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Curves and Surfaces from 3-D MatricesPowerPoint Presentation

Curves and Surfaces from 3-D Matrices

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Presentation Transcript

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

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.

Quadratic Map

This is a tensor multiplication with two vectors!!

Veronese Surface

F(x, y, z)

Veronese Surface

F(x, y, z)

Veronese Surface

F(x, y, z)

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.

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:

AEC = BEC = CEC

Theorem: For any nondegenerate 3x3x3 tensor, the AEC, BEC, and CEC are all projectively equivalent.

This is far from obvious:

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