Projective Geometry: Paradoxes, Polarities, Picturespage.math.tu-berlin.de/~gunn/Documents/Slides/projGeometry2010.pdf · What is Projective Geometry? Projectivities Duality Projective

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Projective Geometry: Paradoxes, Polarities,Pictures

Charles Gunn1

1Institute Of MathematicsTechnical University Berlin

Blossin Summer SchoolJune 24, 2010

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The Historical SettingThe Solution

Perspective Painting

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What is perspective?

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The Mathematical Challenge

Question: In a perspective drawing, how are the points, linesand planes of the world projected onto the points and lines ofthe image plane?

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Central Projection

How is one world plane WP projected onto image plane IP?

World plane

Image Center

plane

Definition of central projection:From each line l through the center of projection P,find the intersection point of l with WP, andassociate it with the intersection point of l with IP.

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Central Projection


World plane

Image Center

plane


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Central Projection


World plane

Image Center

plane


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Central Projection


World plane

Image Center

plane


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Central Projection


World plane

Image Center

plane


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Central Projection


World plane

Image Center

plane


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Central Projection

Demo of central projectionDemo of perspective gridObservations about central projection:

Lines in WP are projected to lines in IP.The horizon line: where the plane through P parallel to WPmeets IP.Parallel world lines in the WP meet – in the IP – at thehorizon line.The role of IP and WP are switched in shadow-casting!

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http://www.math.tu-berlin.de/~gunn/webstart/jnlp/PerspectiveDemo.jnlp

http://www.math.tu-berlin.de/~gunn/webstart/jnlp/PerspectiveGrid.jnlp




Central Projection



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Central Projection



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Central Projection



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Central Projection



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Ideal Points

Question How to pull all these phenomena together ?Answer Extend ordinary geometry with ideal points.

Projective line = ordinary line + ideal point P∞Two parallel lines intersect in an ideal point.Every plane has an ideal line.Now all points of image plane have unique partner in worldplane.

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Ideal Points



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Ideal Points



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Ideal Points



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Ideal Points



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Ideal Points



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Masaccio Revisited

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Drawing consequences

What changes do the ideal elements bring into geometry?

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Partnership

P

l

Perfect partnerships : Intersection and join yield 1:1 maps.Warning: the ideal point to the left is the same as the ideal pointto the right!

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Partnership

P

l

Perfect partnerships : Intersection and join yield 1:1 maps.Warning: the ideal point to the left is the same as the ideal pointto the right!

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Separation

Separation Lines and planes are harder to cut into pieces.

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Separation

Separation Lines and planes are harder to cut into pieces.

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Orientation

Non-orientable Can’t distinguish consistently between left andright.

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Orientation

Non-orientable Can’t distinguish consistently between left andright.

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Measuring distance

Distance: Distance cannot be consistently defined.Coordinates: Ordinary (x , y) coordinates for the plane have tobe extended.

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Measuring distance


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Measuring distance


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Different invariants

Although distance can’t be defined, projective geometry hasinvariants too.

Euclidean The distance of two points P and Q is invariant:|P−Q|.Affine The distance ratio of three points P, Q and R isinvariant: |P−Q|

|P−R| .

Projective The cross ratio of four points P, Q, R and S isinvariant: |P−Q| |R−S|

|P−R| |Q−S| .

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|P−R| .


|P−R| |Q−S| .

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|P−R| .


|P−R| |Q−S| .

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|P−R| .


|P−R| |Q−S| .

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Paradoxes: A Summary

Perfect partnerships : Intersection and join yield 1:1maps.Separation Lines and planes are harder to cut into pieces.Non-orientable Can’t distinguish consistently between leftand right.Distance: Distance cannot be consistently defined.

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Example

What sort of things can one do in projective geometry?

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Desargues’ Theorem

A fundamental theorem of projective geometry, due to ReneDesargues (1591-1661):

Theorem (Desargues 1639)Two triangles are perspective in a point

⇐⇒they are perspective in a line

This dynamic geometry demo illustrates this theorem.

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http://www.math.tu-berlin.de/~gunn/Documents/zirkel/desargues.html




Desargues’ Theorem

A fundamental theorem of projective geometry, due to ReneDesargues (1591-1661):

Theorem (Desargues 1639)Two triangles are perspective in a point

⇐⇒they are perspective in a line

This dynamic geometry demo illustrates this theorem.

Projective Geometry

http://www.math.tu-berlin.de/~gunn/Documents/zirkel/desargues.html



Projectivities

The idea of central projection can be generalized:Basic idea of a projectivity: chain together centralprojections.This produces new partnerships between:

points and points,lines and lines, andpoints and lines.

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Chaining together perspectivities

P

l

Cut/Join: the natural partnership between line pencil P andpoint range l : P Z l .

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world plane

image plane

P

Perspectivity: a partnership between point ranges l and mwith center P: l Z P Z m, OR

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Pl

Perspectivity: the partnership between line pencils P and Qwith axis l : P Z l Z Q

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Projectivity: any concatenation of perspectivities:P Z l Z Q Z m Z R.

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Example: Projective Generation of Conics

What can you do with projectivities?

This dynamic geometry demo explores a projectivityP Z l Z Q Z m Z R.

Theorem (Desargues 1639)

The intersection points of corresponding lines of a projectivitybetween two line pencils lie on a conic [Kegelschnitt].

