# Curves and Surfaces from 3-D Matrices - PowerPoint PPT Presentation

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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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## Curves and Surfaces from 3-D Matrices

Dan Dreibelbis

University of North Florida

### 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.

Det

= 0

### Curving Space

This is a tensor multiplication with two vectors!!

### Tangents from AEC

F(x, y)

AEC maps to the tangent lines of the curvature ellipse.

### Tangents from AEC

F(x, y)

AEC maps to the tangent lines of the curvature ellipse.

### Tangents from AEC

F(x, y)

AEC maps to the tangent lines of the curvature ellipse.

F(x, y, z)

F(x, y, z)

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: