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js/bezierlib.js

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+'use strict';
+import * as Vector from './vectorlib.js';
+/** A 3D vector
+ * @typedef {Object} Vector
+ * @property {number} x
+ * @property {number} y
+ * @property {number} z
+ */
+
+/** Estimate the maximum grade of the bezier.
+ * @param {Vector} p1 - Start
+ * @param {Vector} h1 - Handle 1
+ * @param {Vector} h2 - Handle 2
+ * @param {Vector} p2 - End
+ * @param {number} iterationCount - How many steps along the bezier to check
+ * @returns {number} Grade as a decimal
+ */
+export function estimateMaxGrade(p1, h1, h2, p2, iterationCount = 50){
+    let grade = 0;
+    let totalGrade = 0;
+    for(let i=0; i<=iterationCount; i++){
+        let tan = evaluateVelocity(p1, h1, h2, p2, i/iterationCount);
+        let newGrade = Math.abs(tan.y)/Math.sqrt(tan.x*tan.x + tan.z*tan.z);
+        grade = Math.max(grade, newGrade);
+        totalGrade += newGrade;
+    }
+    return grade;
+}
+
+/** Estimate the length of the bezier.
+ * @param {Vector} p1 - Start
+ * @param {Vector} h1 - Handle 1
+ * @param {Vector} h2 - Handle 2
+ * @param {Vector} p2 - End
+ * @param {number} iterationCount - How many steps along the bezier to check
+ * @returns {number} Length
+ */
+export function estimateLength(p1, h1, h2, p2, iterationCount = 50){
+    let length = 0;
+    let prevPos = p1;
+    let curPos = {};
+    for(let i=1; i<=iterationCount; i++){
+        curPos = evaluatePoint(p1, h1, h2, p2, i/iterationCount);
+        length += Vector.distance(prevPos, curPos);
+        prevPos = curPos;
+    }
+    return length;
+}
+
+/** Estimate the maximum 3D curvature of the bezier.
+ * @param {Vector} p1 - Start
+ * @param {Vector} h1 - Handle 1
+ * @param {Vector} h2 - Handle 2
+ * @param {Vector} p2 - End
+ * @param {number} iterationCount - How many steps along the bezier to check
+ * @returns {number} Curvature, take `1/curvature` to get radius.
+ */
+export function estimateMaxCurvature(p1, h1, h2, p2, iterationCount = 50){
+    let curvature = 0;
+    let t = 0;
+    let vel = {};
+    for(let i=0; i<iterationCount; i++){
+        t = i/iterationCount;
+        vel = evaluateVelocity(p1, h1, h2, p2, t);
+        curvature = Math.max(curvature,
+            Vector.length(Vector.cross(vel, evaluateAcceleration(p1, h1, h2, p2, t)))
+            /Math.pow(Vector.length(vel),3));
+    }
+    return curvature;
+}
+
+/** Evaluate the bezier at `t`, getting position.
+ * @param {Vector} p1 - Start
+ * @param {Vector} h1 - Handle 1
+ * @param {Vector} h2 - Handle 2
+ * @param {Vector} p2 - End
+ * @param {number} t - 0 to 1 of how far into the curve to check.
+ * @returns {Vector}
+ */
+export function evaluatePoint(p1, h1, h2, p2, t){
+    let result = {};
+    let t3 = t*t*t;
+    let t2 = t*t;
+    let p1v = -t3 + 3*t2 - 3*t + 1;
+    let h1v = 3*t3 - 6*t2 + 3*t;
+    let h2v = -3*t3 + 3*t2;
+    for(const axis of ['x', 'y', 'z']){
+        result[axis] = p1[axis]*p1v + h1[axis]*h1v + h2[axis]*h2v + p2[axis]*t3;
+    }
+    return result;
+}
+
+/** Evaluate the bezier's first derivative at `t`, getting velocity.
+ * @param {Vector} p1 - Start
+ * @param {Vector} h1 - Handle 1
+ * @param {Vector} h2 - Handle 2
+ * @param {Vector} p2 - End
+ * @param {number} t - 0 to 1 of how far into the curve to check.
+ * @returns {Vector}
+ */
+export function evaluateVelocity(p1, h1, h2, p2, t){
+    let result = {};
+    let t2 = t*t;
+    let p1v = -3*t2 + 6*t -3;
+    let h1v = 9*t2 - 12*t +3;
+    let h2v = -9*t2 + 6*t;
+    let p2v = 3*t2;
+    for(const axis of ['x', 'y', 'z']){
+        result[axis] = p1[axis]*p1v + h1[axis]*h1v + h2[axis]*h2v + p2[axis]*p2v;
+    }
+    return result;
+}
+
+/** Evaluate the bezier's second derivative at `t`, getting acceleration.
+ * @param {Vector} p1 - Start
+ * @param {Vector} h1 - Handle 1
+ * @param {Vector} h2 - Handle 2
+ * @param {Vector} p2 - End
+ * @param {number} t - 0 to 1 of how far into the curve to check.
+ * @returns {Vector}
+ */
+export function evaluateAcceleration(p1, h1, h2, p2, t){
+    let result = {};
+    let p1v = -6*t + 6;
+    let h1v = 18*t -12;
+    let h2v = -18*t + 6;
+    let p2v = 6*t;
+    for(const axis of ['x', 'y', 'z']){
+        result[axis] = p1[axis]*p1v + h1[axis]*h1v + h2[axis]*h2v + p2[axis]*p2v;
+    }
+    return result;
+}