Transcript MAA Talk - University of North Florida

Curves and Surfaces from 3-D
Matrices
Dan Dreibelbis
University of North Florida
Richard
Goals
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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 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:
AEC=BEC=CEC
4-D Case
4-D AEC, Page 1
4-D AEC, Page 33
AEC
More AEC’s
Thanks!