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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

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 i