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Syllabus
The following is an approximate syllabus for the course. This is meant only
as a guideline; the topics taught on any particular day may change for
various reasons. After each class, I will update this page with what
was taught on a particular day.
- Week 1 (January 6)
- January 6:
Organizational meeting, Background material
on spaces, affine geometry.
- January 8:
Polynomials (monomials), Lagrange polynomials.
Bernstein polynomials and Bezier curves. de Casteljau's
algorithm.
- Week 2 (January 13)
- January 13:
Triangle diagrams.
Blossoming principle.
de Casteljau revisited.
- January 15:
Subdivision. Degree Raising. Functional Bezier curves.
Multilinear blossoming.
- Week 3 (January 20):
- January 20:
Curve derivatives. Curve continuity.
Cubic Hermite interpolation.
- January 22:
Change of basis via de Casteljau evaluation. Fast evaluation.
Numerics, stability of repeated linear interpolation.
- January 24:
Finish stability of repeated linear interpolation.
Triangle diagrams for interpolatory curves; Neville's algorithm.
Root finding with Bezier representation.
- Week 4 (January 27):
- January 27:
No lecture
- January 29:
B-splines. Lane-Riesenfeld.
Knot multiplicity started.
- Week 5 (February 3):
- February 3:
Knot multiplicity finished. Knot insertion.
Triangle diagrams.
Lane-Riesenfeld algorithm revisited.
Lane-Riesenfeld, knots in geometric progression.
- February 5:
Lane-Riesenfeld algorithm extended.
B-spline Basis functions.
- Week 6 (February 10):
- February 10:
Closed B-splines.
Degree raising B-splines.
Degree reduction.
Direct manipulation.
- February 12:
Rational curves. Conic sections.
Generalized barycentric coordinates.
- Reading week (February 17):
- Week 7 (February 24):
- February 24:
Project overview.
Generalized Bernstein polynomials.
Triangular Bézier patches.
Blossoms.
- February 26:
Derivatives.
Storing control points: 3D and 2D arrays.
Simple Triangular Bézier patches evaluation scheme.
- Week 8 (March 2):
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March 2:
Storing control points: 1D array.
Triangular Bézier patches evaluation schemes.
Evaluation of Triangular Bézier patches on a grid of points.
3:1 subdivision, 2:1 subdivision, 4:1 subdivision.
Cracking problems.
Curve evaluation,
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March 4:
C0 continuity. Generalized C1 continuity.
Surfaces above the plane.
Tensor-product patches. Derivatives, C1 continuity.
Tensor-product B-splines.
Surfaces above planes.
- Week 9 (March 9):
- March 9:
Evaluation methods for Tensor-product patches.
Data interpolation: Lagrange surfaces.
Triangular data fitting - Clough-Tocher
- March 11:
Powell-Sabin.
Geometric continuity.
Crossboundary constructions. Farin's conditions.
Chiyokura-Kimura. Triangular data fitting - Shirman-Sequin.
- Week 10 (March 16):
- March 16:
Nielson. Boundary curves. Herron. Gregory-Charrot.
- March 18:
Evaluating surface quality: line drawings, shaded images.
Evaluating surface quality: curvature plots.
More boundary curves (dBHS).
Reflection lines, isophotes.
- Week 11 (March 23):
- March 23:
Subdivision curves.
Subdivision surfaces.
- March 25:
Higher dimensional simplices.
S-patches. A-patches.
Subdivision curves.
- Week 12 (March 30):
- March 30:
Student Presentations.
Topics will include topics such as
- Degree reduction
- The Least
- B-patches
- S-patches
- Wavelets
- A-patches
- Surface Pasting
- Implicit surfaces
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