Abstract We present a simple and effective method for the interactive segmentation of feature regions in a triangular mesh. From the user‐specified radius and click position, the candidate region that contains the desired feature region is defined as geodesic disc on a triangle mesh. A concavity‐aware harmonic field is then computed on the candidate region using the appropriate boundary constraints. An initial isoline is chosen by evaluating the uniformly sampled ones on the harmonic field based on the gradient magnitude. A set of feature points on the initial isoline is selected and the anisotropic geodesics passing through them are then determined as the final segmentation boundary, which is smooth and locally shortest. The experimental results show several segmentation results for various 3D models, revealing the effectiveness of the proposed method.

Polar parameterizations of star-shaped domains are based on the line segments that connect a suitably chosen center point with the points on the domain's boundary. Valid (i.e., regular everywhere except at the center point) polar parameterizations are obtained when choosing a center from the kernel of the domain. Recently, the flexibility of these polar parameterizations has been enhanced by considering so-called arc fibrations (Jüttler et al., 2019), which are polar parameterizations that use circular arcs in order to connect the center with the boundary points. We propose and analyze another generalization of polar parameterizations, which uses parabolic arcs instead of lines or circular arcs. This class of curves is simultaneously simpler (since admitting polynomial parameterizations) and more flexible.

Abstract We present a highly efficient algorithm for computing the minimum distance between two solids of revolution, each of which is defined by a planar cross‐section region and a rotation axis. The boundary profile curve for the cross‐section is first approximated by a bounding volume hierarchy (BVH) of fat arcs. By rotating the fat arcs around the axis, we generate the BVH of fat tori that bounds the surface of revolution. The minimum distance between two solids of revolution is then computed very efficiently using the distance between fat tori, which can be boiled down to the minimum distance computation for circles in the three‐dimensional space. Our circle‐based approach to the solids of revolution has distinctive features of geometric simplification. The main advantage is in the effectiveness of our approach in handling the complex cases where the minimum distance is obtained in non‐convex regions of the solids under consideration. Though we are dealing with a geometric problem for solids, the algorithm actually works in a computational style similar to that of handling planar curves. Compared with conventional BVH‐based methods, our algorithm demonstrates outperformance in computing speed, often 10–100 times faster. Moreover, the minimum distance can be computed very efficiently for the solids of revolution under deformation, where the dynamic reconstruction of fat arcs dominates the overall computation time and takes a few milliseconds.

We present a compact modeling of human lung anatomy (with lung lobes, bronchial trees, and pulmonary blood vessels), and demonstrate a plausible deformation (with collision detection and avoidance) for the whole anatomical structure. For this purpose, we employ a hybrid hierarchy of various bounding volumes (generated by moving spheres). Separation lists and parallel processing are also used for the acceleration of collision detection and avoidance, all implemented in CPUs only. Experimental results show that our modeling scheme compresses conventional lung models (often commercially available as meshes) by around 70 times lighter, and the resulting human lung deformation supports an interactive-speed performance, with frame rate of 10--20 fps.

Abstract We present an interactive tree modeling and deformation system that supports an efficient collision detection and avoidance using a bounding volume hierarchy of sweep surfaces. Starting with conventional tree models (given as meshes), we convert them into sweep surfaces and deform their branches interactively while detecting and avoiding collisions with many other branches. Multiple tree models (sharing the same topology) can be generated with great ease using this sweep‐based approach, and they can serve as a basis for the generation of a multiparameter family of trees. We demonstrate the effectiveness of our approach in an automatic generation of similar trees, the colonization of trees to form a forest, and the tree growth, aging, and withering simulations. Copyright © 2015 John Wiley & Sons, Ltd.

ABSTRACT We present a real‐time algorithm for realistically deforming 3D human blood vessels, while automatically detecting and avoiding interference among a large number of blood vessels under deformation. Sweep surfaces are employed for this purpose. Using a dynamic bounding volume hierarchy, specially designed for sweep surfaces, we support collision detection and other related geometric computations in real time. Copyright © 2013 John Wiley & Sons, Ltd.

