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

TL;DR

This paper presents an exact method for constructing minimal-energy quadratic curves through three points and extends it to create energy-efficient cubic Hermite splines for smooth interpolation of control points.

Contribution

It introduces a novel approach for determining minimal-energy quadratic curves and applies it to develop Hermite splines with lower energy than existing methods.

Findings

Spline curves have lower energy than popular implementations

Exact formulas for minimal-energy quadratic curves are derived

New tangent vector selection method improves spline smoothness

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.