Explore the fascinating world of affine geometry and its connection to Algebraic Calculus in this 37-minute video lecture. Delve into the unique properties of the parabola, characterized projectively as the only conic tangent to the line at infinity. Discover how affine geometry, focused on parallelism and linear algebra, differs from Euclidean geometry. Learn about the parabola's distinction as the sole quadratic de Casteljau Bezier curve and its relevance to Archimedes' parabolic area formula. Examine the implications of a geometry that allows parallel lines but not perpendicular ones. Investigate various properties of the curve, including parallelograms, natural questions, subdividing, and finding R0. Note a correction at 4:20 where the formula should read R0=P0+2tv0+t^2 a1 with a v0, not a v1.