Erich Hartmann

G n-1 -functional splines for interpolation and approximation of curves, surfaces and solids

Implicit surfaces are used for interpolation, approximation, blending surfaces and solids, filling of surface holes and rounding solids. The introduced surfaces can be interpreted as functional splines, which fulfill geometric continuity conditions.

G2 interpolation and blending on surfaces

We introduce a method for curvature-continuous (G2) interpolation of an arbitrary sequence of points on a surface (implicit or parametric) with prescribed tangent and geodesic curvature at every point. The method can also be used forG2 blending of curves on surfaces. The interpolation/blending curve is the intersection curve of the given surface with a functional spline (implicit) surface. For the construction of blending curves, we derive the necessary formulas for the curvature of the surfaces. The intermediate results areG2 interpolation/blending methods in IR2.

Numerical implicitization for intersection and G n -continuous blending of surfaces

Abstract The idea of numerical implicitization allows to treat not only parametric surfaces but also a lot of surfaces of practical use which have per definition no standard (parametric or implicit) representations in a uniform way as implicit surfaces. So it suffices to have a surface/surface intersection algorithm for implicit surfaces. Further more the simple G n -blending techniques for implicit surfaces are applicable. The intersection of a space curve and a surface is just a specialization of a suitable surface/surface intersection problem. At least a simple method for the smooth approximation of a set of intersecting implicit surfaces is extended to more general surfaces.

Eine Klasse nicht einbettbarer reeller Laguerre-Ebenen

AbstractReplace in the parabolic model of the classical Laguerre-Plane the parabolas y=a(x−b)2+c, a≠0, by the curves y=af(x−b)+c with f(x)=

On the convexity of functional splines

x^{r_1 }

Beispiele Nicht Einbettbarer Reeller Minkowski-Ebenen

if x≥0, and f(x)=(−x)r2 if x<0. For each pair r1, r2>1 we obtain again a Laguerre-Plane ℒ(r1,r2).ℒ(r1,r2) can be embedded only if r1=r2=2.

Zykel-und Erzeugendenspiegelungen in Minkowski-Ebenen

Analogous to the wellknown results about the connection between linear transitive groups of collineations and the algebraic description of projective (or affine) planes, we give some statements for Laguerre-planes. In particular we use automorphisms with fixpoints, which induce in the residual plane of a fixpoint dilatations, translations, shears or reflections. We apply these methods in order to characterize certain ovoidal Laguerre-planes.

Über Moufang-Ovale

Abstract With help of results on pseudoconcave functions the convexity of large classes of functional splines is proved.