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Curves and Surfaces from 3-D Matrices Dan Dreibelbis University of North Florida

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Richard

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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

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3-D Matrices

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Vector Multiplication 1

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Vector Multiplication 2

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Vector Multiplication 3

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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.

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AEC Det= 0

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AEC

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Curving Space

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Quadratic Warp

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Quadratic Map This is a tensor multiplication with two vectors!!

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The Curvature Ellipse

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Tangents from AEC F(x, y) AEC maps to the tangent lines of the curvature ellipse.

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Tangents from AEC F(x, y) AEC maps to the tangent lines of the curvature ellipse.

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Tangents from AEC F(x, y) AEC maps to the tangent lines of the curvature ellipse.

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Veronese Surface F(x, y, z)

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Veronese Surface F(x, y, z)

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Veronese Surface F(x, y, z)

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Drawing the AEC

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Cubic Curves

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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.

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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:

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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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AEC=BEC=CEC

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4-D Case

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4-D AEC, Page 1

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4-D AEC, Page 33

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AEC

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More AEC’s

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Thanks!

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