Filling triangular holes by convex combination of surfaces

A surface generation method is presented based on convex combination of surfaces with rational weight functions. The three constituents and the resulting surface are defined over the same triangular domain. The constructed surface matches each component along one of its boundary curves with C0 or C1 continuity depending on the weight functions in the combination. The method can be applied in surface modelling for filling triangular holes

Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

A Generalized Blending Scheme for Arbitrary Order of Continuity

In this thesis, new templates and formulas of blending functions, schemes, and algorithms are derived for solving the scattered data interpolation problem. The resulting data fitting scheme interpolates the positions and derivatives of a triangular mesh, and for each triangle of the mesh blends three triangular sub-surfaces, and creates a triangular patch. Similar to some existing schemes, the resulting surface inherits the derivatives of the sub-surfaces on the boundaries. In contrast with existing schemes, the new scheme has additional properties: The order of interpolated derivatives is extended to arbitrary values, and the restrictions of the sub-surfaces are relaxed. Then based on the properties of the new blending functions, an algorithm for constructing smooth triangular surfaces with global geometric continuity is described. The new blending functions and the scheme are then extended to multi-sided faces. The algorithm using these new blending functions accepts data sites formed by multi-sided polygons

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Designing aesthetically pleasing freeform surfaces in a computer environment

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Architecture, February 2001.Includes bibliographical references (p. 151-160).Statement: If computational tools are to be employed in the aesthetic design of freeform surfaces, these tools must better reflect the ways in which creative designers conceive of and develop such shapes. In this thesis, I studied the design of aesthetically constrained freeform surfaces in architecture and industrial design, formulated a requirements list for a computational system that would aid in the creative design of such surfaces, and implemented a subset of the tools that would comprise such a system. This work documents the clay modeling process at BMW AG., Munich. The study of that process has led to a list of tools that would make freeform surface modeling possible in a computer environment. And finally, three tools from this system specification have been developed into a proof-of-concept system. Two of these tools are sweep modification tools and the third allows a user to modify a surface by sketching a shading pattern desired for the surface. The proof-of-concept tools were necessary in order to test the validity of the tools being presented and they have been used to create a number of example objects. The underlying surface representation is a variational expression which is minimized using the finite element method over an irregular triangulated mesh.by Evan P. Smyth.Ph.D

ShapeWright--finite element based free-form shape design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1990.Includes bibliographical references (p. 179-192).by George Celniker.Ph.D