Thomas Hermann

On the derivatives of second and third degree rational Bézier curves

In this note, an estimate of Floater on the derivatives of rational Bezier curves is improved when the degree of the curve is 2 or 3.

G 2 interpolation of free form curve networks by biquintic Gregory patches

Abstract The problem of interpolating a free form curve network with irregular topology is investigated in order to create a curvature continuous surface. The spanning curve segments are parametric quintic polynomials, the interpolating surface elements are biquintic Gregory patches. A necessary compatibility condition is formulated and proved which need to be satisfied at each node of the curve network. Constraints are derived for assuring G 2 continuity between biquintic Gregory patches, which share a common side or a common corner point. The above conditions still leave certain geometric freedom for defining the entire G 2 surface, so following some analysis a particular construction is presented, by which after computing the principle curvatures at each node the free parameters are locally set for each interpolating Gregory patch.

Geometric criteria on the higher order smoothness of composite surfaces

Abstract A generalization of a theorem by Pegna and Wolter—called Linkage Curve Theorem—is presented. The new theorem provides a condition for joining two surfaces with high order geometric continuity of arbitrary degree n. It will be shown that the Linkage Curve Theorem can be generalized even for the case when the common boundary curve is only G1.

Curve networks compatible with G2 surfacing

Prescribing a network of curves to be interpolated by a surface model is a standard approach in geometric design. Where n curves meet, even when they afford a common normal direction, they need to satisfy an algebraic condition, called the vertex enclosure constraint, to allow for an interpolating piecewise polynomial C^1 surface. Here we prove the existence of an additional, more subtle constraint that governs the admissibility of curve networks for G^2 interpolation. Additionally, analogous to the first-order case but using the Monge representation of surfaces, we give a sufficient geometric, G^2 Euler condition on the curve network. When satisfied, this condition guarantees existence of an interpolating surface.

A geometric criterion for smooth interpolation of curve networks

A key problem when interpolating a network of curves occurs at vertices: an algebraic condition called the vertex enclosure constraint must hold wherever an even number of curves meet. This paper recasts the constraint in terms of the local geometry of the curve network. This allows formulating a new geometric constraint, related to Eulers Theorem on local curvature, that implies the vertex enclosure constraint and is equivalent to it where four curve segments meet without forming an X.

A geometric constraint on curve networks suitable for smooth interpolation

A key problem when interpolating a network of curves occurs at vertices: an algebraic condition, called the vertex enclosure constraint, must hold wherever an even number of curves meet. This paper recasts the vertex enclosure constraint in terms of the local geometry of the curve network. This allows formulating a new geometric constraint, related to Eulers Theorem on local curvature. The geometric constraint implies the vertex enclosure constraint and is equivalent to it where four curve segments meet without forming an X. Also the limiting case of collinear curve tangents is analyzed.

Constraints on curve networks suitable for G 2 interpolation

When interpolating a network of curves to create a C1 surface from smooth patches, the network has to satisfy an algebraic condition, called the vertex enclosure constraint. We show the existence of an additional constraint that governs the admissibility of curve networks for G2 interpolation by smooth patches.

On the smoothness of offset surfaces

It is proved that the geometric continuity of the offset of a composite base surface is the same as that of the base surface.

A new insight into the G n continuity of polynomial surfaces

Abstract A theorem of Degen describes the structure of G1 and G2 connection between two polynomial surfaces. This is based on common vector polynomials and an irreducibility condition of one of these vector functions related to the common boundary. In this paper Degens results are reformulated, a new proof and a new interpretation is given which makes it possible to generalize the previous results to Gn continuity. Surprisingly, in case of Gn continuity, there is no need to add further conditions than for the G 1 G 2 cases, still exists a common Gn continuous polynomial virtual patch with polynomial reparametrizations.

Degree elevation for generalized Poisson functions

In this note a common generalization is given for the Bezier and Poisson functions. Further it is proved that the degree elevated control polygon converges to the function.