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-// Copyright 2007 The Closure Library Authors. All Rights Reserved.
-//
-// Licensed under the Apache License, Version 2.0 (the "License");
-// you may not use this file except in compliance with the License.
-// You may obtain a copy of the License at
-//
-// http://www.apache.org/licenses/LICENSE-2.0
-//
-// Unless required by applicable law or agreed to in writing, software
-// distributed under the License is distributed on an "AS-IS" BASIS,
-// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
-// See the License for the specific language governing permissions and
-// limitations under the License.
-
-
-/**
- * @fileoverview Represents a cubic Bezier curve.
- *
- * Uses the deCasteljau algorithm to compute points on the curve.
- * http://en.wikipedia.org/wiki/De_Casteljau's_algorithm
- *
- * Currently it uses an unrolled version of the algorithm for speed. Eventually
- * it may be useful to use the loop form of the algorithm in order to support
- * curves of arbitrary degree.
- *
- * @author robbyw@google.com (Robby Walker)
- * @author wcrosby@google.com (Wayne Crosby)
- */
-
-goog.provide('goog.math.Bezier');
-
-goog.require('goog.math');
-goog.require('goog.math.Coordinate');
-
-
-
-/**
- * Object representing a cubic bezier curve.
- * @param {number} x0 X coordinate of the start point.
- * @param {number} y0 Y coordinate of the start point.
- * @param {number} x1 X coordinate of the first control point.
- * @param {number} y1 Y coordinate of the first control point.
- * @param {number} x2 X coordinate of the second control point.
- * @param {number} y2 Y coordinate of the second control point.
- * @param {number} x3 X coordinate of the end point.
- * @param {number} y3 Y coordinate of the end point.
- * @constructor
- */
-goog.math.Bezier = function(x0, y0, x1, y1, x2, y2, x3, y3) {
- /**
- * X coordinate of the first point.
- * @type {number}
- */
- this.x0 = x0;
-
- /**
- * Y coordinate of the first point.
- * @type {number}
- */
- this.y0 = y0;
-
- /**
- * X coordinate of the first control point.
- * @type {number}
- */
- this.x1 = x1;
-
- /**
- * Y coordinate of the first control point.
- * @type {number}
- */
- this.y1 = y1;
-
- /**
- * X coordinate of the second control point.
- * @type {number}
- */
- this.x2 = x2;
-
- /**
- * Y coordinate of the second control point.
- * @type {number}
- */
- this.y2 = y2;
-
- /**
- * X coordinate of the end point.
- * @type {number}
- */
- this.x3 = x3;
-
- /**
- * Y coordinate of the end point.
- * @type {number}
- */
- this.y3 = y3;
-};
-
-
-/**
- * Constant used to approximate ellipses.
- * See: http://canvaspaint.org/blog/2006/12/ellipse/
- * @type {number}
- */
-goog.math.Bezier.KAPPA = 4 * (Math.sqrt(2) - 1) / 3;
-
-
-/**
- * @return {!goog.math.Bezier} A copy of this curve.
- */
-goog.math.Bezier.prototype.clone = function() {
- return new goog.math.Bezier(this.x0, this.y0, this.x1, this.y1, this.x2,
- this.y2, this.x3, this.y3);
-};
-
-
-/**
- * Test if the given curve is exactly the same as this one.
- * @param {goog.math.Bezier} other The other curve.
- * @return {boolean} Whether the given curve is the same as this one.
- */
-goog.math.Bezier.prototype.equals = function(other) {
- return this.x0 == other.x0 && this.y0 == other.y0 && this.x1 == other.x1 &&
- this.y1 == other.y1 && this.x2 == other.x2 && this.y2 == other.y2 &&
- this.x3 == other.x3 && this.y3 == other.y3;
-};
-
-
-/**
- * Modifies the curve in place to progress in the opposite direction.
- */
-goog.math.Bezier.prototype.flip = function() {
- var temp = this.x0;
- this.x0 = this.x3;
- this.x3 = temp;
- temp = this.y0;
- this.y0 = this.y3;
- this.y3 = temp;
-
- temp = this.x1;
- this.x1 = this.x2;
- this.x2 = temp;
- temp = this.y1;
- this.y1 = this.y2;
- this.y2 = temp;
-};
-
-
-/**
- * Computes the curve at a point between 0 and 1.
- * @param {number} t The point on the curve to find.
- * @return {!goog.math.Coordinate} The computed coordinate.
