1. The Projective
Plane
Motivation, The
projective plane, Points and lines, Projective line coordinates, Projective
plane coordinates, Ranges and pencils, Pappus' theorem, Menelaos'
Theorem, Duality, The extended affine plane, Topological aspects,
Parametric lines, Lines and line segments, Projective threespace,
Problems,
2. Projective
Maps
Perspectivities,
Cross ratios, Projectivities, Moebius transformations, Collineations,
Constructing Collineations, Affine maps, Problems,
3. Conics
Line and point
conics, Conic constructions, Points and lines, Pascal's theorem, Affine
conics, Problems,
4. Conics in
Parametric Form
Parametric curves
in P2, The Bernstein form of a conic, The Dual Form, Derivatives,
The implicit form, Implicit interpolating conics, Parametric interpolating
conics, Blossoms and polars, Reparametrization, The complementary
segment, Parametric affine conics, Problems
5. Rational
Quadratic Conics
The rational quadratic
form, Conic classification, Properties of rational quadratics, Derivatives,
Control vectors, The parabola, The circle, Problems,
6. Conic splines
C1 conditions,
Parametric continuity for rational conic splines, Conic Bsplines,
Curvature continuity, G2 Conic splines, Interpolation with G2 conic
splines, Higher order smoothness, Problems
7. Rational
Bezier curves
The Bernstein form,
The de Casteljau algorithm, Degree elevation, Degree reduction, Reparametrization,
Nonparametric curves, Base points, Derivatives, Smoothness, Blossoms,
Control vectors, Duality, Hybrid Bezier curves, Straight lines as
Bezier curves, Problems
8. Rational
Cubics
Rational cubic
conics, Embedding cones, The embedding bilinear patch, Asymptotes,
The Hermite form, Cubic Lagrange interpolation, Osculatory interpolation,
Problems
9. Pythagorean
Curves
Circles and the
stereographic projection, Pythagorean hodograph curves, Rational PH
offsets, Rational offsets, Rational curves on the sphere, Problems
10. Rectangular
Patches
Bilinear patches,
Bezier patches, De Casteljau algorithms and blossoms, The weight points,
Derivatives and continuity, Reparametrization, Rectangular NURBS surfaces,
Blossoms, Surfaces of revolution, Developable surfaces, Problems
11. Rational
Bezier Triangles
Projective Bezier
triangles, Degree elevation, Blossoms and derivatives, Barycentric
coordinates, Rational Bezier triangles, Problems
12. Quadrics
Quadrics, Quadric
classification, The stereographic projection, Quartics on quadrics,
Problems
13. Gregory
Patches
Bicubic Gregory
patches, The rational Bezier form, RBG patches, Gregory triangles,
Base points and multisided patches, Problems
14. Examples
and standards
IGES, A semicircle,
A hyperbolic paraboloid, A quarter of a cylinder, A rational quadratic
patch on a sphere, An octant of a sphere, A sixth of a sphere, A sixteenth
of a torus, Problems
