Split a transform matrix into orhhogonal matrix and scale matrix

Question

If I have a matrix from scale, translate, and rotation transform. I want to split this matrix to two matrix. One is rotation+translation matrix, the other is scale matrix.

Because I want to compute the correct normal vector transform, so I only need orthogonal matrix to do the computation for surface normal vector

Any ideas?

I'm confused when you say you want to compute the correct normal vector transform. If your matrix is a scale, translate, and rotate, then it is orthogonal. — Matt
If the matrix is <2,0>,<0,1>. The vector <x,y> will be transformed to <2x,y>. The vector direction is not right. — giggle
I don't have time for a proper answer, but this kind of decomposition is QR decomposition (look it up on wikipedia). In addition the normal should transform according to the transpose of the inverse of your transformation matrix (although it will no longer be unit length). — Adam Bowen
QR is a bit overkill, and will only yield the desired answer if the operation is scale, then rotate+translate. An RQ decomposition is needed if the operation is rotate+translate, then scale. — David Hammen

David Hammen David Hammen · Accepted Answer · 2011-06-29T08:40:38

If I have a matrix from scale, translate, and rotation transform. I want to split this matrix to two matrix. One is rotation+translation matrix, the other is scale matrix.

I'm assuming this matrix you are talking about is a 4x4 matrix that is widely used by some, widely despised by others, with the fourth row being 0,0,0,1.

I'll cause these two operations "scale" and "rotate+translate". Note well: These operations are not commutative. Scaling a 3-vector and then rotating/translating this scaled vector yields a different result than you would get by reversing the order of operations.

Case 1, operation is "rotate+translate", then "scale".
Let SR=S*R, where S is a 3x3 diagonal matrix with positive diagonal elements (a scaling matrix) and R is a 3x3 orthonormal rotation matrix. The rows of matrix SR will be orthogonal to one another, but the columns will not be orthogonal. The scale factors are the square root of the norms of the rows of the matrix SR.

Algorithm: Given 4x4 matrix A, produce 4x4 scaling matrix S, 4x4 rotation+translation matrix T

A = [  SR(3x3)  Sx(3x1) ]
    [   0(1x3)     1    ]

Partition A into a 3x3 matrix SR and a 3 vector Sx as depicted above.
Construct the scaling matrix S. The first three diagonal elements are the vector norms of the rows of matrix SR; the last diagonal element is 1.
Construct the 4x4 rotation+translation matrix T by dividing each row of A by the corresponding scale factor.

Case 2, operation is "scale", then "rotate+translate".
Now consider the case RS=R*S. Here the columns of A will be orthogonal to one another, but the rows will not be orthogonal. In this case the scale factors are the square root of the norms of the columns of the matrix RS.

Algorithm: Given 4x4 matrix A, produce 4x4 rotation+translation matrix T, 4x4 scaling matrix S

A = [  RS(3x3)  x(3x1) ]
    [   0(1x3)    1    ]

Partition A into a 3x3 matrix RS and a 3 vector x as depicted above.
Construct the scaling matrix S. The first three diagonal elements are the vector norms of the columns of matrix RS; the last diagonal element is 1.
Construct the 4x4 rotation+translation matrix T by dividing each row of A by the corresponding scale factor.

If the scaling is not uniform (e.g., scale x by 2, y by 4, z by 1/2), you can tell the order of operations by looking at the inner products of the rows and columns of the upper 3x3 matrix with one another. Scaling last (my case 1) means the row inner products will be very close to zero but the column inner products will be non zero. Scaling first (my case 2) reverses the situation. If the scaling is uniform there is no way to tell which case is which. You need to know beforehand.

Split a transform matrix into orhhogonal matrix and scale matrix

3 Answers