- */
-goog.math.Bezier.prototype.getPoint = function(t) {
- // Special case start and end
- if (t == 0) {
- return new goog.math.Coordinate(this.x0, this.y0);
- } else if (t == 1) {
- return new goog.math.Coordinate(this.x3, this.y3);
- }
-
- // Step one - from 4 points to 3
- var ix0 = goog.math.lerp(this.x0, this.x1, t);
- var iy0 = goog.math.lerp(this.y0, this.y1, t);
-
- var ix1 = goog.math.lerp(this.x1, this.x2, t);
- var iy1 = goog.math.lerp(this.y1, this.y2, t);
-
- var ix2 = goog.math.lerp(this.x2, this.x3, t);
- var iy2 = goog.math.lerp(this.y2, this.y3, t);
-
- // Step two - from 3 points to 2
- ix0 = goog.math.lerp(ix0, ix1, t);
- iy0 = goog.math.lerp(iy0, iy1, t);
-
- ix1 = goog.math.lerp(ix1, ix2, t);
- iy1 = goog.math.lerp(iy1, iy2, t);
-
- // Final step - last point
- return new goog.math.Coordinate(goog.math.lerp(ix0, ix1, t),
- goog.math.lerp(iy0, iy1, t));
-};
-
-
-/**
- * Changes this curve in place to be the portion of itself from [t, 1].
- * @param {number} t The start of the desired portion of the curve.
- */
-goog.math.Bezier.prototype.subdivideLeft = function(t) {
- if (t == 1) {
- return;
- }
-
- // Step one - from 4 points to 3
- var ix0 = goog.math.lerp(this.x0, this.x1, t);
- var iy0 = goog.math.lerp(this.y0, this.y1, t);
-
- var ix1 = goog.math.lerp(this.x1, this.x2, t);
- var iy1 = goog.math.lerp(this.y1, this.y2, t);
-
- var ix2 = goog.math.lerp(this.x2, this.x3, t);
- var iy2 = goog.math.lerp(this.y2, this.y3, t);
-
- // Collect our new x1 and y1
- this.x1 = ix0;
- this.y1 = iy0;
-
- // Step two - from 3 points to 2
- ix0 = goog.math.lerp(ix0, ix1, t);
- iy0 = goog.math.lerp(iy0, iy1, t);
-
- ix1 = goog.math.lerp(ix1, ix2, t);
- iy1 = goog.math.lerp(iy1, iy2, t);
-
- // Collect our new x2 and y2
- this.x2 = ix0;
- this.y2 = iy0;
-
- // Final step - last point
- this.x3 = goog.math.lerp(ix0, ix1, t);
- this.y3 = goog.math.lerp(iy0, iy1, t);
-};
-
-
-/**
- * Changes this curve in place to be the portion of itself from [0, t].
- * @param {number} t The end of the desired portion of the curve.
- */
-goog.math.Bezier.prototype.subdivideRight = function(t) {
- this.flip();
- this.subdivideLeft(1 - t);
- this.flip();
-};
-
-
-/**
- * Changes this curve in place to be the portion of itself from [s, t].
- * @param {number} s The start of the desired portion of the curve.
- * @param {number} t The end of the desired portion of the curve.
- */
-goog.math.Bezier.prototype.subdivide = function(s, t) {
- this.subdivideRight(s);
- this.subdivideLeft((t - s) / (1 - s));
-};
-
-
-/**
- * Computes the position t of a point on the curve given its x coordinate.
- * That is, for an input xVal, finds t s.t. getPoint(t).x = xVal.
- * As such, the following should always be true up to some small epsilon:
- * t ~ solvePositionFromXValue(getPoint(t).x) for t in [0, 1].
- * @param {number} xVal The x coordinate of the point to find on the curve.
- * @return {number} The position t.
- */
-goog.math.Bezier.prototype.solvePositionFromXValue = function(xVal) {
- // Desired precision on the computation.
- var epsilon = 1e-6;
-
- // Initial estimate of t using linear interpolation.
- var t = (xVal - this.x0) / (this.x3 - this.x0);
- if (t <= 0) {
- return 0;
- } else if (t >= 1) {
- return 1;
- }
-
- // Try gradient descent to solve for t. If it works, it is very fast.
- var tMin = 0;
- var tMax = 1;
- for (var i = 0; i < 8; i++) {
- var value = this.getPoint(t).x;
- var derivative = (this.getPoint(t + epsilon).x - value) / epsilon;
- if (Math.abs(value - xVal) < epsilon) {
- return t;
- } else if (Math.abs(derivative) < epsilon) {
- break;
- } else {
- if (value < xVal) {
- tMin = t;
- } else {
- tMax = t;
- }
- t -= (value - xVal) / derivative;
- }
- }
-
- // If the gradient descent got stuck in a local minimum, e.g. because
- // the derivative was close to 0, use a Dichotomy refinement instead.
- // We limit the number of interations to 8.
- for (var i = 0; Math.abs(value - xVal) > epsilon && i < 8; i++) {
- if (value < xVal) {
- tMin = t;
- t = (t + tMax) / 2;
- } else {
- tMax = t;
- t = (t + tMin) / 2;
- }
- value = this.getPoint(t).x;
- }
- return t;
-};
-
-
-/**
- * Computes the y coordinate of a point on the curve given its x coordinate.
- * @param {number} xVal The x coordinate of the point on the curve.
- * @return {number} The y coordinate of the point on the curve.
- */
-goog.math.Bezier.prototype.solveYValueFromXValue = function(xVal) {
- return this.getPoint(this.solvePositionFromXValue(xVal)).y;
-};