Unity动画TA：一些算法 Cheat Sheet
备忘录，可能持续更新。位移、旋转、缩放转矩阵
实现与Matrix4x4自带同名方法相同的效果
public static Matrix4x4 Rotate(Quaternion q) {
q.Normalize();
float q_x = q.x, q_y = q.y, q_z = q.z, q_w = q.w;
float x2 = q_x * q_x, y2 = q_y * q_y, z2 = q_z * q_z;
float xy = q_x * q_y, xz = q_x * q_z, yz = q_y * q_z, xw = q_x * q_w, yw = q_y * q_w, zw = q_z * q_w;
Matrix4x4 rotMat = new Matrix4x4(
new Vector4(1 - 2 * y2 - 2 * z2, 2 * xy + 2 * zw, 2 * xz - 2 * yw, 0),
new Vector4(2 * xy - 2 * zw, 1 - 2 * x2 - 2 * z2, 2 * yz + 2 * xw, 0),
new Vector4(2 * xz + 2 * yw, 2 * yz - 2 * xw, 1 - 2 * x2 - 2 * y2, 0),
new Vector4(0, 0, 0, 1));
return rotMat;
}
public static Matrix4x4 Translate(Vector3 t) {
Matrix4x4 transMat = new Matrix4x4(new Vector4(1,0,0,0), new Vector4(0,1,0,0), new Vector4(0,0,1,0), new Vector4(t.x, t.y, t.z,1));
return transMat;
}
public static Matrix4x4 Scale(Vector3 s) {
Matrix4x4 scaleMat = new Matrix4x4(new Vector4(s.x, 0, 0, 0), new Vector4(0, s.y, 0, 0), new Vector4(0, 0, s.z, 0), new Vector4(0, 0, 0, 1));
return scaleMat;
}
public static Matrix4x4 TRS(Vector3 t, Quaternion r, Vector3 s) {
Matrix4x4 trs = Rotate(r) * Scale(s);
trs.SetColumn(3, new Vector4(t.x, t.y, t.z, 1));
return trs;
//return Translate(t) * Rotate(r) * Scale(s);
}
Cubic Hermite Curve用矩阵实现
/// <summary>
/// Cubic Hermite2D，实际上把Vector2类型的参数换成Vector3就是Hermite3D
/// </summary>
/// <param name=&#34;t&#34;>参数t</param>
/// <param name=&#34;P1&#34;>第一个点</param>
/// <param name=&#34;P2&#34;>第二个点</param>
/// <param name=&#34;T1&#34;>第一个点切线</param>
/// <param name=&#34;T2&#34;>第二个点切线</param>
/// <returns></returns>
public static Vector4 Hermite2D(float t, Vector2 P1 ,Vector2 P2, Vector2 T1, Vector2 T2) {
//永远不变的矩阵，应该提到外面去写成readonly
Matrix4x4 H = new Matrix4x4(new Vector4(1, 0, 0, 0), new Vector4(0, 0, 1, 0), new Vector4(-3, 3, -2, -1), new Vector4(2, -2, 1, 1));
//每条Hermite只需要计算一次，应该搬到外边
Matrix4x4 P = new Matrix4x4(P1, P2, T1, T2);
float t_2 = t * t;
float t_3 = t_2 * t;
Vector4 T = new Vector4(1, t, t_2, t_3);
return P * H * T;
}
四个2D点实现拉格朗日插值
用矩阵写是为了以后迁移到python做准备。
/// <summary>
/// 平面四个点做拉格朗日插值
/// </summary>
/// <param name=&#34;xy1&#34;>数据点1</param>
/// <param name=&#34;xy2&#34;>数据点2</param>
/// <param name=&#34;xy3&#34;>数据点3</param>
/// <param name=&#34;xy4&#34;>数据点4</param>
/// <param name=&#34;x&#34;>x</param>
/// <returns>y</returns>
public static float LagrangeInterpolationCubic(Vector2 xy1, Vector2 xy2, Vector2 xy3, Vector2 xy4, float x) {
//跟python风格对齐的话，可以直接传入X和Y
Vector4 X = new Vector4(xy1.x, xy2.x, xy3.x, xy4.x);
Vector4 Y = new Vector4(xy1.y, xy2.y, xy3.y, xy4.y);
Matrix4x4 X4 = new Matrix4x4(X, X, X, X);
//每两个元素之间减一次，存在4x4矩阵中
Matrix4x4 X4SubX4T = MatrixSubtract(X4, X4.transpose);
//把自己减自己的地方置为1
X4SubX4T = MatrixAdd(X4SubX4T, Matrix4x4.identity);
Vector4 xCap = new Vector4(x, x, x, x);
Vector4 cap = xCap - X;
