Euler B-spline for Curve Interpolation and Data Fitting

Abstract

Euler spirals are a kind of fair curves that have linear curvatures in terms of arc length. However, Euler spirals cannot be represented by polynomials exactly. In this paper, we propose an algorithm for approximating a segment of Euler spiral which interpolates two specified points and two tangents at the points by a B-spline curve. The approximating B-spline curve is obtained as the solution to a differential equation which satisfies the boundary conditions. Similar to the exact interpolating Euler spiral, the interpolating B-spline curve is also fair and has approximate linear curvature. Furthermore, an algorithm for fitting data points by a sequence of smoothly connected fair B-spline curves is also developed.