Graphics Reference
In-Depth Information
The approach taken in our implementation is not the only one possible. Our
approach depends on
transformations
as the primitive notion, but it's quite pos-
sible and reasonable to think instead of
coordinate systems
as the fundamental
entity. Just as in Chapter 10 we discussed the interpretation of a linear transfor-
mation as a change of coordinates, you can approach much of graphics with this
point of view. You end up having coordinate frames for vector spaces (for a 2D
space, you have two independent vectors; for a 3D space, you have three inde-
pendent vectors), for affine spaces (typically a coordinate frame for a 2D affine
space consists of three points, from which you determine barycentric coordinates,
but you can also build a frame from one point and two vectors), and for projective
spaces (where in 2D, a projective frame is represented by four points in “general
position,” which we'll discuss shortly). This coordinate frame approach is taken
by Mann et al. [MLD97].
The predefined
Point
and
Vector
classes in WPF already implement the main
ideas we've discussed (for two dimensions): There are operators defined so that
you can add one
Point
and one
Vector
to get a new
Point
, but there is no operator
for adding two
Point
s, for instance. Certain convenience operations have been
included, like the dot product of
Vector
s.
There are idiosyncrasies in the design of the classes, however. The two coor-
dinates of a point
P
are
P.X
and
P.Y
; there's no way to refer to them as elements
of an array of length 2, nor even a predefined “cast” operation to convert to a
double[2]
. There is, however, a predefined
CrossProduct
operation for
Vector
s,
which treats the vectors as lying in the
xy
-plane of 3-space, computes the 3D cross
product (which always points along the
z
-axis), and returns the
z
-component of the
resultant vector. In deference to the convenience of having data types that work
well with the remainder of WPF, we've ignored these idiosyncrasies and simply
used the parts of the
Point
and
Vector
classes (and their 3D analogs) that we like.
We've also added some geometric functions to our
LIN_ALG
namespace (in which
all the transformation classes reside) to compute things like the two-dimensional
cross product of one vector.
While WPF also has a class called
Matrix
, its peculiarities made it unsuitable
for our use. Furthermore, we wanted to build a library in which the fundamental
idea was that of a
transformation
rather than the matrix used to represent it. We
therefore define four classes.
•
MatrixTransformation2
: A parent class for linear, affine, and projective
transformations. Since all three can be represented by 3
×
3 matrices, a
MatrixTransformation
holds a 3
3 matrix and provides certain support
procedures to multiply and invert such matrices.
×
•
LinearTransformation2
: A transformation that takes
Vector
sto
Vector
s.
•
AffineTransformation2
: A transformation that can operate on both
Point
s and
Vector
s.
