Posts in category 'Conics & Analytic Loci'

F.1 Albert’s Parabola

Given a parabola y = x^2 and a point P = (P_x, P_y) on it, the tangent line T and normal line N at P satisfy y = 2 P_x x - P_x^2 and y = -x/(2 P_x) + P_x^2 + 1/2; requiring N to pass through (M, 0) yields the depressed cubic P_x^3 + (1/2) P_x - M/2 = 0.

F.17 Ellipse

Given semi-major axis a and semi-minor axis b, a four-center circular-arc oval approximating the ellipse x^2/a^2 + y^2/b^2 = 1 is constructed by a 7-step five-circle compass procedure, while the exact conic satisfies F = a*e, d = a/e, and vertex mean proportional V = a = sqrt(d*F).

F.22 Functions

The standard normal cumulative distribution Phi(x) and error function erf(x) satisfy Phi(x) = 1/2 + (1/2) erf(x/sqrt(2)), where erf(x) is approximated to 1.5e-7 accuracy by Abramowitz and Stegun’s 5-term rational-exponential formula.

F.37 Euler’s Inradius Parabola

For a fixed circumradius R, Euler’s relation d^2 = R^2 - 2 r R defines a downward-opening parabola r(d) = R/2 - d^2/(2R) = (R - d)(R + d)/(2R) with vertex (0, R/2) at the equilateral triangle and roots at the degenerate limits d = +/- R.