矩阵变换问题 - Z 轴旋转倾斜

Question

对于我正在制作的一个简单的 2d 游戏，我尝试使用矩阵绕 z 轴旋转精灵。我显然做错了什么，因为当我尝试旋转精灵时，它看起来像是围绕屏幕原点（底部，左侧）而不是精灵原点旋转。我很困惑，因为我的四边形已经在原点了，所以我认为我不需要翻译 ->旋转并平移回来。这是一个代码片段和一个小视频或错误的转换

void MatrixMultiply(
MATRIX      &mOut,
const MATRIX    &mA,
const MATRIX    &mB);
/*!***************************************************************************
@Function           TransTransformArray
@Output         pTransformedVertex  Destination for transformed vectors
@Input              pV                  Input vector    array
@Input              nNumberOfVertices   Number of vectors to transform
@Input              pMatrix             Matrix to transform the vectors of input vector (e.g. use 1 for position, 0 for normal)
@Description        Transform all vertices in pVertex by pMatrix and store them in
                pTransformedVertex
                - pTransformedVertex is the pointer that will receive transformed vertices.
                - pVertex is the pointer to untransformed object vertices.
                - nNumberOfVertices is the number of vertices of the object.
                - pMatrix is the matrix used to transform the object.
*****************************************************************************/
void TransTransformArray(
VECTOR3     * const pTransformedVertex,
const VECTOR3   * const pV,
const int     nNumberOfVertices,
const MATRIX    * const pMatrix);


RenderQuad CreateRenderQuad(
    const Texture2D & texture,
    float x,
    float y,
    float scaleX, 
    float scaleY,
    float rotateRadians,
    int   zIndex,
    const Color & color,
    const Quad2 & textureCoord,
    const char * name
) {
    MATRIX mT;
    MATRIX mS;
    MATRIX concat;  
    MATRIX mR;

    MatrixTranslation(mT, x, y, 0.0f);
    MatrixRotationZ(mR, rotateRadians);
    MatrixScaling(mS, scaleX, scaleY, 1.0f);

    VECTOR3 quad[] = {
        {-0.5f, 0.5f, 0.f}, //tl
        {0.5f, 0.5f, 0.f}, //tr
        {-0.5, -0.5f, 0.0f}, //bl
        {0.5f, -0.5f, 0.0f}, //br
    };

    MatrixMultiply(concat, mR, mT);
    MatrixMultiply(concat, concat, mS);
    // apply to all the points in the quad
    TransTransformArray(quad, quad, 4, &concat);

==更新：

这里是结构和渲染代码：

我正在使用 oolongengine code.google.com/p/oolongengine/source/browse/trunk/Oolong%20Engine2/Math/Matrix.cpp 中的矩阵类，

我转换所有四边形，然后使用OpenGL。这是我的数据结构和渲染代码：

typedef struct _RenderData {
    VECTOR3        vertex;
    RenderColor3D      color;
    RenderTextureCoord textureCoord;
    float              zIndex;
    GLuint             textureId;
} RenderData;

typedef struct _RenderQuad {
    //! top left
    RenderData  tl;
    //! top right
    RenderData  tr;
    //! bottom left
    RenderData  bl;        
    //! bottom right
    RenderData  br;

    float zIndex;

    Texture2D * texture; // render quad draws a source rect from here

    ESpriteBlendMode blendMode;

} RenderQuad ;

/// Draw
class QuadBatch {
   GLushort *              m_indices;
   const Texture2D *       m_texture;
   GLuint                   m_vbos[2];
   RenderData *            m_vertices; 
};

QuadBatch::Draw () {
    int offset = (int)&m_vertices[startIndex];

            // vertex
            int diff = offsetof( RenderData, vertex);
            glVertexPointer(3, GL_FLOAT, kRenderDataSize, (void*) (offset + diff) );

            // color
            diff = offsetof( RenderData, color);
            glColorPointer(4, GL_FLOAT, kRenderDataSize, (void*)(offset + diff));

            // tex coords
            diff = offsetof( RenderData, textureCoord);
            glTexCoordPointer(2, GL_FLOAT, kRenderDataSize, (void*)(offset + diff));

            // each quad has 6 indices

            glDrawElements(GL_TRIANGLES, vertexCount * elementMultiplier, GL_UNSIGNED_SHORT, m_indices);

Answer 1

根据定义，“旋转”是围绕原点 (0,0,0) 的。如果您想要不同的旋转轴，则必须应用平移组件。假设您要围绕 a 轴应用旋转 R。应用于任意向量 x 的变换为：

x--> a + R(x - a) = Rx + (a - Ra)

（这可能需要一些凝视才能消化）。因此，应用旋转后（正如您所观察到的，旋转围绕原点旋转），您必须添加常数向量 (a - Ra)。

[编辑：]这个答案与语言和平台无关——无论你看什么，数学都是一样的。特定的库包含不同的结构和 API 来应用转换。例如，DirectX 和 OpenGL 都维护 4x4 矩阵变换，以将旋转和平移统一为单个矩阵乘法（通过称为齐次坐标的装置）。