We present a compact representation for the bounding volume hierarchy (BVH) of freeform NURBS surfaces using Coons patches. Following the Coons construction, each subpatch can be bounded very efficiently using the bilinear surface determined by the four corners. The BVH of freeform surfaces is represented as a hierarchy of Coons patch approximation until the difference is reduced to within a given error bound. Each leaf node contains a single Coons patch, where a detailed BVH for the patch can be represented very compactly using two lists (containing curve approximation errors) of length proportional only to the height of the BVH. We demonstrate the effectiveness of our compact BVH representation using several experimental results from real-time applications in collision detection and minimum distance computation for freeform models.

Abstract We present a layered geometric approach to the skinning of character animation. On top of a global shape deformation, B‐spline surface patches are attached to various body parts for local control. The patches are directly manipulated so as to generate example local shapes at some important poses. During character animation, the B‐spline control points move continuously by kinematically interpolating a set of example patches, and at the same time they can also move dynamically by elastic simulation. The movement of control points changes the shape of a local patch, and consequently skin vertices move according to the patch deformation, finally generating the desired local deformation. We demonstrate the effectiveness of our approach by generating natural local details at various poses including muscular effects and elastic deformation. Moreover, dealing with only a small number of control points, the proposed method is very efficient, while generating hundreds to thousands frames per second. Copyright © 2009 John Wiley & Sons, Ltd.

We present an accurate and efficient algorithm for continuous collision detection between two moving ellipsoids under rational Euclidean or affine motion. We start with a highly optimized implementation of interference testing between two stationary ellipsoids based on an algebraic condition described in terms of the signs of roots of the characteristic equation of two ellipsoids. Then we derive a time-dependent characteristic equation for two moving ellipsoids, which enables us to develop an efficient algorithm for computing the time intervals in which two moving ellipsoids collide. The effectiveness of our approach is demonstrated with several practical examples.

Abstract We propose a variational approach to computing an optimal segmentation of a 3D shape for computing a union of tight bounding volumes. Based on an affine invariant measure of e‐tightness, the resemblance to ellipsoid, a novel functional is formulated that governs an optimization process to obtain a partition with multiple components. Refinement of segmentation is driven by application‐specific error measures, so that the final bounding volume meets pre‐specified user requirement. We present examples to demonstrate the effectiveness of our method and show that it works well for computing ellipsoidal bounding volumes as well as oriented bounding boxes.

Abstract We introduce a sweep‐based hand shape adaptation algorithm to fit a generic sweep‐based hand model to the shape of an individual's hand, presented as a single photograph. The sweep trajectory curves of the generic hand model are modified to interpolate a sequence of keyframes determined by target features. Details of the real hand can be transferred to the model by adjusting its sweep displacement map. Palm lines are also acquired from sketches drawn on the photograph. The bespoke model inherits the fully animatable structure of the generic model. We demonstrate the effectiveness of our sweep‐based approach using several examples of reconstructing animatable bespoke hand models. Copyright © 2007 John Wiley & Sons, Ltd.

Abstract We propose a sweep‐based approach to the freeform deformation of three‐dimensional objects. Instead of using a volume enclosing the whole object, we approximate only its deformable parts using sweep surfaces. The vertices on the object boundary are bound to the sweep surfaces and follow their deformation. Several sweep surfaces can be organized into a hierarchy so that they interact with each other in a controlled manner. Thus we can support intuitively plausible shape deformation of objects of arbitrary topology with multiple control handles. A sweep‐based approach also provides important advantages such as volume preservation. We demonstrate the effectiveness of our technique in several examples. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computational Geometry and Object Modeling]: Curve, surface, solid, and object representations

Abstract We present a new approach to realistic hand modeling and deformation with real‐time performance. We model the underlying shape of a human hand by means of sweeps which follow a simplified skeleton. The resulting swept surfaces are blended, and an auxiliary surface is then bound to the swept representation in the palm region. In the areas of this palm‐control surface where bulges occur in certain poses of a real hand, the vertices are given their own trajectories, so that the palm forms realistic shapes as the joints bend. Palm lines can also be modeled as valleys in the skin by sketching them on a displacement map on the palm‐control surface, and activitating them when appropriate joint movements take place. Self‐intersections and collisions are detected using geometric primitives that are automatically generated from, and deform with, the sweeps and palm surface. Our algorithm runs in real time, and the naturalism of its results are demonstrated by comparative images of modeled and real hands, including several challenging poses. Copyright © 2006 John Wiley & Sons, Ltd.

