Review of Mathematics for Computer Graphics

Vectors

Vectors form a space with the operations of

• commutativity
• associativity
• identity element
• inverse

• commutativity
• associativity
• identity element

Once an origin is defined we can use the parallelogram rule for the addition of vectors.

The span of vectors is the union of all linear combinations of the vectors.

Contrast a vector space to an Affine Space that is a collection of points with no inherent origin but that does have an associated vector space and two operations with associativity and an identity element.

P-Q = v a vector
P+v = R a new point

Define an affine combination to be
P + t(Q-P)

which leads to the equation of a line in affine space (or parametric space) to be:

(1-t)P + tQ

In a similar way, the equation of the plane is
(1-s)((1-t)P + tQ) + sR

A convex combination is when the sum of the parametric coefficients is less than or equal to one. Under this condition, all points will be within the convex hull of the three noncollinear points that define the plane.

S is a linear subspace if aV and v + w are in the same set for all v and w in the space.

An affine subspace is similarly defined. This is an important concept for geometrical transformations from the text, but a picture serves best to describe it.

Other notions from vector spaces

• linear independence of vectors if you can't make one from the others by linear combination
• basis is the minimal span of the vectors defining the space
• standard basis of R3 is (1 0 0) (0 1 0) and (0 0 1)
• orthonormal basis is where u.v is zero for all pairs of basis vectors

Dot Product

Know how to calculate x1y1 + x2y2 + x3y3 and that the length of a vector is sqrt(v.v)

nice properties, symmetric, nondegenerate, and bilinear in that v.(v+aW) = v.u + a(v.w)

normalize a vector by taking v/||v||

angle between two vectors given by cos(theta) = v.w/||v|| ||w||

The equation of a parametric plane is
(X-P).v = 0

Cross Product

The cross product of two vectors produces a third at right angles that has length related to the sin of the angle between the two vectors.

use the determinant i,j,k method for remembering how to compute.

cross products give rise to the notion of an orientation that is based on the sign of the determinant. So called right and left handed coordinate systems.

From Arizona State University -- see details 18.12.1996