Last class we looked at the cone as related to projection from a central light source and the curves determined by conic sections.

Finish watch video on conics.

The characterization of points on a conic by measurements are:

ellipse: AC + BC = constant

parabola: EC = EF

hyperbola: |CB-CA| = constant.

Rope Conic Activity.parabola: EC = EF

hyperbola: |CB-CA| = constant.

Using a rope form a human ellipse.

After that demonstrate how to use a rope to form a hyperbola and then a parabola.

String drawing conic activity.

ellipse: AC + BC = constant

parabola: EC = EF

hyperbola: |CB-CA| = constant.

Key background: Tangents to a circle are equal in length.

In right triangle,

This ratio is

If the angle is designated by the vertex A in a right triangle ABC where the right angle is at vertex C, then sin(A) = BC/AB.

Examples:

If A is a 45 degree angle, sin(A) = 1/ sqrt(2).

If A is a 30 degree angle, sin(A) = 1/2.

Show "Conics" video.

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.

Center of projection/ perspective.

[ compare with similarity].

Correspondences: Activity:Circle to Line:

How to see the Infinite using Projective geometry.

Circle to line and line to Circle!

Line to Line

Perspective and projective relations between points on lines.

Perpective between planes in Space!

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

How parallel lines "meet" on the horizon.

Equations for conics in coordinate geometry.

Using coordinates to draw in perspective.

Activity: Creating coordinates in the projective plane and using them to transfer a circle to the projective plane as an ellipse.

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

Using coordinates to draw in perspective.

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