {x:pad,y:cy},
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// final curve control point
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{x:pad,y:cy-kr}, {x:cx-kr,y:pad}
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];
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api.lpts = pts;
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},
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movePointsQuadraticLD: function(api, i) {
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// ...we need to move _everything_
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var anchor, fixed, toMove;
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for(var p=1; p<4; p++) {
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anchor = i + (2*p - 2) + api.lpts.length;
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anchor = api.lpts[anchor % api.lpts.length];
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fixed = i + (2*p - 1);
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fixed = api.lpts[fixed % api.lpts.length];
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toMove = i + 2*p;
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toMove = api.lpts[toMove % api.lpts.length];
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toMove.x = fixed.x + (fixed.x - anchor.x);
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toMove.y = fixed.y + (fixed.y - anchor.y);
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}
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},
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movePointsCubicLD: function(api, i) {
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var toMove, fixed;
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if (i%3 === 1) {
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fixed = i-1;
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fixed += (fixed < 0) ? api.lpts.length : 0;
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toMove = i-2;
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toMove += (toMove < 0) ? api.lpts.length : 0;
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} else {
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fixed = (i+1) % api.lpts.length;
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toMove = (i+2) % api.lpts.length;
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}
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fixed = api.lpts[fixed];
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toMove = api.lpts[toMove];
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toMove.x = fixed.x + (fixed.x - api.mp.x);
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toMove.y = fixed.y + (fixed.y - api.mp.y);
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},
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linkDerivatives: function(evt, api) {
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if (api.mp) {
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var quad = api.lpts.length === 8;
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var i = api.mp_idx;
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if (quad && i%2 !== 0) { this.movePointsQuadraticLD(api, i); }
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else if(i%3 !== 0) { this.movePointsCubicLD(api, i); }
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}
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},
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movePointsQuadraticDirOnly: function(api, i) {
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// ...we need to move _everything_ ...again
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var anchor, fixed, toMove;
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// move left and right
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[-1,1].forEach(v => {
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anchor = api.mp;
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fixed = i + v + api.lpts.length;
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fixed = api.lpts[fixed % api.lpts.length];
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toMove = i + 2*v + api.lpts.length;
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toMove = api.lpts[toMove % api.lpts.length];
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var a = atan2(fixed.y - anchor.y, fixed.x - anchor.x),
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dx = toMove.x - fixed.x,
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dy = toMove.y - fixed.y,
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d = sqrt(dx*dx + dy*dy);
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toMove.x = fixed.x + d*cos(a);
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toMove.y = fixed.y + d*sin(a);
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});
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// then, the furthest point cannot be computed properly!
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toMove = i + 4;
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toMove = api.lpts[toMove % api.lpts.length];
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api.problem = toMove;
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},
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movePointsCubicDirOnly: function(api, i) {
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var toMove, fixed;
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if (i%3 === 1) {
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fixed = i-1;
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fixed += (fixed < 0) ? api.lpts.length : 0;
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toMove = i-2;
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toMove += (toMove < 0) ? api.lpts.length : 0;
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} else {
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fixed = (i+1) % api.lpts.length;
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toMove = (i+2) % api.lpts.length;
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}
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fixed = api.lpts[fixed];
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toMove = api.lpts[toMove];
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var a = atan2(fixed.y - api.mp.y, fixed.x - api.mp.x),
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dx = toMove.x - fixed.x,
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dy = toMove.y - fixed.y,
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d = sqrt(dx*dx + dy*dy);
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toMove.x = fixed.x + d*cos(a);
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toMove.y = fixed.y + d*sin(a);
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},
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linkDirection: function(evt, api) {
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if (api.mp) {
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var quad = api.lpts.length === 8;
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var i = api.mp_idx;
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if (quad && i%2 !== 0) { this.movePointsQuadraticDirOnly(api, i); }
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else if(i%3 !== 0) { this.movePointsCubicDirOnly(api, i); }
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}
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},
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bufferPoints: function(evt, api) {
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api.bpts = JSON.parse(JSON.stringify(api.lpts));
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},
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moveQuadraticPoint: function(api, i) {
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this.moveCubicPoint(api,i), anchor, toMove, fixed;
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// then move the other control points
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[-1,1].forEach(v => {
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anchor = i - v + api.lpts.length;
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anchor = api.lpts[anchor % api.lpts.length];
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fixed = i - 2*v + api.lpts.length;
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fixed = api.lpts[fixed % api.lpts.length];
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toMove = i - 3*v + api.lpts.length;
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toMove = api.lpts[toMove % api.lpts.length];
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var a = atan2(fixed.y - anchor.y, fixed.x - anchor.x),
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dx = toMove.x - fixed.x,
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dy = toMove.y - fixed.y,
