Re: 3D Game Engine Design Book Q2


And what will be your "proof"? In my definition, the domain of the
transformation A is a set of points. According to Axiom 5 on page 15,
V = a1*P1 + ... + ak*Pk is a vector, since the sum of the coefficients is 0.
The left-hand side of your equation is A(V), where the input is a vector,
not a point. The right-hand side has terms like A(P1), where the input
is a point.

From my understanding of previous literature, an affine transformation
maps elements from an affine space to an affine space. Thus vectors
would be in the domain of an affine transformation. In "Geometric
Tools" the definition is: "An affine transformation is a map taking
points and vectors in one affine space to points and vectors,
respectively, in another affine space." This book also proves that an
affine transformation is linear with respect to vectors.

If the domain of an affine transformation does not include vectors, I
don't see how it can be written that:

A(Q-P) = A(Q)-A(P), because Q-P is a vector, which is what is written
in "Geometric Tools".

In affine algebra, points and vectors are not the same
concept, yet your equation insists somehow that the transformation A
can have inputs that are both points and vectors.

Why can't it input both points and vectors? The affine space includes
both points and vectors, so a mapping from one affine space to another
should be able to handle both? You could define the function
piecewise to act a certain way for points and a certain way for
vectors, just like you can define a real valued function to act a
certain way for rationals and a certain way for irrationals. Of are
you saying an affine map is just a point map?



Notice on page 29:
"Let Q = A(P) be a transformation that maps points to points." Nowhere
is there an allowance for the inputs or outputs to be vectors.

--
Dave Eberlyhttp://www.geometrictools.com


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