Posts tagged 'parabola'

W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas

A conic is the zero locus of a quadratic polynomial proportion C = <a:b:c:d:e:f>; metrically, Q(X, l) = R defines a ribbon, Q(X, F_0) = Q(X, l_dir) defines a parabola, the squared-sum/difference bifocal locus A(Q(X, F_1), Q(X, F_2), S) = 0 defines a quadrola (unifying ellipses and hyperbolas into one irreducible algebraic curve), and Q(X, l_1) +/- Q(X, l_2) = R defines a grammola.

W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity

A Galilean projectile launched from O with initial speed-squared v^2 and elevation spread s traces a parabola whose apex height h = v^2*s/(2*g), horizontal range quadrance Q_range = 16*(v^4/(4*g^2))*s*(1 - s) = (v^4/g^2)*S_2(s), and focus-directrix geometry depend rationally on s; Snell’s refraction law becomes n_1^2 * s_1 = n_2^2 * s_2 in spreads.