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d = sqrt(dx*dx + dy*dy);
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toMove.x = fixed.x + d*cos(a);
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toMove.y = fixed.y + d*sin(a);
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});
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// then signal a problem
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var toMove = i + 4;
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toMove = api.lpts[toMove % api.lpts.length];
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api.problem = toMove;
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},
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moveCubicPoint: function(api, i) {
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var op = api.bpts[i],
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np = api.lpts[i],
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dx = np.x - op.x,
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dy = np.y - op.y,
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len = api.lpts.length,
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l = i-1+len,
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r = i+1,
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// original left and right
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ol = api.bpts[l % len],
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or = api.bpts[r % len],
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// current left and right
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nl = api.lpts[l % len],
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nr = api.lpts[r % len];
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// update current left
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nl.x = ol.x + dx;
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nl.y = ol.y + dy;
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// update current right
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nr.x = or.x + dx;
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nr.y = or.y + dy;
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return {x:dx, y:dy};
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},
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modelCurve: function(evt, api) {
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if (api.mp) {
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var quad = api.lpts.length === 8;
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var i = api.mp_idx;
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if (quad) {
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if (i%2 !== 0) { this.movePointsQuadraticDirOnly(api, i); }
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else { this.moveQuadraticPoint(api, i); }
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}
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else {
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if(i%3 !== 0) { this.movePointsCubicDirOnly(api, i); }
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else { this.moveCubicPoint(api, i); }
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}
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}
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},
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draw: function(api, curves) {
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api.reset();
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var pts = api.lpts;
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var quad = pts.length === 8;
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var c1 = new api.Bezier(pts[0],pts[1],pts[2]);
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api.drawSkeleton(c1);
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var c1 = quad ? new api.Bezier(pts[0],pts[1],pts[2]) : new api.Bezier(pts[0],pts[1],pts[2],pts[3]);
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api.drawSkeleton(c1, false, true);
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api.drawCurve(c1);
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var c2 = new api.Bezier(pts[2],pts[3],pts[4]);
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api.drawSkeleton(c2);
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var c2 = quad ? new api.Bezier(pts[2],pts[3],pts[4]) : new api.Bezier(pts[3],pts[4],pts[5],pts[6]);
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api.drawSkeleton(c2, false, true);
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api.drawCurve(c2);
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var c3 = new api.Bezier(pts[4],pts[5],pts[6]);
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api.drawSkeleton(c3);
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var c3 = quad ? new api.Bezier(pts[4],pts[5],pts[6]) : new api.Bezier(pts[6],pts[7],pts[8],pts[9]);
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api.drawSkeleton(c3, false, true);
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api.drawCurve(c3);
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var c4 = new api.Bezier(pts[6],pts[7],pts[0]);
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api.drawSkeleton(c4);
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var c4 = quad ? new api.Bezier(pts[6],pts[7],pts[0]) : new api.Bezier(pts[9],pts[10],pts[11],pts[0]);
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api.drawSkeleton(c4, false, true);
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api.drawCurve(c4);
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if (api.problem) {
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api.setColor("red");
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api.drawCircle(api.problem,5);
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}
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},
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render: function() {
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@@ -56,22 +244,37 @@ var PolyBezier = React.createClass({
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<SectionHeader {...this.props} />
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<p>Much like lines can be chained together to form polygons, Bézier curves can be chained together
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to form poly-Béziers, and the only trick required is to make sure that: A) the end point of each
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section is the starting point of the following section, and B) the derivatives across that
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dual point line up. Unless, of course, you want discontinuities; then you don't even need (B).</p>
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to form poly-Béziers, and the only trick required is to make sure that:</p>
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<ol>
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<li>the end point of each section is the starting point of the following section, and</li>
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<li>the derivatives across that dual point line up.</li>
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</ol>
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<p>Unless, of course, you want discontinuities; then you don't even need 2.</p>
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<p>We'll cover three forms of poly-Bézier curves in this section. First, we'll look at the kind
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that enforces "the outgoing derivative is the same as the incoming derivative" across sections:</p>
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that just follows point 1. where the end point of a segment is the same point as the start point
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of the next segment. This leads to poly-Béziers that are pretty hard to work with, but they're
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the easiest to implement:</p>
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<Graphic preset="poly" title="Loosely connected quadratic poly-Bézier" setup={this.setupQuadratic} draw={this.draw}/>
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<Graphic preset="poly" title="Loosely connected cubic poly-Bézier" setup={this.setupCubic} draw={this.draw}/>
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<p>Dragging the control points around only affects the curve segments that the control point belongs
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to, and moving an on-curve point leaves the control points where they are, which is not the most useful
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for practical modelling purposes. So, let's add in the logic we need to make things a little better.
