Posts tagged 'conics'

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).