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Curves and Surfaces from 3-D Matrices

Curves and Surfaces from 3-D Matrices. Dan Dreibelbis University of North Florida. Richard. Goals. What is a 3-D matrix? Vector multiplication with a tensor Geometric objects from tensors Motivation Pretty pictures Richard’s work More pretty pictures. 3-D Matrices.

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Curves and Surfaces from 3-D Matrices

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  1. Curves and Surfaces from 3-D Matrices Dan Dreibelbis University of North Florida

  2. Richard

  3. Goals • What is a 3-D matrix? • Vector multiplication with a tensor • Geometric objects from tensors • Motivation • Pretty pictures • Richard’s work • More pretty pictures

  4. 3-D Matrices

  5. Vector Multiplication 1

  6. Vector Multiplication 2

  7. Vector Multiplication 3

  8. AEC, BEC, CEC • Define the AEC of a tensor as the zero set of all vectors such that the contraction with respect to the first index is a singular matrix. • Similar for BEC and CEC. • We can get this by doing the vector multiplication, taking the determinant of the result, then setting it equal to zero. • The result is a homogeneous polynomial whose degree and number of variables are both the same as the size of the tensor.

  9. AEC Det = 0

  10. AEC

  11. Curving Space

  12. Quadratic Warp

  13. Quadratic Warp

  14. Quadratic Warp

  15. Quadratic Map This is a tensor multiplication with two vectors!!

  16. The Curvature Ellipse

  17. Tangents from AEC F(x, y) AEC maps to the tangent lines of the curvature ellipse.

  18. Tangents from AEC F(x, y) AEC maps to the tangent lines of the curvature ellipse.

  19. Tangents from AEC F(x, y) AEC maps to the tangent lines of the curvature ellipse.

  20. Veronese Surface F(x, y, z)

  21. Veronese Surface F(x, y, z)

  22. Veronese Surface F(x, y, z)

  23. Drawing the AEC

  24. Cubic Curves

  25. Normalizing the Curve Two AEC are equivalent if there is a change of coordinates that takes one form into another. Goal: Find a representative of each equivalence class.

  26. Normal Form Theorem:Any nondegenerate 3x3x3 tensor is equivalent to a tensor of the form: for some c and d. The AEC for this tensor is:

  27. AEC = BEC = CEC Theorem: For any nondegenerate 3x3x3 tensor, the AEC, BEC, and CEC are all projectively equivalent. This is far from obvious:

  28. AEC=BEC=CEC

  29. 4-D Case

  30. 4-D AEC, Page 1

  31. 4-D AEC, Page 33

  32. AEC

  33. More AEC’s

  34. Thanks!

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