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We'll start by linking up control points by ensuring that the "incoming" derivative at an on-curve
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point is the same as it's "outgoing" derivative:</p>
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<p>\[
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B'(1)_n = B'(0)_{n+1}
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\]</p>
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<p>We can actually guarantee this really easily, because we know that the vector from a curve's
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last control point to its last on-curve point is equal to the derivative vector. If we want to
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ensure that the first control point of the next curve matches that, all we have to do is mirror
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that last control point through the last on-curve point. And mirroring any point A through any
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point B is really simple:</p>
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<p>We can effect this quite easily, because we know that the vector from a curve's last control point
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to its last on-curve point is equal to the derivative vector. If we want to ensure that the first control
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point of the next curve matches that, all we have to do is mirror that last control point through the
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last on-curve point. And mirroring any point A through any point B is really simple:</p>
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<p>\[
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Mirrored = \left [
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@@ -82,20 +285,13 @@ var PolyBezier = React.createClass({
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\]</p>
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<p>So let's implement that and see what it gets us. The following two graphics show a quadratic
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and a cubic poly-Bézier curve; both consist of multiple sub-curves, but because of our constraint,
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not all points on the curves can be moved around freely. Some points, when moved, will move other
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points by virtue of changing the curve across sections.</p>
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and a cubic poly-Bézier curve again, but this time moving the control points around moves others,
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too. However, you might see something unexpected going on for quadratic curves...</p>
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<Graphic preset="poly" title="Forming a quadratic poly-Bézier" setup={this.setupQuadratic} draw={this.draw}/>
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<textarea className="sketch-code" data-sketch-preset="poly" data-sketch-title="Forming a cubic poly-Bézier">
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void setupCurve() {
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setupDefaultCubicPoly();
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}
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void movePoint(PolyBezierCurve p, int pt, int mx, int my) {
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p.movePointConstrained(pt, mx, my);
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}</textarea>
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<Graphic preset="poly" title="Loosely connected quadratic poly-Bézier" setup={this.setupQuadratic} draw={this.draw}
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onMouseMove={this.linkDerivatives}/>
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<Graphic preset="poly" title="Loosely connected cubic poly-Bézier" setup={this.setupCubic} draw={this.draw}
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onMouseMove={this.linkDerivatives}/>
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<p>As you can see, quadratic curves are particularly ill-suited for poly-Bézier curves, as all
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the control points are effectively linked. Move one of them, and you move all of them. This means
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@@ -104,66 +300,40 @@ var PolyBezier = React.createClass({
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||||
that the derivatives are linked means we can't manipulate curves as well as we might if we
|
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relaxed the constraints a little.</p>
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||||
<p>So: let's relax them!</p>
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||||
<p>So: let's relax the requirement a little.</p>
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||||
<p>We can change the constraint so that we still preserve the angle of the derivatives across
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sections (so transitions from one section to the next will still look natural), but give up
|
||||
the requirement that they should also have the same vector length. Doing so will give us
|
||||
a much more a useful kind of poly-Bézier curve:</p>
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<p>We can change the constraint so that we still preserve the <em>angle</em> of the derivatives across
|
||||
sections (so transitions from one section to the next will still look natural), but give up the
|
||||
requirement that they should also have the same <em>vector length</em>. Doing so will give us a much
|
||||
more useful kind of poly-Bézier curve:</p>
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||||
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||||
<textarea className="sketch-code" data-sketch-preset="poly" data-sketch-title="A half-constrained quadratic poly-Bézier">
|
||||
void setupCurve() {
|
||||
setupDefaultQuadraticPoly();
|
||||
}
|
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<Graphic preset="poly" title="Loosely connected quadratic poly-Bézier" setup={this.setupQuadratic} draw={this.draw} onMouseMove={this.linkDirection}/>
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<Graphic preset="poly" title="Loosely connected cubic poly-Bézier" setup={this.setupCubic} draw={this.draw} onMouseMove={this.linkDirection}/>
