needle-like triangles, matrices, and lewis carroll alan edelman mathematics computer science &...
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Needle-like Triangles, Matrices, and Lewis Carroll
Alan EdelmanMathematics
Computer Science & AI Labs
Gilbert StrangMathematics
Computer Science & AI Laboratories
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A note passed during a lecture
Can you do this integral in R6 ? It will tell us the probability a random triangle is acute!
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What do triangles look like?
Popular triangles as measured by Google are all acute
Textbook “any old” triangles are always acute
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What is the probability that a random triangle is acute?
January 20, 1884
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Depends on your definition of random: One easy case!
Uniform (with respect to area) on the space(Angle 1)+(Angle 2)+(Angle 3)=180o
(0,180,0)
(0,0,180) (180,0,0)(90,0, 90)
(90,90,0)(0,90, 90) (45,90,45)
(45,45,90) (90,45,45)
(120,30,30)
Acute
Obtuse
ObtuseObtuse
Right Right
(60.60.60)
(30,120,30)
(30,30,120)
Right
Prob(Acute)=¼
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Random Triangles with coordinates from the Normal Distribution
A 10x10 Table of Random Triangles
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An interesting experiment
Compute side lengths normalized to a2+b2+c2=1Plot (a2,b2,c2) in the plane x+y+z=1
Black=Obtuse Blue=Acute Dot density largest near the perimeter
Dot density = uniform on hemisphere as it appears to the eye from above
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What is the z coordinate?Answer:Area *
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Kendall and others, “Shape Space”
Kendall “Father of modern probability theory in Britiain.
Explore statistically: historical sites are nearly colinear?
Shape Theory quotients out rotations and scalings
Kendall knew that triangle space with Gaussian measure was uniform on hemisphere
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Connection to Numerical Linear Algebra
The problem is equivalent to knowing the condition number distribution of a random 2x2 matrix of normals normalized to Frobenius norm 1.
Page 9Identify M with the triangle
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Connection to Shape Theory
svd(M):Latitude on the Hemisphere =Longitude on the Hemisphere = 2(rotation angle of Singular Vectors)
right^
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Area of a Triangle
s=(a+b+c)/2
a2+b2+c2=1
Heron of Alexandria
Marcus Baker139 Formulas
Annals of Math1884/1885
Kahan of Berkeley (Toronto really)
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a ≥b≥ c
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Conditioning
Condition(Area(a,b,c))=
Kahan: For acute triangles Condition(Area) ≤ 2
Condition(f(x)) = Condition()=2 Condition(Area(Square))=2
Perturbations = Scalings + ShapeChanges
Interpreting Kahan: For acute, ShapeChanges≤ScalingsPage 12
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Perturbation Theory in Shape Space
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Cube neighborhood projects onto a hexagon in shape space.
Some hexagons penetrate the perimeter=numerical violation of triangle inequality
Needle-like acuteTriangle have neighborhoodstangent to the latitude line
“head-on”view removes scalings
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Triangle Shape Points on the Hemisphere 2x2 Matrices Normalized through SVD
Conclusion
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A Northern Hemisphere Map: Points mapped to angles
Acute Territory
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HH11: Granlibakken
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Angle Density (A+B+C=180)
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100,000 triangles in 100 binstheory
Not Uniform!
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Please (in your mind) imagine a triangle
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Another case/same answer: normals! P(acute)=¼
3 vertices x 2 coordinates = 6 independent Standard Normals
Experiment: A=randn(2,3)
=triangle vertices
Not the same probability measure!
Open problem:give a satisfactory explanation of why both measures should give the same answer
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Shape Theory Conditioning vs Non Shape Theory for LargeAreas
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Tiny Area Triangles
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Condition
Longitude
Condition over a circle of latitude (Area=0.0024)
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Random Tetrahedra
Page 23(Generalization uses randn(m,n)*Helmert Matrix)
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Random “Gems”Convex Hulls (m=3, n=100)
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Construction of Triangle Shape
The three triangles with bases = parallelians through the a point on the sphere and its vertical projection are similar. They share the same height (in blue).
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An interesting experiment
Compute side lengths normalized to a2+b2+c2=1Plot (a2,b2,c2) when obtuse in the triangle x+y+z=1, x,y,z≥0.
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Uniform?
Distribution of radii:
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I remembered that the uniform distribution on the sphere means uniform Cartesian coordinates
This picture wants to be on a hemisphere looking down
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In Terms of Singular Values
A=(2x2 Orthogonal)(Diagonal)(Rotation(θ))
Longitude on hemisphere = 2θz-coordinate on hemisphere = determinant
Condition Number density (Edelman 89) =
Or the normalized determinant is uniform:
Also ellipticity statistic in multivariate statistics!Page 29
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Triangle can be calculated but also can be geometrically constructed using parallelians
Parallelians through P
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Question: For (n,m) what are the statistics for number of points in convex hull? Seems very small
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Opportunities to use latest technology of random matrix theory
• Zonal polynomials and hypergeometric functions of matrix argument
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Generalized Approach with Helmart Matrix (Kendall)
• What is a good way to construct the vertices of a regular simplex in n-dimensions?
• Answer: Matrix orthogonal to (1,1,…,1)/sqrt(n)
• Helmert Matrix:
• randn(m,n-1)∆n=n points in Rm
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