几何变换中用到的矩阵知识

Mon Nov 17 2025

仿射变换通用矩阵（Affine Transform）

[x′y′1]=[abdxcddy001][xy1]\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} a & b & dx \\ c & d & dy \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}
  • a, b, c, d：控制旋转、缩放、斜切
  • dx, dy：平移

平移

[10dx01dy001]\begin{bmatrix} 1 & 0 & dx \\ 0 & 1 & dy \\ 0 & 0 & 1 \end{bmatrix} [x′,y′]=(x+dx,y+dy)[x', y'] = (x + dx, y + dy)

缩放

[sx000sy0001]\begin{bmatrix} sx & 0 & 0 \\ 0 & sy & 0 \\ 0 & 0 & 1 \end{bmatrix} [x′,y′]=(sx∗x,sy∗y)[x', y'] = (sx * x, sy * y)

旋转

[cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001]\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} [x′,y′]=(cos⁡θ∗x−sin⁡θ∗y,sin⁡θ∗x+cos⁡θ∗y)[x', y'] = (\cos\theta * x -\sin\theta * y, \sin\theta * x + \cos\theta * y)

Compose使用的4x4矩阵

[x′y′z′w]=[m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33][xyz1]\begin{bmatrix} x' \\ y' \\ z' \\ w \end{bmatrix} = \begin{bmatrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}

点映射

w=m03x+m13y+m33x′=m00x+m10y+m30wy′=m01x+m11y+m31w\begin{aligned} w &= m_{03}x + m_{13}y + m_{33} \\[6pt] x' &= \frac{m_{00}x + m_{10}y + m_{30}}{w} \\[6pt] y' &= \frac{m_{01}x + m_{11}y + m_{31}}{w} \end{aligned} 
    fun map(point: Offset): Offset {
        // See top-level comment
        if (values.size < 16) return point
 
        val v00 = this[0, 0]
        val v01 = this[0, 1]
        val v03 = this[0, 3]
        val v10 = this[1, 0]
        val v11 = this[1, 1]
        val v13 = this[1, 3]
        val v30 = this[3, 0]
        val v31 = this[3, 1]
        val v33 = this[3, 3]
 
        val x = point.x
        val y = point.y
        val z = v03 * x + v13 * y + v33
        val inverseZ = 1 / z
        val pZ = if (inverseZ.fastIsFinite()) inverseZ else 0f
 
        return Offset(x = pZ * (v00 * x + v10 * y + v30), y = pZ * (v01 * x + v11 * y + v31))
    }

缩放

[sx0000sy0000sz00001][m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33]\begin{bmatrix} s_x & 0 & 0 & 0 \\ 0 & s_y & 0 & 0 \\ 0 & 0 & s_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{bmatrix} 
    /** Scale this matrix by [x], [y], [z] */
    fun scale(x: Float = 1f, y: Float = 1f, z: Float = 1f) {
        // See top-level comment
        if (values.size < 16) return
        this[0, 0] *= x
        this[0, 1] *= x
        this[0, 2] *= x
        this[0, 3] *= x
        this[1, 0] *= y
        this[1, 1] *= y
        this[1, 2] *= y
        this[1, 3] *= y
        this[2, 0] *= z
        this[2, 1] *= z
        this[2, 2] *= z
        this[2, 3] *= z
    }

位移

[000000000000dxdydz1][m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33]\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ d_x & d_y & d_z & 1 \end{bmatrix} \begin{bmatrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{bmatrix} 
    /** Translate this matrix by [x], [y], [z] */
    fun translate(x: Float = 0f, y: Float = 0f, z: Float = 0f) {
        // See top-level comment
        if (values.size < 16) return
        val t1 = this[0, 0] * x + this[1, 0] * y + this[2, 0] * z + this[3, 0]
        val t2 = this[0, 1] * x + this[1, 1] * y + this[2, 1] * z + this[3, 1]
        val t3 = this[0, 2] * x + this[1, 2] * y + this[2, 2] * z + this[3, 2]
        val t4 = this[0, 3] * x + this[1, 3] * y + this[2, 3] * z + this[3, 3]
        this[3, 0] = t1
        this[3, 1] = t2
        this[3, 2] = t3
        this[3, 3] = t4
    }