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||||
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void movePoint(PolyBezierCurve p, int pt, int mx, int my) {
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p.movePointHalfConstrained(pt, mx, my);
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||||
}</textarea>
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||||
<p>Cubic curves are now better behaved when it comes to dragging control points around, but the
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quadratic poly-Bézier has a problem: in the example shape, moving one control points will move
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the control points around it properly, but they in turn define "the next" control point and they
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do so in incompatible ways! This is one of the many reasons why quadratic curves are not really
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useful to work with.</p>
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||||
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||||
<textarea className="sketch-code" data-sketch-preset="poly" data-sketch-title="A half-constrained cubic poly-Bézier">
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||||
void setupCurve() {
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||||
setupDefaultCubicPoly();
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||||
}
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<p>Finally, we also want to make sure that moving the on-curve coordinates preserves the relative
|
||||
positions of the associated control points. With that, we get to the kind of curve control that you
|
||||
might be familiar with from applications like Photoshop, Inkscape, Blender, etc.</p>
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||||
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||||
void movePoint(PolyBezierCurve p, int pt, int mx, int my) {
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p.movePointHalfConstrained(pt, mx, my);
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||||
}</textarea>
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||||
<p>Quadratic curves are still silly, but cubic curves are now much more controllable.</p>
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||||
<p>If we want even more control, we could just abandon the derivative constraints entirely,
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||||
and simply assure that the end point of one section is the same as the start point of the next section,
|
||||
and then keep it at that. This gives us the greatest degree of freedom when it comes to modelling
|
||||
shapes, but also means that our poly-Bézier constructs are no longer continuous curves. Sometimes
|
||||
this is exactly what you want (because it lets you add corners to a shape, while still only using
|
||||
Bézier curves).</p>
|
||||
|
||||
<textarea className="sketch-code" data-sketch-preset="poly" data-sketch-title="An unconstrained quadratic poly-Bézier">
|
||||
void setupCurve() {
|
||||
setupDefaultQuadraticPoly();
|
||||
}
|
||||
|
||||
void movePoint(PolyBezierCurve p, int pt, int mx, int my) {
|
||||
p.movePoint(pvt, mx, my);
|
||||
}</textarea>
|
||||
|
||||
<textarea className="sketch-code" data-sketch-preset="poly" data-sketch-title="An unconstrained cubic poly-Bézier">
|
||||
void setupCurve() {
|
||||
setupDefaultCubicPoly();
|
||||
}
|
||||
|
||||
void movePoint(PolyBezierCurve p, int pt, int mx, int my) {
|
||||
p.movePoint(pvt, mx, my);
|
||||
}</textarea>
|
||||
|
||||
<p>When doing any kind of modelling, you generally don't want a poly-Bézier that will only let you
|
||||
pick one of the three forms for all your points; most graphics applications that deal with Bézier
|
||||
curves will actually let you pick, per on-curve point, how to deal with the control points around it:
|
||||
fully constrained, loosely constrained, or completely unconstrained. The best shape modelling comes
|
||||
from having a curve that will let you pick what you need, when you need it, without having to start
|
||||
a new poly-Bézier curve.</p>
|
||||
<Graphic preset="poly" title="Loosely connected quadratic poly-Bézier" setup={this.setupQuadratic} draw={this.draw}
|
||||
onMouseDown={this.bufferPoints} onMouseMove={this.modelCurve}/>
|
||||
<Graphic preset="poly" title="Loosely connected cubic poly-Bézier" setup={this.setupCubic} draw={this.draw}
|
||||
onMouseDown={this.bufferPoints} onMouseMove={this.modelCurve}/>
|
||||
|
||||
<p>Again, we see that cubic curves are now rather nice to work with, but quadratic curves have a
|
||||
serious problem: we can move an on-curve point in such a way that we can't compute what needs to
|
||||
"happen next". Move the top point down, below the left and right points, for instance. There
|
||||
is no way to preserve correct control points without a kink at the bottom point. Quadratic curves:
|
||||
just not that good...</p>
|
||||
|
||||
<p>A final improvement is to offer fine-level control over which points behave which, so that you can
|
||||
have "kinks" or individually controlled segments when you need them, with nicely well-behaved curves
|
||||
for the rest of the path. Implementing that, is left as an excercise for the reader.</p>
|
||||
</section>
|
||||
);
|
||||
}
|
||||
|
||||
@@ -46,6 +46,7 @@ var Projections = React.createClass({
|
||||
api.drawCurve(curve);
|
||||
if (api.mousePt) {
|
||||
api.setColor("red");
|
||||
api.setFill("red");
|
||||
api.drawCircle(api.mousePt, 3);
|
||||
// naive t value
|
||||
var t = this.findClosest(api._lut, api.mousePt, api.utils.dist);
|
||||
@@ -53,6 +54,7 @@ var Projections = React.createClass({
|
||||
var p = curve.get(t);
|
||||
api.drawLine(p, api.mousePt);
|
||||
api.drawCircle(p, 3);
|
||||
api.text("t = "+api.utils.round(t,2), p, {x:10, y:3});
|
||||
}
|
||||
},
|
||||
|
||||
|
||||
108
images/latex/2fb1287989c59780ba3c9db8af20ed1438dd3db7.svg
Normal file
108
images/latex/2fb1287989c59780ba3c9db8af20ed1438dd3db7.svg
Normal file
@@ -0,0 +1,108 @@
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After Width: | Height: | Size: 12 KiB |
38
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38
images/latex/65b818f3b5f44da349d9b4e157374e9f1d500ab5.svg
Normal file
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After Width: | Height: | Size: 4.6 KiB |
Reference in New Issue
Block a user