Math
105

Tuesday, May 2

Tuesday, May 2

Review from last class:

"Seeing
the Infinite"

How to see the Infinite using "Projective geometry".

We will try to understand

- why we see the horizon on a plane....

- why the horizon is a line composed of ideal/infinite points.

- why parallel lines "meet" on the horizon.

Center of projection/ perspective.

[compare with similarity and geometric series].

Correspondences:

Two Line segments of different length:

Different lengths- 1:1 correspondence of points!

Semicircle to line and line to semicircle!

Two points at "infinity"

Circle to line and line to Circle!

One point at "infinity"

How to see the Infinite on a line using Projective geometry.

Line to Circle to Line

Use Rope and lines on floor with an image plane to see parallels meet on an "image" plane ---

where is the horizon?

Perspective between planes in Space!

Source plane ( Flatland) ----> Image plane (Canvass)

Seeing the horizon on a plane.... a line of ideal/infinite points.

How parallel lines "meet" on the horizon.

Drawing with perspective:

one point- two point (some java examples from UIUC)

da Vinci: Last Supper.

Veronese: The Marriage at Cana

RAPHAEL: School of Athens

Hogarth: Perspective absurdities

Durer: Unterweisung der Messung 1525

from Albrecht Durer’s “Treatise on measurement with compasses and straightedge

(

Using coordinates to draw in perspective.

Activity: Creating coordinates in the affine plane.

Question: What is on the other side of the horizon?

The affine plane -

The Horizon is a special line- not a part of the plane.

The projective plane - The horizon is just another line- not special!

Activity: Using coordinates to transfer a circle to the affine plane as an ellipse.

Equations for conics in coordinate geometry.

Activity: Using coordinates to draw in perspective.

What do the conics look like in a perspective drawing or in the projective plane?

The Geometry of Perspective Drawing on the Computer

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:

We define a perspective relation:

Two points

Another aspect of Projection: Desargues' Theorem in 3-space
and the plane.

**Some final remarks on the projective plane
and visualizing Flatland and Space.:
**

**What is on the other side of the horizon?****The projective plane as a surface.**

**Visualizing the projective plane on a disc with the boundary identified as the horizon.****The projective plane as a non-euclidean flatland.**

The End!