Hexagrammum Mysticum 3 | Affine and projective geometry and a proof of Pappus' theorem | Wild Egg

Описание: We want to explore one of the most remarkable developments of 19th century geometry -- the Hexagrammum Mysticum arising from Pascal's theorem viewed more symmetrically. A good place to start is with the corresponding situation for the precursor to Pascal's theorem, namely Pappus' theorem. We will in fact uncover a wide range of new previously undiscovered phenomenon here, with plenty of scope for amateur investigations!

Here we develop some preliminary material for this. We look at some basic facts about affine and projective geometries in the 2D situation, for example the definitions of points and lines, and the usefulness of projective coordinates even in the affine case.

We discuss also symmetries of the affine plane. And then we give a simple proof of Pappus' theorem, using the analytic freedom that symmetries provide, allowing us to choose the two initial lines in a particularly simple and pleasant fashion.

The more complete playlist is available to Members at Hexagrammum Mysticum :    • Плейлист  

Video Contents:

0:00 – Introduction: Pappus' Theorem & Projective Geometry
4:25 – Incidence in Projective Geometry
7:23 – The Power of Projective Coordinates for Computation
12:36 – Duality: Join of Points and Meet of Lines
17:25 – Translations as Simple Parallelism-Preserving Transformations
23:09 – Linear Transformation to Align Lines with Coordinate Axes
26:25 – Checking Collinearity of Points c1, c2, c3 Using a 3x3 Determinant
27:17 – Efficient Computations Using Coordinate Transformations and Computers

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