Other marching direction of third order

The main motivation of this work is the problem of compute intersection curve between two surface. The surface/surface intersection, is a fundamental problem in computational geometry and geometric modelling of complex shapes. In general surface intersections, the most commonly used methods include subdivision and marching. Marching-based algorithms begin by finding a starting point on the intersection curve, and proceed to march along the curve. Local differential geometric properties are applied to determine the marching directions and steps. The marching direction is used for estimating the next tracing point. Because of the tradeoff between efficiency and accuracy, that point is usually not on the intersection curve. Newton iterations are required to improve the accuracy of reached points at each step. It is clear that the near lies the point to the curve, the less iterations are necessary to improved its coordinates in relation to the exact intersection curve. The marching directions can be: tangent [2], circle [5], parabola [3], helix and (locally canonical) polynomial form of a curve of third order [4]. When the parametric form of a curve is known its local properties, such as tangent, curvature, normal, binormal, curvature, and torsion, can be derived exactly. However, in the case of marching schemes, these properties are used for determining the unknown curve. Motivated by the applications of differential properties in determination of marching directions [1, 6] proposed algorithms based on Differential Geometry to compute the local properties of the intersection curve as long as the intersection points are obtained. In this work we present an alternative way for marching directions of the third order: marching along osculating sphere. According to our experiments, our spherical marching algorithm is efficient and robust. We could observed that this direction is equivalent in efficiency with the others of third order, however among the three marching of third order, the polynomial is lightly superior in the sense of needing less Newton iterations.