Projective Geometry: Paradoxes, Polarities, gunn/Documents/Slides/projGeometry2010.pdf ·...

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Transcript of Projective Geometry: Paradoxes, Polarities, gunn/Documents/Slides/projGeometry2010.pdf ·...

  • 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 i