/** This class demonstrates the code discussed in these two articles: http://devmag.org.za/2011/04/05/bzier-curves-a-tutorial/ http://devmag.org.za/2011/06/23/bzier-path-algorithms/ Use this code as you wish, at your own risk. If it blows up your computer, makes a plane crash, or otherwise cause damage, injury, or death, it is not my fault. @author Herman Tulleken, dev.mag.org.za */ using System.Collections.Generic; namespace UnityEngine.UI.Extensions { /** Class for representing a Bezier path, and methods for getting suitable points to draw the curve with line segments. */ public class BezierPath { public int SegmentsPerCurve = 10; public float MINIMUM_SQR_DISTANCE = 0.01f; // This corresponds to about 172 degrees, 8 degrees from a straight line public float DIVISION_THRESHOLD = -0.99f; private List controlPoints; private int curveCount; //how many bezier curves in this path? /** Constructs a new empty Bezier curve. Use one of these methods to add points: SetControlPoints, Interpolate, SamplePoints. */ public BezierPath() { controlPoints = new List(); } /** Sets the control points of this Bezier path. Points 0-3 forms the first Bezier curve, points 3-6 forms the second curve, etc. */ public void SetControlPoints(List newControlPoints) { controlPoints.Clear(); controlPoints.AddRange(newControlPoints); curveCount = (controlPoints.Count - 1) / 3; } public void SetControlPoints(Vector2[] newControlPoints) { controlPoints.Clear(); controlPoints.AddRange(newControlPoints); curveCount = (controlPoints.Count - 1) / 3; } /** Returns the control points for this Bezier curve. */ public List GetControlPoints() { return controlPoints; } /** Calculates a Bezier interpolated path for the given points. */ public void Interpolate(List segmentPoints, float scale) { controlPoints.Clear(); if (segmentPoints.Count < 2) { return; } for (int i = 0; i < segmentPoints.Count; i++) { if (i == 0) // is first { Vector2 p1 = segmentPoints[i]; Vector2 p2 = segmentPoints[i + 1]; Vector2 tangent = (p2 - p1); Vector2 q1 = p1 + scale * tangent; controlPoints.Add(p1); controlPoints.Add(q1); } else if (i == segmentPoints.Count - 1) //last { Vector2 p0 = segmentPoints[i - 1]; Vector2 p1 = segmentPoints[i]; Vector2 tangent = (p1 - p0); Vector2 q0 = p1 - scale * tangent; controlPoints.Add(q0); controlPoints.Add(p1); } else { Vector2 p0 = segmentPoints[i - 1]; Vector2 p1 = segmentPoints[i]; Vector2 p2 = segmentPoints[i + 1]; Vector2 tangent = (p2 - p0).normalized; Vector2 q0 = p1 - scale * tangent * (p1 - p0).magnitude; Vector2 q1 = p1 + scale * tangent * (p2 - p1).magnitude; controlPoints.Add(q0); controlPoints.Add(p1); controlPoints.Add(q1); } } curveCount = (controlPoints.Count - 1) / 3; } /** Sample the given points as a Bezier path. */ public void SamplePoints(List sourcePoints, float minSqrDistance, float maxSqrDistance, float scale) { if (sourcePoints.Count < 2) { return; } Stack samplePoints = new Stack(); samplePoints.Push(sourcePoints[0]); Vector2 potentialSamplePoint = sourcePoints[1]; int i = 2; for (i = 2; i < sourcePoints.Count; i++) { if ( ((potentialSamplePoint - sourcePoints[i]).sqrMagnitude > minSqrDistance) && ((samplePoints.Peek() - sourcePoints[i]).sqrMagnitude > maxSqrDistance)) { samplePoints.Push(potentialSamplePoint); } potentialSamplePoint = sourcePoints[i]; } //now handle last bit of curve Vector2 p1 = samplePoints.Pop(); //last sample point Vector2 p0 = samplePoints.Peek(); //second last sample point Vector2 tangent = (p0 - potentialSamplePoint).normalized; float d2 = (potentialSamplePoint - p1).magnitude; float d1 = (p1 - p0).magnitude; p1 = p1 + tangent * ((d1 - d2) / 2); samplePoints.Push(p1); samplePoints.Push(potentialSamplePoint); Interpolate(new List(samplePoints), scale); } /** Caluclates a point on the path. @param curveIndex The index of the curve that the point is on. For example, the second curve (index 1) is the curve with controlpoints 3, 4, 5, and 6. @param t The paramater indicating where on the curve the point is. 0 corresponds to the "left" point, 1 corresponds to the "right" end point. */ public Vector2 CalculateBezierPoint(int curveIndex, float t) { int nodeIndex = curveIndex * 3; Vector2 p0 = controlPoints[nodeIndex]; Vector2 p1 = controlPoints[nodeIndex + 1]; Vector2 p2 = controlPoints[nodeIndex + 2]; Vector2 p3 = controlPoints[nodeIndex + 3]; return CalculateBezierPoint(t, p0, p1, p2, p3); } /** Gets the drawing points. This implementation simply calculates a certain number of points per curve. */ public List GetDrawingPoints0() { List drawingPoints = new List(); for (int curveIndex = 0; curveIndex < curveCount; curveIndex++) { if (curveIndex == 0) //Only do this for the first end point. //When i != 0, this coincides with the //end point of the previous segment, { drawingPoints.Add(CalculateBezierPoint(curveIndex, 0)); } for (int j = 1; j <= SegmentsPerCurve; j++) { float t = j / (float)SegmentsPerCurve; drawingPoints.Add(CalculateBezierPoint(curveIndex, t)); } } return drawingPoints; } /** Gets the drawing points. This implementation simply calculates a certain number of points per curve. This is a lsightly different inplementation from the one above. */ public List GetDrawingPoints1() { List drawingPoints = new List(); for (int i = 0; i < controlPoints.Count - 3; i += 3) { Vector2 p0 = controlPoints[i]; Vector2 p1 = controlPoints[i + 1]; Vector2 p2 = controlPoints[i + 2]; Vector2 p3 = controlPoints[i + 3]; if (i == 0) //only do this for the first end point. When i != 0, this coincides with the end point of the previous segment, { drawingPoints.Add(CalculateBezierPoint(0, p0, p1, p2, p3)); } for (int j = 1; j <= SegmentsPerCurve; j++) { float t = j / (float)SegmentsPerCurve; drawingPoints.Add(CalculateBezierPoint(t, p0, p1, p2, p3)); } } return drawingPoints; } /** This gets the drawing points of a bezier curve, using recursive division, which results in less points for the same accuracy as the above implementation. */ public List GetDrawingPoints2() { List drawingPoints = new List(); for (int curveIndex = 0; curveIndex < curveCount; curveIndex++) { List bezierCurveDrawingPoints = FindDrawingPoints(curveIndex); if (curveIndex != 0) { //remove the fist point, as it coincides with the last point of the previous Bezier curve. bezierCurveDrawingPoints.RemoveAt(0); } drawingPoints.AddRange(bezierCurveDrawingPoints); } return drawingPoints; } List FindDrawingPoints(int curveIndex) { List pointList = new List(); Vector2 left = CalculateBezierPoint(curveIndex, 0); Vector2 right = CalculateBezierPoint(curveIndex, 1); pointList.Add(left); pointList.Add(right); FindDrawingPoints(curveIndex, 0, 1, pointList, 1); return pointList; } /** @returns the number of points added. */ int FindDrawingPoints(int curveIndex, float t0, float t1, List pointList, int insertionIndex) { Vector2 left = CalculateBezierPoint(curveIndex, t0); Vector2 right = CalculateBezierPoint(curveIndex, t1); if ((left - right).sqrMagnitude < MINIMUM_SQR_DISTANCE) { return 0; } float tMid = (t0 + t1) / 2; Vector2 mid = CalculateBezierPoint(curveIndex, tMid); Vector2 leftDirection = (left - mid).normalized; Vector2 rightDirection = (right - mid).normalized; if (Vector2.Dot(leftDirection, rightDirection) > DIVISION_THRESHOLD || Mathf.Abs(tMid - 0.5f) < 0.0001f) { int pointsAddedCount = 0; pointsAddedCount += FindDrawingPoints(curveIndex, t0, tMid, pointList, insertionIndex); pointList.Insert(insertionIndex + pointsAddedCount, mid); pointsAddedCount++; pointsAddedCount += FindDrawingPoints(curveIndex, tMid, t1, pointList, insertionIndex + pointsAddedCount); return pointsAddedCount; } return 0; } /** Caluclates a point on the Bezier curve represented with the four controlpoints given. */ private Vector2 CalculateBezierPoint(float t, Vector2 p0, Vector2 p1, Vector2 p2, Vector2 p3) { float u = 1 - t; float tt = t * t; float uu = u * u; float uuu = uu * u; float ttt = tt * t; Vector2 p = uuu * p0; //first term p += 3 * uu * t * p1; //second term p += 3 * u * tt * p2; //third term p += ttt * p3; //fourth term return p; } } }