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http://www.math.tu-berlin.de/~gunn/Documents/zirkel/pointConic.html



Example: Projective Generation of Conics

What can you do with projectivities?

This dynamic geometry demo explores a projectivityP Z l Z Q Z m Z R.

Theorem (Desargues 1639)

The intersection points of corresponding lines of a projectivitybetween two line pencils lie on a conic [Kegelschnitt].

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http://www.math.tu-berlin.de/~gunn/Documents/zirkel/pointConic.html



Example: Pascal’s Theorem

Pascal discovered this theorem when he was 16 years old!

Theorem (Pascal 1640)The intersection points of opposite sides of a hexagoninscribed in a conic are collinear.

This dynamic geometry demo extends the previous demo.

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http://www.math.tu-berlin.de/~gunn/Documents/zirkel/pascaltheorem.html







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Duality Warm-up: Cube and Octahedron

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Dictionary of duality (in space)

term dual termpoint plane

line linelie on pass through

intersect joinmove along rotate around

... ...

Note: Duality is sometimes referred to as polarity.

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... ...


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... ...


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... ...


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... ...


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... ...


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... ...


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... ...


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Exercises

Dualize:

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Exercises

Dualize:

Four faces meet in a vertex of the octahedron.

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Exercises

Dualize:

Four faces meet in a vertex of the octahedron.Four vertices lie in a face of the cube.

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Exercises

Dualize:

A point moves along an edge of the octahedron from one vertexto the next.

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Exercises

Dualize:

A point moves along an edge of the octahedron from one vertexto the next.A plane rotates around an edge of the cube from one face tothe next.

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Exercises

Dualize:

The joining lines of opposite vertices of the octahedron meet inthe center point of the octahedron.

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Exercises

Dualize:

The joining lines of opposite vertices of the octahedron meet inthe center point of the octahedron.The intersecting lines of opposite faces of the cube lie in thecenter plane of the cube.

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Exercises

Dualize:

The joining lines of opposite vertices of the octahedron meet inthe center point of the octahedron.The intersecting lines of opposite faces of the cube lie in thecenter plane of the cube. (???)

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Archimedian solids and their duals

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Principle of Duality

DefinitionThe dual of a statement (configuration) in projective geometryis the statement (configuration) obtained by translating it bymeans of the dictionary of duality.

Theorem (Poncelet 1822)The dual of a true statement in projective geometry is also true.

Duality is a deep property of the fundamental axioms ofprojective geometry.Simultaneous with rebirth of projective geometry (Poncelet,1822).

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Example: Brianchon’s Theorem

A similar dictionary of duality exists for the projective plane.Point and line are dual in the plane.

Theorem (Pascal 1640)The intersection points of opposite sides of a hexagon[Sechseck] inscribed in a conic lie in a line.

Theorem (Brianchon ∼1830)The joining lines of opposite vertices of a hexahedron[Sechsseite] circumscribed around a conic pass through apoint.

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Brianchon’s Theorem

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Non-euclidean metric geometryProjective models

The discovery of elliptic and hyperbolic geometry

18th century: questioning the foundations of Euclideangeometry.Attempts to prove Euclid’s Axiom V:

Axiom (Euclid V)In a plane, through a given point there is exactly one lineparallel to a given line.

Possible alternatives that still allow measurement?The sphere: almost a non-euclidean geometry.

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The Sphere

A sphere is almost a non-euclidean geometry.

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The Sphere


We define the lines of this geometry to be the great circles.

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The Sphere


Distance is the central angle formed by two points.

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The Sphere


Sum of angles of a triangle is always greater than 180 degrees.

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The Sphere


There are no parallel lines...

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The Sphere


but every two lines intersect in two points, not one.

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Mission Accomplished

Two alternative, consistent geometries were discovered!All the other postulates remain valid.

The three classic metric geometriesName #|| Σ(angles) Discoverer

hyperbolic ∞ > 180◦ LBG∗

euclidean 1 180◦ Euclidelliptic 0 < 180◦ Riemann

*) LBG: Lobachevski, Bolyai, Gauss independently discovered.

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Pictures

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Pictures

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Pictures

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Existence of projective models

Theorem (Cayley-Klein ∼1860)There are projective models for hyperbolic, elliptic, andeuclidean geometry in all dimensions.

G MHyperbolic D2 (unit disk)

Elliptic P2

Euclidean P2 \ P1

Measuring distances involves the cross ratio.Dynamic demo of triangle tessellations in differentgeometries

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http://www.math.tu-berlin.de/~gunn/webstart/jnlp/TriangleGroup.jnlp







Elliptic P2

Euclidean P2 \ P1


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Elliptic P2

Euclidean P2 \ P1


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Elliptic P2

Euclidean P2 \ P1


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3D non-euclidean tessellations

Figure: Tessellations of elliptic and hyperbolic space by (different)regular dodecahedra

Hardware acceleration on modern GPU’s: projective geometryis all geometry!

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Closing Thoughts

600 years ago: discovery of perspective painting.400 years ago: Desargues invents projective geometry(conics).200 years ago: Poncelet invents projective geometry again(duality).Now: what is the next stage of development for projectivegeometry?

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Closing Thoughts


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Closing Thoughts


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Closing Thoughts


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Closing Thoughts


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References

For links to these slides and the live demos contained inthis talk:http://www.math.tu-berlin.de/ gunn

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http://www.math.tu-berlin.de/~gunn