# Projective Geometry: Paradoxes, Polarities,...

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### Transcript of Projective Geometry: Paradoxes, Polarities,...

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Perspective Painting

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Perspective Painting

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Perspective Painting

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Perspective Painting

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

What is perspective?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

What is perspective?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

What is perspective?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

What is perspective?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Central Projection

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

World plane

Image Center

plane

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Central Projection

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

World plane

Image Center

plane

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Central Projection

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

World plane

Image Center

plane

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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!

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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!

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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!

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Central Projection

Demo of central projectionDemo of perspective gridObservations about central projection:

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Central Projection

Demo of central projectionDemo of perspective gridObservations about central projection:

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Ideal Points

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

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Ideal Points

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

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Ideal Points

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

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Ideal Points

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Ideal Points

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Ideal Points

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Masaccio Revisited

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Drawing consequences

What changes do the ideal elements bring into geometry?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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!

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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!

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Separation

Separation Lines and planes are harder to cut into pieces.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Separation

Separation Lines and planes are harder to cut into pieces.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Orientation

Non-orientable Cant distinguish consistently between left andright.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Orientation

Non-orientable Cant distinguish consistently between left andright.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Measuring distance

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Measuring distance

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Measuring distance

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Different invariants

Although distance cant be defined, projective geometry hasinvariants too.

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

Projective The cross ratio of four points P, Q, R and S isinvariant: |PQ| |RS||PR| |QS| .

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Different invariants

Although distance cant be defined, projective geometry hasinvariants too.

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

Projective The cross ratio of four points P, Q, R and S isinvariant: |PQ| |RS||PR| |QS| .

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Different invariants

Although distance cant be defined, projective geometry hasinvariants too.

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

Projective The cross ratio of four points P, Q, R and S isinvariant: |PQ| |RS||PR| |QS| .

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Different invariants

Although distance cant be defined, projective geometry hasinvariants too.

Projective The cross ratio of four points P, Q, R and S isinvariant: |PQ| |RS||PR| |QS| .

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Paradoxes: A Summary

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

Example

What sort of things can one do in projective geometry?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

The Historical SettingThe Solution

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Chaining together perspectivities

P

l

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Chaining together perspectivities

world plane

image plane

P

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Chaining together perspectivities

Pl

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Chaining together perspectivities

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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].

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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].

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Pascals 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.

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Pascals 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.

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Pascals 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.

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Duality Warm-up: Cube and Octahedron

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Duality Warm-up: Cube and Octahedron

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Duality Warm-up: Cube and Octahedron

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Exercises

Dualize:

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Exercises

Dualize:

Four faces meet in a vertex of the octahedron.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Exercises

Dualize:

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Exercises

Dualize:

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Exercises

Dualize:

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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. (???)

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Archimedian solids and their duals

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

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).

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons 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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons 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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons 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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Example: Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Brianchons Theorem

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The discovery of elliptic and hyperbolic geometry

18th century: questioning the foundations of Euclideangeometry.Attempts to prove Euclids 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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The discovery of elliptic and hyperbolic geometry

18th century: questioning the foundations of Euclideangeometry.Attempts to prove Euclids 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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The discovery of elliptic and hyperbolic geometry

18th century: questioning the foundations of Euclideangeometry.Attempts to prove Euclids 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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The discovery of elliptic and hyperbolic geometry

18th century: questioning the foundations of Euclideangeometry.Attempts to prove Euclids 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.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The Sphere

A sphere is almost a non-euclidean geometry.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The Sphere

A sphere is almost a non-euclidean geometry.

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The Sphere

A sphere is almost a non-euclidean geometry.

Distance is the central angle formed by two points.

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The Sphere

A sphere is almost a non-euclidean geometry.

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The Sphere

A sphere is almost a non-euclidean geometry.

There are no parallel lines...

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

The Sphere

A sphere is almost a non-euclidean geometry.

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

Mission Accomplished

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

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

hyperbolic > 180 LBGeuclidean 1 180 Euclid

elliptic 0 < 180 Riemann

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

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

Pictures

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

Pictures

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

Pictures

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

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

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

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

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

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

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

Existence of projective models

G MHyperbolic D2 (unit disk)

Elliptic P2

Euclidean P2 \ P1

Projective Geometry

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

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

3D non-euclidean tessellations

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

Hardware acceleration on modern GPUs: projective geometryis all geometry!

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

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?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

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?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

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?

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

Closing Thoughts

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

Closing Thoughts

Projective Geometry

What is Projective Geometry?Projectivities

DualityProjective Geometry is All Geometry

Non-euclidean metric geometryProjective models

References

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

Projective Geometry

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

What is Projective Geometry?The Historical SettingThe Solution

ProjectivitiesDualityProjective Geometry is All GeometryNon-euclidean metric geometryProjective models