Mathematics 431 Lecture Notes |
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Geometry and Computer Graphics |
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| Course Overview | |||||||||||||
| The lectures are available for download as Mathematica notebooks. Clicking on a notebook link below should cause your browser to download the document as a text file which can be opened by Mathematica 6.0 or Mathematica Player. Mathematica Player is a free program distributed by Wolfram Research that displays Mathematica documents.
When viewed using Mathematica, computations may be performed and animations may be run. Computations may not be performed when using Mathematica Player.
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| Download Mathematica Player | |||||||||||||
Packages: |
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| Peano.m (Revised 12/07) | |||||||||||||
Lecture Notes: |
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| 1. Homogeneous Coordinates in the Plane. Points, lines and the projective plane. Displaying curves and surfaces. Transformations in homogeneous coordinates. Download Notebook.
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| 2. The Join. Permutations and determinants. Definition of the join. The extensor-subspace correspondence. General projective spaces. Direction flats and parallel flats. The Peano algebra, Plucker coordinates, and the bracket. Envelopes and tangents. Transformations of flats. Download Notebook. |
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| 3. The Meet. The intersection of subspaces. The meet of multivectors. The meet of flats. Intercepts and covectors. Direction flats and parallelism. Envelopes. Transformations. Download Notebook. |
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| 4. Projections. Projections. Projection matrices, adjugates and Cramer's rule. Two Examples. Case Study: Mathematica. Download Notebook. |
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| 5. Duality and Orthogonality. The duality map. Duality for flats. The extensor A+. Normal flats. Download Notebook.
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| 6.Two-Sided Projective Space. Two sided projective space. The two sides of a hyperplane and clipping. Transformations, Oriented angles. Shading I—diffuse reflection. Download Notebook. |
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| 7. Coordinate Systems. Defining a coordinate system for a flat. Affine Coordinates. Projective transformations. Affine transformations. The Z-list. Brezier curves. Brezier Surfaces. Download Notebook. |
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| 8. Rotations and Reflections. The Clifford product. Congruences. Reflections. Shading II—specular reflection. Translations. Complex numbers, quaternions, and spinors. Rotations. Euler Angles. More rotation examples. Generating surfaces with transformations. Download Notebook. |
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Appendices: |
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| A. Mathematica Selecta. Notebooks. Common problems. Entering special symbols. Loading packages. Graphics. Common constructions. Download Notebook. | |||||||||||||
| B. Some Curves and Surfaces. Plane curves. Space curves. Surfaces. Download Notebook. | |||||||||||||
| C. Some Linear Algebra. Vector spaces. Linear transformations. Matrices. Inner products. Orthogonal Transformations. Download Notebook. | |||||||||||||
| D. Notation and Set Theory. Notation. Equivalence relations. Download Notebook. | |||||||||||||
| References. Further reading.. Download Notebook. | |||||||||||||