Kai Hormann

Surface Parameterization: a Tutorial and Survey

This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another.

Mesh parameterization: theory and practice

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

PolyCube-Maps

Standard texture mapping of real-world meshes suffers from the presence of seams that need to be introduced in order to avoid excessive distortions and to make the topology of the mesh compatible to the one of the texture domain. In contrast, cube maps provide a mechanism that could be used for seamless texture mapping with low distortion, but only if the object roughly resembles a cube. We extend this concept to arbitrary meshes by using as texture domain the surface of a polycube whose shape is similar to that of the given mesh. Our approach leads to a seamless texture mapping method that is simple enough to be implemented in currently available graphics hardware.

Mean value coordinates for arbitrary planar polygons

Barycentric coordinates for triangles are commonly used in computer graphics, geometric modeling, and other computational sciences because they provide a convenient way to linearly interpolate the data that is given at the corners of a triangle. The concept of barycentric coordinates can also be extended in several ways to convex polygons with more than three vertices, but most of these constructions break down when used in the nonconvex setting. Mean value coordinates offer a choice that is not limited to convex configurations, and we show that they are in fact well-defined for arbitrary planar polygons without self-intersections. Besides their many other important properties, these coordinate functions are smooth and allow an efficient and robust implementation. They are particularly useful for interpolating data that is given at the vertices of the polygons and we present several examples of their application to common problems in computer graphics and geometric modeling.

A general construction of barycentric coordinates over convex polygons

Barycentric coordinates are unique for triangles, but there are many possible generalizations to convex polygons. In this paper we derive sharp upper and lower bounds on all barycentric coordinates over convex polygons and use them to show that all such coordinates have the same continuous extension to the boundary. We then present a general approach for constructing such coordinates and use it to show that the Wachspress, mean value, and discrete harmonic coordinates all belong to a unifying one-parameter family of smooth three-point coordinates. We show that the only members of this family that are positive, and therefore barycentric, are the Wachspress and mean value ones. However, our general approach allows us to construct several sets of smooth five-point coordinates, which are positive and therefore barycentric.

The point in polygon problem for arbitrary polygons

A detailed discussion of the point in polygon problem for arbitrary polygons is given. Two concepts for solving this problem are known in literature: the even–odd ruleand the winding number, the former leading to ray-crossing, the latter to angle summationalgorithms. First we show by mathematical means that both concepts are very closely related, thereby developing a first version of an algorithm for determining the winding number. Then we examine how to accelerate this algorithm and how to handle special cases. Furthermore we compare these algorithms with those found in literature and discuss the results.  2001 Elsevier Science B.V. All rights reserved.

Efficient clipping of arbitrary polygons

Clipping 2D polygons is one of the basic routines in computer graphics. In rendering complex 3D images it has to be done several thousand times. Efficient algorithms are therefore very important. We present such an efficient algorithm for clipping arbitrary 2D-polygons. The algorithm can handle arbitrary closed polygons, specifically where the clip and subject polygons may self-intersect. The algoirthm is simple and faster that Vattis (1992) algorithm, which was designed for the general case as well. Simple modifications allow determination of union and set-theoretic differences of two arbitrary polygons.

Mesh parameterization: theory and practice Video files associated with this course are available from the citation page

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

Barycentric rational interpolation with no poles and high rates of approximation

It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval, regardless of the distribution of the points. These interpolants depend linearly on the data and include a construction of Berrut as a special case.

Maximum entropy coordinates for arbitrary polytopes

Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangles vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper we introduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. The coordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving a convex optimization problem with Newtons method, and experimental evidence indicates that they are smooth inside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra and higher‐dimensional polytopes.