Game Development Reference
In-Depth Information
Chapter 6
More on Matrices
Man's mind stretched to a new idea
never goes back to its original dimensions.
— Oliver Wendell Holmes Jr. (1841-1935)
Chapter 4 presented a few of the most of the important properties and
operations of matrices and discussed how matrices can be used to ex-
press geometric transformations in general. Chapter 5 considered matri-
ces and geometric transforms in detail. This chapter completes our cov-
erage of matrices by discussing a few more interesting and useful matrix
operations.
•
Section 6.1
covers the determinant of a matrix.
•
Section 6.2
covers the inverse of a matrix.
•
Section 6.3
discusses orthogonal matrices.
•
Section 6.4
introduces homogeneous vectors and 4 × 4 matrices, and
shows how they can be used to perform a
ne transformations in
3D.
•
Section 6.5
discusses perspective projection and shows how to do it
with a 4 × 4 matrix.
6.1
Determinant of a Matrix
For square matrices, there is a special scalar called the determinant of the
matrix. The determinant has many useful properties in linear algebra, and
it also has interesting geometric interpretations.
As is our custom, we first discuss some math, and then make some
geometric interpretations.
Section 6.1.1
introduces the notation for deter-
minants and gives the linear algebra rules for computing the determinant of
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