Insights into Mathematics | Calculus and affine geometry of the magical parabola | Algebraic Calc and dCB curves 3 | Wild Egg @njwildberger | Uploaded 3 years ago | Updated 1 hour ago
Algebraic Calculus naturally lives in affine geometry, not Euclidean geometry. Affine geometry is the geometry of parallelism, or (almost the same thing) --- the geometry of pure linear algebra. The parabola is characterized projectively in this geometry as the unique conic which is tangent to the line at infinity, or in purely affine terms, which has a distinguished axis direction. Other metrical definitions of a parabola, relying on focus and directrix, will not be needed here. However the characterization of a parabola as the unique quadratic de Casteljau Bezier curve is key!

We introduce this under-rated geometry here, focusing on the implications of allowing parallel lines but not perpendicular ones. And we discuss several pleasant properties of the curve that are relevant to appreciate Archimedes' parabolic area formula, and its establishment through the Algebraic Calculus.

Correction: At 4:20, the formula should be R0=P0+2tv0+t^2 a1 with a v0, not a v1. (Thanks Me Too)