Axioms, Duality and Projections
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Video Lecture 6 of 16
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### Lecture Description

We discuss how projective geometry can be formalized in different ways, and then focus upon the ideas of perspective and projection. The interesting duality/polarity between points and lines also becomes apparent.
In particular, we approach the following issues:

The projective plane as an extension of the euclidean plane.

The projective plane, described by homogeneous coordinates.

Axioms for projective geometry
(here I refer to the document:

Projective Geometry, Finite and Infinite
Brendan Hassett
www.math.rice.edu/~hassett/RUSMP3.pdf
)

The definition of perspectives, with respect to points and lines.

The definition of a projection.

How to find a projection between any 3 co-linear points and any other 3 co-linear points.

How to find a projection between any 3 concurrent lines and any other 3 concurrent lines.

This gets us close to the point of being able to discuss the fundamental theorem of projective geometry.

### Course Description

Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.