Tatiana Surazhsky

Fast exact and approximate geodesics on meshes

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact single source, all destination algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.

A comparison of Gaussian and mean curvatures estimation methods on triangular meshes

Estimating intrinsic geometric properties of a surface from a polygonal mesh obtained from range data is an important stage of numerous algorithms in computer and robot vision, computer graphics, geometric modeling, industrial and biomedical engineering. This work considers different computational schemes for local estimation of intrinsic curvature geometric properties. Five different algorithms and their modifications were tested on triangular meshes that represent tessellations of synthetic geometric models. The results were compared with the analytically computed values of the Gaussian and mean curvatures of the non-uniform rational B-spline (NURBs) surfaces, these meshes originated from. This work manifests the best algorithms suited for that indeed different algorithms should be employed to compute the Gaussian and mean curvatures.

Parameterization for Curve Interpolation

Abstract A common task in geometric modelling is to interpolate a sequence of points or derivatives, sampled from a curve, with a parametric polynomial or spline curve. To do this we must first choose parameter values corresponding to the interpolation points. The important issue of how this choice affects the accuracy of the approximation is the focus of this paper. The underlying principle is that full approximation order can be achieved if the parameterization is an accurate enough approximation to arc length. The theory is illustrated with numerical examples.

Blending polygonal shapes with different topologies

Abstract In this paper, we propose a new method for morphing between two polygonal, possibly non-simply connected, shapes in the plane. The method is based on reconstructing an xy -monotone surface whose extreme cross-sections coincide with the given shapes. The surface generated by our algorithm does not contain any self-intersections, does not change the topologies of the input slices, does not contain any horizontal triangles, and guarantees that all the topology changes occur at a mid-height which is a degenerate form of both input topologies. All these properties are highly desirable for blending shapes of different topologies.

METAMORPHOSIS OF PLANAR PARAMETRIC CURVES VIA CURVATURE INTERPOLATION

This work considers the problem of metamorphosis interpolation between two freeform planar curves. Given two planar parametric curves, the curvature signature of the two curves is linearly blended, yielding a gradual change that is not only smooth but also employs intrinsic curvature shape properties, and hence is highly appealing. In order to be able to employ this curvature blending, we present a constructive scheme to derive curvature signatures of parameter curves. Additionally, we propose a scheme to reconstruct a curve from its curvature signature.

Matching free-form surfaces

Abstract In this work, the matching problem between two free-form parametric surfaces, S 0 (u, v) and S 1 (u, v) , has been considered in the context of a first stage of the metamorphosis process, which is defined as gradual and continuous transformation of one key shape into another. A method is presented to approximate the two reparametrization functions, r(u, v) and t(u, v), via a discrete sampled set of N2 grid samples on the surfaces. The N2 samples on the source surface are matched against the N2 samples on the target surface, employing a resemblance metric between their Gauss fields as a criterion. The method preserves the connectivity of the two key surfaces as well. Thus, the resulting reparameterization functions, interpolating the discrete solution must be diffeomorphism. The algorithm works with simple open surfaces, i.e. the surfaces that are homeomorphic to a disk.

Artistic Surface Rendering Using Layout of Text

An artistic rendering method of free‐form surfaces with the aid of half‐toned text that is laid‐out on the given surface is presented. The layout of the text is computed using symbolic composition of the free‐form parametric surface S(u, v) with cubic or linear Bézier curve segments C(t) = {cu (t), cv (t)}, comprising the outline of the text symbols. Once the layout is constructed on the surface, a shading process is applied to the text, affecting the width of the symbols as well as their color, according to some shader function. The shader function depends on the surface orientation and the view direction as well as the color and the direction or position of the light source.

Arbitrary precise orientation specification for layout of text

A novel method for the layout of text strings over some given free-form parametric base curves is considered. Each letter of the string is represented by a collection of cubic and linear Bezier curves. The layout of the string over the free-form parametric curve is derived as a symbolic composition of the string geometry (i.e. a sequence of Bezier curves) and a free-form parametric surface S(u, v) with the parameters u, v between zero and one, and S(u, 0) given by the base curve. This method has proven to provide great flexibility and give high quality results in layout of text.