旋转

[cos⁡θsin⁡θ00−sin⁡θcos⁡θ0000100001][m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33]\begin{bmatrix} \cos\theta & \sin\theta & 0 & 0 \\ -\sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{bmatrix} 
    fun rotateZ(degrees: Float) {
        // See top-level comment
        if (values.size < 16) return
 
        val r = degrees * (PI / 180.0)
        val s = sin(r).toFloat()
        val c = cos(r).toFloat()
 
        val a00 = this[0, 0]
        val a10 = this[1, 0]
        val v00 = c * a00 + s * a10
        val v10 = -s * a00 + c * a10
 
        val a01 = this[0, 1]
        val a11 = this[1, 1]
        val v01 = c * a01 + s * a11
        val v11 = -s * a01 + c * a11
 
        val a02 = this[0, 2]
        val a12 = this[1, 2]
        val v02 = c * a02 + s * a12
        val v12 = -s * a02 + c * a12
 
        val a03 = this[0, 3]
        val a13 = this[1, 3]
        val v03 = c * a03 + s * a13
        val v13 = -s * a03 + c * a13
 
        this[0, 0] = v00
        this[0, 1] = v01
        this[0, 2] = v02
        this[0, 3] = v03
        this[1, 0] = v10
        this[1, 1] = v11
        this[1, 2] = v12
        this[1, 3] = v13
    }
[m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33][cos⁡θ0−sin⁡θ00100sin⁡θ0cos⁡θ00001]\begin{bmatrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{bmatrix} \begin{bmatrix} \cos\theta & 0 & -\sin\theta & 0 \\ 0 & 1 & 0 & 0 \\ \sin\theta & 0 & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} 
    fun rotateY(degrees: Float) {
        // See top-level comment
        if (values.size < 16) return
 
        val r = degrees * (PI / 180.0)
        val s = sin(r).toFloat()
        val c = cos(r).toFloat()
 
        val a00 = this[0, 0]
        val a02 = this[0, 2]
        val v00 = a00 * c + a02 * s
        val v02 = -a00 * s + a02 * c
 
        val a10 = this[1, 0]
        val a12 = this[1, 2]
        val v10 = a10 * c + a12 * s
        val v12 = -a10 * s + a12 * c
 
        val a20 = this[2, 0]
        val a22 = this[2, 2]
        val v20 = a20 * c + a22 * s
        val v22 = -a20 * s + a22 * c
 
        val a30 = this[3, 0]
        val a32 = this[3, 2]
        val v30 = a30 * c + a32 * s
        val v32 = -a30 * s + a32 * c
 
        this[0, 0] = v00
        this[0, 2] = v02
        this[1, 0] = v10
        this[1, 2] = v12
        this[2, 0] = v20
        this[2, 2] = v22
        this[3, 0] = v30
        this[3, 2] = v32
    }
[m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33][10000cos⁡θsin⁡θ00−sin⁡θcos⁡θ00001]\begin{bmatrix} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & \sin\theta & 0 \\ 0 & -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} 
    fun rotateX(degrees: Float) {
        // See top-level comment
        if (values.size < 16) return
 
        val r = degrees * (PI / 180.0)
        val s = sin(r).toFloat()
        val c = cos(r).toFloat()
 
        val a01 = this[0, 1]
        val a02 = this[0, 2]
        val v01 = a01 * c - a02 * s
        val v02 = a01 * s + a02 * c
 
        val a11 = this[1, 1]
        val a12 = this[1, 2]
        val v11 = a11 * c - a12 * s
        val v12 = a11 * s + a12 * c
 
        val a21 = this[2, 1]
        val a22 = this[2, 2]
        val v21 = a21 * c - a22 * s
        val v22 = a21 * s + a22 * c
 
        val a31 = this[3, 1]
        val a32 = this[3, 2]
        val v31 = a31 * c - a32 * s
        val v32 = a31 * s + a32 * c
 
        this[0, 1] = v01
        this[0, 2] = v02
        this[1, 1] = v11
        this[1, 2] = v12
        this[2, 1] = v21
        this[2, 2] = v22
        this[3, 1] = v31
        this[3, 2] = v32
    }

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