Page 102 - Computer Graphics Handout

M is called the matrix representation of the change of frames. We can also use M to compute the changes in the representations
directly. Suppose that a and b are the homogeneous-coordinate representations either of two points or of two vectors in the two

frames. Then

When we work with representations, as is usually the case, we are interested in MT, which is of the form and is determined by 12
coefficients. There are other advantages to using homogeneous coordinates that we explore extensively in later chapters. Perhaps
the most important is that all affine (linepreserving) transformations can be represented as matrix multiplications in homogeneous
coordinates. Although we have to work in four dimensions to solve threedimensional problems when we use homogeneous-
coordinate representations, less arithmetic work is involved. The uniformrepresentation of all affine transformations makes carrying
out successive transformations (concatenation) far easier than in three-dimensional space. In addition, modern hardware
implements homogeneouscoordinate operations directly, using parallelism to achieve high-speed calculations.

3.3.5 Example Change in Frames
Consider again the example of Section 3.3.3. If we again start with the basis vectors v1, v2, and v3 and convert to a basis determined
by the same u1, u2, and u3, then the three equations are the same:
u1= v1,u2= v1+ v2,
u3= v1+ v2+ v3.
The reference point does not change, so we add the equation
Q0= P0.
Thus, the matrices in which we are interested are the matrix its transpose, and their inverses.

Suppose that in addition to changing the basis vectors, we also want to move the reference point to the point that has the
representation (1, 2, 3, 1) in the original system. The displacement vector v = v1+ 2v2+ 3v3 moves P0 to Q0. The fourth component
identifies this entity as a point. Thus, we add to the three equations from the previous example the equation
Q0= P0+ v1+ 2v2+ 3v3,
T
and the matrix M becomes

Its inverse is

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