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MultiplyPoint3x4 in Unity

01/21/2020

What is the difference between MultiplyPoint and MultiplyPoint3x4 in Unity3D?

What steps of a matrix multiplication could Unity possibly be skipping to make an optimization? The optimization is possible because a scaling and rotation needs only a 3x3 matrix and another vector for the translation, so part of the full 4x4 matrix is unused and can be ignored.

Here is the source code taken from the decompilation here:

public Vector3 MultiplyPoint(Vector3 v)
{
    Vector3 vector3;
    vector3.x =  ( m00 *  v.x +  m01 *  v.y +  m02 *  v.z) + m03;
    vector3.y =  ( m10 *  v.x +  m11 *  v.y +  m12 *  v.z) + m13;
    vector3.z =  ( m20 *  v.x +  m21 *  v.y +  m22 *  v.z) + m23;
    float num = 1f / ( ( m30 *  v.x +  m31 *  v.y +  m32 *  v.z) + m33); // this is 1/1=1 when m30, m31, m32 = 0 and m33 = 1
    vector3.x *= num; // so then multiplying by 1 is pointless..
    vector3.y *= num;
    vector3.z *= num;
    return vector3;
}

This is a full 4x4 * 4x1 matrix multiplication, and then scaling the x y z of the result by the w of the result.

 | m_0_0 m_0_1 m_0_2 m_0_3 |     | x |
 | m_1_0 m_1_1 m_1_2 m_1_3 |  *  | y |
 | m_2_0 m_2_1 m_2_2 m_2_3 |     | z |
 | m_3_0 m_3_1 m_3_2 m_3_3 |     | 1 |

This w necessary for projective transformations, but not for rotation and scaling. This is why they have the faster 3x4 version. It assumes the w component of each column is 0 and skips that bottom row of multiplication and scaling, because the resulting w is always 1.

public Vector3 MultiplyPoint3x4(Vector3 v)
{
    Vector3 vector3;
    vector3.x =  ( m00 *  v.x +  m01 *  v.y +  m02 *  v.z) + m03;
    vector3.y =  ( m10 *  v.x +  m11 *  v.y +  m12 *  v.z) + m13;
    vector3.z =  ( m20 *  v.x +  m21 *  v.y +  m22 *  v.z) + m23;
    return vector3;
}

This function is equivalent to multiplying matrices which look like:

 | m_0_0 m_0_1 m_0_2 m_0_3 |     | x |
 | m_1_0 m_1_1 m_1_2 m_1_3 |  *  | y |
 | m_2_0 m_2_1 m_2_2 m_2_3 |     | z |
 | 0     0     0     1     |     | 1 |