3.5 Summary

3.5 Summary

• The fundamental transformation matrices — scaling, rotation, and translation — are given by:

• 

• We use 4×4 matrices to represent transformations and 1×4 homogeneous coordinates to describe points and vectors, where we denote a point by setting the fourth component to w=1 and we denote a vector by setting w=0. In this way, translations are applied to points but not to vectors.

• A matrix is orthogonal if all of its row vectors are of unit length and mutually orthogonal. An orthogonal matrix has the special property that its inverse is equal to its transpose, thereby making the inverse easy and efficient to compute. All the rotation matrices are orthogonal.

• From the associative property of matrix multiplication, we can combine several transformation matrices into one transformation matrix that represents the net effect of applying the individual matrices sequentially.

• Let Q_B, u_B, v_B, and w_B describe the origin and x-, y-, and z-axes of frame A with coordinates relative to frame B, respectively. If a vector/point p has coordinates p_A = (x, y, x) relative to frame A, then the same vector/point relative to frame B has coordinates:

  pB = (x′, y′, z′) = xuB + yvB + zwB	For vectors (direction and magnitude)
  pB = (x′, y′, z′) = QB + xuB + yvB + zwB	For position vectors (points)

  These change of coordinate transformations can be written in terms of matrices using homogeneous coordinates.

• Suppose we have three frames, F, G, and H, and let A be the change of frame matrix from F to G, and let B be the change of frame matrix from G to H. Using matrix-matrix multiplication, the matrix C = AB can be thought of as the change of frame matrix F directly into H; that is, matrix-matrix multiplication combines the effects of A and B into one net matrix, and so we can write: p_F(AB) = p_H.

• If the matrix M maps frame A coordinates into frame B coordinates, then the matrix M⁻¹ maps frame B coordinates into frame A coordinates.