Polynomial surfaces interpolating arbitrary triangulations

Abstract

Triangular Bézier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bézier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occur with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: the tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.

Detail preserving deformation of B-spline surfaces with volume constraint

“…It is well known that G 1 continuity for surfaces of arbitrary topology is a quite difficult problem. In fact, at a patch corner with more than four patches, the so-called "vertex compatibility problem" has to be solved [Pet93,HB03]. In a classical tensor-product setting, where four patches meet at a corner, the G 1 continuity can be expressed by linear equations with respect to the control points.…”

Smooth adaptive fitting of 3D models using hierarchical triangular splines

“…All computational details of that scheme can be found in [15]. ¿From a coarse mesh M C a smooth piecewise triangular spline is constructed interpolating the mesh vertices V C .…”

“…It starts by computing an initial fitting spline surface over the coarse mesh using [15]. A piecewise triangular G 1 continuous polynomial surface is thus constructed that interpolates the vertices of the coarse mesh.…”

“…We use instead the previous work of Hahmann and Bonneau [15] for fitting the initial smooth surface S 0 . This method interpolates the coarse mesh M C with G 1 continuity.…”

“…Since several free parameters (see Table 1) are available, their optimal values for fitting the samples are found such that the sum of the squared distances of S 0 to the target points p i are minimized. Since the boundary curves of the macro patches of S 0 depend non-linearly on some of the free parameters (see [15], p. 102) a global minimization would lead to a timeconsuming non-linear optimization. Instead we split the optimization into two linear problems which are successively solved.…”