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Staring at Infinity – The Disk Model of the Projective Plane
Jim Hatton September 2012
The Plan
Projective Plane– 40 minutes Operations Research - 5 minutes
Jim’s Fractured Career - 5 minutes
Jim Hatton September 2012
Looking for Infinity
Jim Hatton September 2012
Interactive Graph
Can’t Get There from Here
Cool BookA guide to Plane Algebraic Curves
by Keith Kendig
Jim Hatton September 2012
What do we Expect?
Jim Hatton September 2012
Intersecting Lines
One “Point” – A “Loop”
Map the Number Line onto [-1,1]
Jim Hatton September 2012
Map Number Line Take 1
“captured” point at infinity .
Gave up “normal” distance metric
Map the Number Line onto [-1,1]
Jim Hatton September 2012
Map Number Line Take 2
All “points” the same.
Identify all antipodal points
Map the Number Line onto [-1,1]
Jim Hatton September 2012
Map Number Line Take 3
But we want to be looking at only one point
Map the Number Line onto [-1,1]
Jim Hatton September 2012
Map Number Line Take 4
So far, so good
Extend to Mapping the plane to the disk
The 3D Version – Plane onto a Sphere
Jim Hatton September 2012
Plane on a Sphere
We can draw a plane curve on a sphere
Now extend by mapping the plane to the disk
The 3D Version – Plane onto a Sphere – Part II
Jim Hatton September 2012
Plane on a Sphere- Part II
We can draw a plane curve on a sphere onto a disk
The Disk Model of the Projective Plane
Jim Hatton September 2012
Antipodal Points are Identified!
The Disk Model of the Projective Plane
Jim Hatton September 2012
We just stared at infinity!
Now compare behavior at infinity.
Parabola
The Disk Model of the Projective Plane
Jim Hatton September 2012
Even degree Polynomials
The Disk Model of the Projective Plane
Jim Hatton September 2012
Odd degree Polynomials
The Disk Model of the Projective Plane
Jim Hatton September 2012
Inverse Function
The Disk Model of the Projective Plane
Jim Hatton September 2012
Rational Function I
The Disk Model of the Projective Plane
Jim Hatton September 2012
Rational Function II
Tools used for this Demonstration
Jim Hatton September 2012
Analytic Geometry
The End – Now OR
Jim Hatton September 2012
Operations ResearchSometimes called Industrial Engineering, Management Science, Quantitative management…
Some of the tools used are statistics, optimization, probability, queuing theory, game theory, graph theory, decision analysis, simulation, and computer applications. (Wikipedia)
Types of problems solved: project planning, factory layout, efficient reliable telecommunications networks, traffic flow, bus routes, computer chip layout, supply chains, robotics, freight transportation, scheduling, blending … (Wikipedia)
Jim Hatton September 2012
Classes Hatton took: • Linear Programming• Microeconomics• Linear Algebra• Game Theory• Algorithmic Processes• Network Flows• Macroeconomics• Mathematical Programming• Stochastic Processes• Elementary Statistical Inferences• Reliability Models• Inventory-Production Control• Queuing Theory• Mathematical Systems Theory• Dynamic Programming• Advanced Probability• Large Scale Systems• Decision Theory
• Combinatorial Analysis• Applied Probability• Modern Algebra,• Advanced Stochastic Processes• Combinatorial Optimization.
Jim Hatton September 2012
Now, Jim’s Fractured Career-
• Kirkwood High School - St. Louis, Missouri• Rice University - Houston, Texas• Stanford University - Palo Alto, California• U.S. Army - St. Louis, Missouri• Stanford University• Massachusetts; Mt Shasta, California• College of the Siskiyous - Weed, California• SOU
Jim Hatton September 2012
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