Re: affine transform,
- From: "Dave Eberly" <dNOSPAMeberly@xxxxxxxxxxxxxxx>
- Date: Sat, 15 Jul 2006 13:25:46 GMT
"vsgdp" <hello@xxxxxxxx> wrote in message
news:TfZtg.6686$5K2.5606@xxxxxxxxxxxxx
In books like "Geometric Tools for Computer Graphics" it seems that they
assume a transformation is affine to construct the matrix representation
without proving it is affine first. In particular, they often use the
step:
T(P) = T(Q + v) = T(Q) + T(v) = Q + T(v), where T(Q) = Q because Q is
invariant under the transform.
Chapters 1 through 4 of that book are somewhat loose with
the mathematics and with the notation (not my choice).
The first part of the book's definition (page 98) for an affine
transformation T 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 does not conform to the formal definition a mathematician
would use: Affine transformations map points to points; they
do not map vectors to vectors. See my discussion later in this
post, marked (***).
The second part of the book's definition is that T preserves
affine combinations:
T(a1*P1+...+an*Pn) = a1*T(P1)+...+an*T(Pn)
where a1+...+an = 1. The P1 through Pn are points in some
affine space A and the T(P1) through T(Pn) are the transformed
points in some affine space B.
On page 100, there is a discussion about how the affine
transformation operates on vectors. Given two points P
and Q, the difference is a vector v = Q - P. The displayed
equation on that page is
T(v) = T(Q - P) = T(Q) - T(P)
Because the book's definition insists that affine transformations
apply to vectors as well as points, the displayed equation
must be accepted as *the definition* for how T applies to a
vector v. It is *not* a consequence of T preserving affine
combinations (as the book leads you to believe). Indeed,
the preservation of affine combinations cannot be used to
conclude T(Q - P) = T(Q) - T(P), because Q - P = 1*Q + (-1)*P
and the coefficient sum is (1) + (-1) = 0, not 1.
But how is T(Q + v) = T(Q) + T(v) legal until they have proved it is an
affine transform?
If you use the book's definition for an affine transformation,
and if you define P = Q + v, then v = P - Q and
T(v) = T(P) - T(Q)
which is rewritten as
T(P) = T(Q) + T(v)
and finally, using the definition for P
[eqn. 1] T(Q + v) = T(Q) + T(v)
The mathematical statement is:
[If T is an affine transformation, then T(Q + v) = T(Q) + T(v)]
The *hypothesis* is that T is an affine transformation. Under
that hypothesis, the conclusion is that T(Q + v) = T(Q) + T(v).
There is no "proving" that T is affine. The statement is a
hypothesis.
(***) To be abstract, Let E be an n-dimensional vector space and
let A be a set of elements called points. The set A is said to be
an n-dimensional affine space with the "difference space" E when
1. For each ordered pair of points P and Q in A, there is a unique
vector v in E associated with P and Q. Call this vector <P,Q>.
2. For each point P in A and for every vector v in E, there is a
exactly one point Q in A such that <P,Q> = v.
3. If P, Q, and R are points in A, then <P,Q> + <Q,R> = <P,R>
Note that the "+" in item 3 is the addition that is defined in the
vector space E. The usual notation for <P,Q> is suggestive of
subtraction, namely, v = P - Q.
Let A be an n-dimensional affine space with vector space E and
let B be an m-dimensional affine space with vector space F. The
transformation T : A -> B is said to be an affine transformation
when
1. <P1,Q1> = <P2,Q2> implies <T(P1),T(Q1)> = <T(P2),T(Q2)>
2. The function L : E -> F defined by L(<P,Q>) = <T(P),T(Q)>
is a linear transformation.
To make this appear more familiar, let's switch to the subtractcion
notation.
1. P1 - Q1 = P2 - Q2 implies T(P1) - T(Q1) = T(P2) - T(Q2)
2. L(P - Q) = T(P) - T(Q) is a linear transformation
If you define vector v = P - Q, then item 2 says L(v) = T(P) - T(Q),
or
T(P) = T(Q) + L(v)
or
[eqn. 2] T(Q + v) = T(Q) + L(v)
Notice that [eqn. 1] says T(Q + v) = T(Q) + T(v) but [eqn. 2]
says T(Q + v) = T(Q) + L(v). The latter equation makes it clear
that an affine transformation T has an associated linear
transformation L. In short, affine transformation T applies to
points and linear transformation L applies to vectors.
--
Dave Eberly
http://www.geometrictools.com
.
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