M103 Notes 11-13

Thursday, November 13

Symmetry in conics activity.

Last class

Projective geometry: The study of properties of figures that are related by projections... perspectivities and projectivities.

Examples: The projection of a line is a line. The point of intersection of two lines will project to the point of intersection of the projected lines.

Desargues' Theorem in Space:

Build Desargues' Configuration with straws and pipecleaners.

Desargues' Theorem in the plane can be justified using the result in space.
"Pull the planar configuration into space, then project the Desargues' line from space back to the plane. "

Duality again!
Duality in graphical configurations in the plane : points and lines.

The triangle and n-gons.

[Use Wingeom]

3 points - 3 lines.

N points - N lines.

Other popular configurations:

The complete quadrangle and quadrangle.

The Complete Quadrangle:
4 points {A,B,C,D} determine 6 lines {AB,AC,AD, BC, BD, CD}
and three additional points {X,Y,Z}.

The Complete Quadilateral:
4 lines {AB, BC, CD,AD}determine 6 points {A,B, C, D, X,Y}
with three additional lines{AC, BD, XY} .

The 7-7 Configuration.

3 points on each "line", 3 "lines" through each point.

Desargues 10-10 Configuration:

Planar Configuration

3 points on each line	3 lines through each point
Triangles in Perspective from a point	Triangles in Perpective from a line

Desargue's Spatial Configuration

Pascal Configuration and [the dual] Brianchon Configuration.

Pascal: 9 points - 7 lines
A Point Conic- The conic is described by a set of points

Brianchon: 9 lines - 7 points
A Line Conic-The conic is described by a set of lines

Pascal's Theorem - Brianchon's Theorem.
Show video on projective generation of conics.
This shows how to construct a conic using Pascal's Theorem and it converse.

Comment about the Principle of Duality in Projective geometry:
If you can justify a result of the form ...."line"...."point"....
then the same result will be true if you switch the words line and point appropriately. ...."point"...."line"....

This principle works because the key properties off the projective geometry satisfy this quality. E.g. "Two points determine a unique line" corresponds to "Two lines determine a unique point" which is also true in projective geometry, but not in Euclidean (flatland) geometry.