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Some Mathematical Elements of
Some Mathematical Elements of Graphics
Ray tracing primitives
Mathematical preliminaries
Coordinate systems
Vector algebra
Equation of a ray
Equations for the primitives
Sphere
Cylinder
Cone
Torus
Plane
Polygon
Disc
Intersections with arbitrarily positioned primitives
Transforming the primitives
Transforming normal vectors
Converting the primitives to polygons
Bezier curves
Ways of thinking about Bezier curves
Continuity
Bezier surfaces
B-splines
The knot vector
Uniform knot vector
Things you can change about a uniform B-spline
Open uniform knot vector
The difference between uniform and open uniform
Non-uniform knot vector
Use of non-uniform knot vectors
What can you do to control the shape of a B-spline?
What should the defaults be?
B-spline patches
Rational B-splines
An example: a circle defined by NURBS
Sweeping with NURBS
Subdivision curves and surfaces
Mathematical details: curves
Mathematical details: surfaces
About this document ...
Neil Dodgson
2000-09-25