Staring at Infinity The Disk Model of the Projective Plane Jim Hatton September 2012.

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



Staying aliveJohn Travolta Rule

http://youtu.be/I_izvAbhExY

Looking for Infinity


Interactive Graph

Can’t Get There from Here

Cool BookA guide to Plane Algebraic Curves

by Keith Kendig


What do we Expect?


Intersecting Lines

One “Point” – A “Loop”

Map the Number Line onto [-1,1]


Map Number Line Take 1

“captured” point at infinity .

Gave up “normal” distance metric




All “points” the same.

Identify all antipodal points




But we want to be looking at only one point




So far, so good

Extend to Mapping the plane to the disk

The 3D Version – Plane onto a Sphere


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


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


Antipodal Points are Identified!



We just stared at infinity!

Now compare behavior at infinity.

Parabola



Even degree Polynomials



Odd degree Polynomials



Inverse Function



Rational Function I



Rational Function II

Tools used for this Demonstration


Analytic Geometry

The End – Now OR


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)


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.


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


Staring at Infinity The Disk Model of the Projective Plane Jim Hatton September 2012.

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