B-Spline Approximation Of Euler Spiral And Data Fitting

Curve and surface modeling are important research contents in Computer Graphics and Computer Aided Geometric Design. Particularly, high quality curves and surfaces are expected by designers, and curve fairing and surface fairing are important ways to generate high quality curves and surfaces. In particular, Euler Spiral is a kind of fair curve whose curvature is linear with respect to the arc length of the curve. However, Euler curve defined by the Fresnel integral is a transcendental curve, and should be approximated by other discrete or continuous curves for practical engineering applications.In this paper, we propose an algorithm for approximating Euler spiral by B-spline and another algorithm for data fitting by the approximate Euler spirals. By the definition of Euler spiral, we can see that Euler spiral is just the solution to a differential equation which satisfies the boundary conditions. We can obtain an approximation of the Euler spiral by using B-spline as the approximate solution to the differential equation. Similar to the exact Euler spiral, the approximate B-spline curve has approximate linear curvature and is fair, too. Therefore, the fair B-spline curve can be used as a characteristics spline curve for data fitting.In this paper, the concrete calculation formulae and their derivations are given for the two proposed algorithms. The algorithm for B-spline approximation of Euler Spiral can not only construct the approximation of Euler spirals but also can be used for fairing of existing B-spline curves. The given data points can be fitted by one piece of approximate Euler spiral or piecewise approximates spirals represented by B-spline. Several examples are presented to validate the effectiveness of the proposed algorithms.