6.2: Perspectivities

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Review the motivation for projective geometry and answer the following.

If you view a tree from 10 meters away and 100 meters away (viewing from the same side), what is the difference in the objects?
As you continue to move away from the tree, toward what does the mapping of the tree approach?
Look up the term 'vanishing point' in an art book. Explain its relationship to this.

Definition: Perspectivity

A transformation is a perspectivity if and only if it maps the points of a line to the points of another line such that all lines from points to their images are incident in a single point.

Draw two distinct lines. Choose a point not on either line. Choose three points on one of the lines and find the points on the second line to which they are mapped by the perspectivity defined by your chosen point.

Investigate the composition of perspectivities as follows.

Draw three distinct lines ℓ₁, ℓ₂, ℓ₃.
Choose a point A not on any of the lines.
Select three points P₁, Q₁, R₁ on ℓ₁ and map them to P₂, Q₂, R₂ ON ℓ₂ by the perspectivity defined by A.
Choose a point distinct from all other points.
Map P₂, Q₂, R₂ to P₃, Q₃, R₃ on ℓ₃ by the perspectivity defined by B.
Determine if P₁, Q₁, R₁are mapped to P₃, Q₃, R₃by any perspectivity.

Definition: Projectivity

A transformation is a Projectivity if and only if it can be written as a composition of perspectivities.

Definition: Triangle

A set of points is a triangle if and only if it is size three.

Definition: Trilateral

A set of lines is a trilateral if and only if it is size three.

Definition: Perspective from a point

Two triangles are perspective with respect to a point if and only if the lines connecting corresponding pairs of vertices are incident in a point.

Definition: Perspective from a line

Two trilaterals are perspective with respect to a line if and only if the corresponding sides are incident on a line.

Theorem: Desargues Theorem

Triangles perspective from a point are perspective from a line.

Theorem: Fundamental Theorem of Projective Geometry

A projectivity is uniquely defined by three points and their images.