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

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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

Richard


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


3 d matrices

3-D Matrices


Vector multiplication 1

Vector Multiplication 1


Vector multiplication 2

Vector Multiplication 2


Vector multiplication 3

Vector Multiplication 3


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.


Curves and surfaces from 3 d matrices

AEC

Det

= 0


Curves and surfaces from 3 d matrices

AEC


Curving space

Curving Space


Quadratic warp

Quadratic Warp


Quadratic warp1

Quadratic Warp


Quadratic warp2

Quadratic Warp


Quadratic map

Quadratic Map

This is a tensor multiplication with two vectors!!


The curvature ellipse

The Curvature Ellipse


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.


Veronese surface

Veronese Surface

F(x, y, z)


Veronese surface1

Veronese Surface

F(x, y, z)


Veronese surface2

Veronese Surface

F(x, y, z)


Drawing the aec

Drawing the AEC


Cubic curves

Cubic Curves


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:


Aec bec cec2

AEC=BEC=CEC


4 d case

4-D Case


4 d aec page 1

4-D AEC, Page 1


4 d aec page 33

4-D AEC, Page 33


Curves and surfaces from 3 d matrices

AEC


More aec s

More AEC’s


Thanks

Thanks!


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