//以下计算L的部分如果用numpy还可以继续向量化
float L1 = (cap.y * cap.z * cap.w) / Prod(X4SubX4T.GetRow(0));
float L2 = (cap.x * cap.z * cap.w) / Prod(X4SubX4T.GetRow(1));
float L3 = (cap.x * cap.y * cap.w) / Prod(X4SubX4T.GetRow(2));
float L4 = (cap.x * cap.y * cap.z) / Prod(X4SubX4T.GetRow(3));
Vector4 L = new Vector4(L1, L2, L3, L4);
float res = Vector4.Dot(Y, L);
return res;
}
/// <summary>
/// 所有元素连乘
/// </summary>
/// <param name=&#34;x&#34;>X</param>
/// <returns></returns>
public static float Prod(Vector4 x) {
return x.x * x.y * x.z * x.w;
}
/// <summary>
/// 逐元素矩阵加法
/// </summary>
/// <param name=&#34;a&#34;>矩阵a</param>
/// <param name=&#34;b&#34;>矩阵b</param>
/// <returns>a+b</returns>
public static Matrix4x4 MatrixAdd(Matrix4x4 a, Matrix4x4 b) {
Matrix4x4 add = new Matrix4x4(a.GetColumn(0) + b.GetColumn(0), a.GetColumn(1) + b.GetColumn(1), a.GetColumn(2) + b.GetColumn(2), a.GetColumn(3) + b.GetColumn(3));
return add;
}
/// <summary>
/// 逐元素矩阵减法
/// </summary>
/// <param name=&#34;a&#34;>矩阵a</param>
/// <param name=&#34;b&#34;>矩阵b</param>
/// <returns>a-b</returns>
public static Matrix4x4 MatrixSubtract(Matrix4x4 a, Matrix4x4 b)
{
Matrix4x4 sub = new Matrix4x4(a.GetColumn(0) - b.GetColumn(0), a.GetColumn(1) - b.GetColumn(1), a.GetColumn(2) - b.GetColumn(2), a.GetColumn(3) - b.GetColumn(3));
return sub;
}
以上拉格朗日插值实现的效果
稍作修改得到双线性二次插值。
/// <summary>
/// 双线性二次插值
/// </summary>
/// <param name=&#34;xy1&#34;>数据点1</param>
/// <param name=&#34;xy2&#34;>数据点2</param>
/// <param name=&#34;xy3&#34;>数据点3</param>
/// <param name=&#34;xy4&#34;>数据点4</param>
/// <param name=&#34;x&#34;>x</param>
/// <returns>y</returns>
public static float LagrangeInterpolationSquare(Vector2 xy1, Vector2 xy2, Vector2 xy3, Vector2 xy4, float x) {
//跟python风格对齐的话，可以直接传入X和Y
Vector4 X = new Vector4(xy1.x, xy2.x, xy3.x, xy4.x);
//Vector4 Y = new Vector4(xy1.y, xy2.y, xy3.y, xy4.y);
Matrix4x4 X4 = new Matrix4x4(X, X, X, X);
//每两个元素之间减一次，存在4x4矩阵中
Matrix4x4 X4SubX4T = MatrixSubtract(X4, X4.transpose);
//把自己减自己的地方置为1
X4SubX4T = MatrixAdd(X4SubX4T, Matrix4x4.identity);
Vector4 xCap = new Vector4(x, x, x, x);
Vector4 cap = xCap - X;
//以下计算L的部分如果用numpy还可以继续向量化
float L1 = (cap.y * cap.z) / (X4SubX4T * X4SubX4T);
float L2 = (cap.x * cap.z) / (X4SubX4T * X4SubX4T);
float L3 = (cap.x * cap.y) / (X4SubX4T * X4SubX4T);
Vector3 lPrev = new Vector3(L1, L2, L3);
Vector3 yPrev = new Vector3(xy1.y, xy2.y, xy3.y);
float resPrev = Vector3.Dot(lPrev, yPrev);
L1 = (cap.z * cap.w) / (X4SubX4T * X4SubX4T);
L2 = (cap.y * cap.w) / (X4SubX4T * X4SubX4T);
L3 = (cap.y * cap.z) / (X4SubX4T * X4SubX4T);
Vector3 lNext = new Vector3(L1, L2, L3);
Vector3 yNext = new Vector3(xy2.y, xy3.y, xy4.y);
float resNext = Vector3.Dot(lNext, yNext);
float t = (x - X.y) / (X.z - X.y);
return Mathf.Lerp(resPrev, resNext, t);
}
双线性二次插值
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