Projective Coordinates for Points and Lines | Algebraic Calculus One | Wild Egg

Dr Anna Tomskova explains a more modern framework for projective geometry where the extra coordinate often associated with infinity is the first coordinate in a projective vector. This gives us a uniform way to associate to affine points and lines projective points and lines, with the advantage that now higher dimensional vectors become extensions, not rewrites of lower dimensional vectors, and equations of lines, and eventually also curves, respect the natural increasing order of degree.

The Join of Points and Meet of Lines formulas, which are the workhorses for planar analytic geometry, are pleasantly symmetrical, and the dual role of points and lines should be clear. This is an important computational tool for the Algebraic Calculus, as it generally allows us to clear denominators more efficiently, meaning that rational number arithmetic is reduced to integer arithmetic --which we generally all prefer!

Also this gives us a simple algebraic way of talking about points at infinity and the line at infinity.

This video is a reworking of an earlier video in the series.

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