A minimum-energy quadratic curve through three points and corresponding cubic Hermite spline

Abstract: We demonstrate a method for exact determination of the quadratic curve of minimal energy and minimal curvature variation through three non-colinear points in the plane, including methods to determine the tangent vector and curvature at any point along the curve and an exact expression for the arc length of the curve between the first and last points. We then extended this to a novel method of selecting tangent vectors for use in constructing Hermite splines to smoothly interpolate ordered sets of control points. Our results are spline curves of lower energy than that of many popular spline implementations in most cases, which a series of examples demonstrate.