Projective Geometry: Paradoxes, Polarities,...

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What is Projective Geometry? Projectivities Duality Projective Geometry is All Geometry Projective Geometry: Paradoxes, Polarities, Pictures Charles Gunn 1 1 Institute Of Mathematics Technical University Berlin Blossin Summer School June 24, 2010 Projective Geometry
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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

    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

    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:

    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

    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

    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

    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.

    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

    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

    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

    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

    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

    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

    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

    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

    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

    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