curves and surfaces from 3-d matrices dan dreibelbis university of north florida
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Curves and Surfaces from 3-D Matrices
Dan DreibelbisUniversity 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 Warp
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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 ofcoordinates 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!