curves and surfaces from 3 d matrices
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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

Curves and Surfaces from 3-D Matrices

Dan Dreibelbis

University of North Florida

goals
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
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.
slide9
AEC

Det

= 0

quadratic map
Quadratic Map

This is a tensor multiplication with two vectors!!

tangents from aec
Tangents from AEC

F(x, y)

AEC maps to the tangent lines of the curvature ellipse.

tangents from aec1
Tangents from AEC

F(x, y)

AEC maps to the tangent lines of the curvature ellipse.

tangents from aec2
Tangents from AEC

F(x, y)

AEC maps to the tangent lines of the curvature ellipse.

normalizing the curve
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
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 cec1
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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