Piecewise Reconstruction of 3D Euler Spirals From Planar Polygonal Curves

Abstract

In this article, we propose a method for reconstructing approximate piecewise 3D Euler spirals from planar polygonal curves. The method computes the 3D coordinates of approximate Euler spiral such that its orthogonal projection onto the 2D plane is the closest possible to the input curve. To achieve this, a dataset is created, comprising Euler spiral segments and their orthogonal projections. Given an input curve, it is sampled and split into segments. Each segment is matched with the closest Euler spiral segments from the dataset, forming a pool of candidates. The optimal set of connected Euler spiral segments is then selected to reconstruct the approximate piecewise 3D Euler spiral. The selection prioritizes smoothness continuity at connecting points while minimizing the distance between the orthogonal projection and the input curve. We evaluate our method against synthetic 3D Euler spirals by applying our reconstruction to the orthogonal projection of the synthetic Euler spirals.