Curves on a polygonal mesh are quite useful for geometric modeling and processing such as mesh-cutting and segmentation. In this paper, an effective method for constructing C1 piecewise cubic curves on a triangular mesh M while interpolating the given mesh points is presented. The conventional Hermite interpolation method is extended such that the generated curve lies on M. For this, a geodesic vector is defined as a straightest geodesic with symmetric property on edge intersections and mesh vertices, and the related geodesic operations between points and vectors on M are defined. By combining cubic Hermite interpolation and newly devised geodesic operations, a geodesic Hermite spline curve is constructed on a triangular mesh. The method follows the basic steps of the conventional Hermite interpolation process, except that the operations between the points and vectors are replaced with the geodesic. The effectiveness of the method is demonstrated by designing several sophisticated curves on triangular meshes and applying them to various applications, such as mesh-cutting, segmentation, and simulation.

A shape detail transfer is the process of extracting the geometric details of a source region and transferring it onto a target region. In this paper, we present a simple and effective method, called GeoStamp, for transferring shape details using a Poisson equation. First, the mean curvature field on a source region is computed by using the Laplace–Beltrami operator and is defined as the shape details of the source region. Subsequently, the source and target regions are parameterized on a common 2D domain, and a mean curvature field on the target region is interpolated by the correspondence between two regions. Finally, we solve the Poisson equation using the interpolated mean curvature field and the Laplacian matrix of the target region. Consequently, the mean curvature field of the target region is replaced with that of the source region, which results in the transfer of shape details from the source region to the target region. We demonstrate the effectiveness of our technique by showing several examples and also show that our method is quite useful for adding shape details to a surface patch filling a hole in a triangular mesh.