Curves and Surfaces from 3-D Matrices

Dan DreibelbisUniversity 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

Vector Multiplication 1

Vector Multiplication 2

Vector Multiplication 3

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.

AEC

Det = 0

AEC

Curving Space

Quadratic Warp

Quadratic Warp

Quadratic Warp

Quadratic Map

This is a tensor multiplication with two vectors!!

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.

Tangents from AEC

F(x, y)

AEC maps to the tangent lines of the curvature ellipse.

Veronese Surface

F(x, y, z)

Veronese Surface

F(x, y, z)

Veronese Surface

F(x, y, z)

Drawing the AEC

Cubic Curves

Normalizing the Curve

Two AEC are equivalent if there is a change ofcoordinates 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:

AEC=BEC=CEC

4-D Case

4-D AEC, Page 1

4-D AEC, Page 33

AEC

More AEC’s

Thanks!