Collision detection and avoidance are important in robotics. Compared with commonly used circular disks, elliptic disks provide a more compact shape representation for robots or other vehicles confined to move in the plane. Furthermore, elliptic disks allow a simpler analytic representation than rectangular boxes, which makes it easier to perform continuous collision detection (CCD). We shall present a fast and accurate method for CCD between two moving elliptic disks, which avoids any need to sample the time domain of the motion, thus avoiding the possibility of missing collisions between time samples. Based on some new algebraic conditions on the separation of two ellipses, we reduce collision detection for two moving ellipses to the problem of detecting real roots of a univariate equation, which is the discriminant of the characteristic polynomial of the two ellipses. Several techniques are investigated for robust and accurate processing of this univariate equation for two classes of commonly used motions: planar cycloidal motions and planar rational motions. Experimental results demonstrate the efficiency, accuracy, and robustness of our method.

We present an algorithm for generating the Voronoi cells for a set of rational C1-continuous planar closed curves, which is precise up to machine precision. Initially, bisectors for pairs of curves, (C(t), Ci(r)), are generated symbolically and represented as implicit forms in the tr-parameter space. Then, the bisectors are properly trimmed after being split into monotone pieces. The trimming procedure uses the orientation of the original curves as well as their curvature fields, resulting in a set of trimmed-bisector segments represented as implicit curves in a parameter space. A lower-envelope algorithm is then used in the parameter space of the curve whose Voronoi cell is sought. The lower envelope represents the exact boundary of the Voronoi cell.

We propose an algorithm for contouring k-manifolds (k = 1,2) embedded in an arbitrary n-dimensional space. We assume (n -k) geometric constraints are represented as polynomial equations in n variables. The common zero-set of these (n-k) equations is computed as an 1-or 2-manifold, respectively, for k = 1 or k = 2. In the case of 1-manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitly-defined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2-manifolds, the algorithm is similar to techniques for contouring iso-surfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 -2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zero-set as a piecewise linear 1or 2-manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.

We introduce a robustness measure which allows to analyze implicitly defined curves and surfaces with respect to their stability. It can be used to bound the maximal position error, which is introduced by small perturbations of the coefficients of a curve or surface. It is shown that the robustness of an implicitly defined curve or surface can be enhanced by multiplying it with auxiliary factors.

We present efficient and robust algorithms for intersecting a freeform surface with a ringed surface or a ruled surface. A ringed surface is given as a one-parameter family of circles. By computing the intersection between a freeform surface and each circle in the family, we can solve the intersection problem. We propose two approaches which are closely related to each other. The first approach detects certain critical points; and the intersection curve is constructed by connecting them in a correct topology. The second approach converts the intersection problem to that of finding the zero set of two polynomial equations in the parameter space. The intersection between a freeform surface and a ruled surface can be computed in a similar way.

In this paper, we describe an exact method for detecting collision between two moving ellipsoids under pre-specified rational motions. Our method is based on an algebraic condition that determines the separation status of two static ellipsoids - the condition itself is described by the signs of roots of the characteristic equation of the two ellipsoids. To deal with moving ellipsoids, we derive a bivariate function whose zero-set possesses a special topological structure. By analyzing the zero-set of this function, we are able to tell whether or not two moving ellipsoids under pre-specified rational motions are collision-free; and if not, we can determine the intervals in which they overlap.

We present a new approach to the modeling and deformation of a human or virtual character's arm and legs. Each limb is represented as a set of ellipsoids of varying size interpolated along a skeleton curve. A base surface is generated by approximating these ellipsoids with a swept ellipse, and the difference between that and the detailed shape of the arm or leg is represented as a displacement map. We demonstrate that the natural bending of arms and legs can be emulated using this approach, and show its effectiveness by articulating the limbs of a scanned human body and those of a virtual character.

We present an algorithm for computing Minkowski sums among surfaces of revolution and surfaces of linear extrusion, generated by slope-monotone closed curves. The special structure of these simple surfaces allows the process of normal matching between two surfaces to be expressed as an explicit equation. Based on this insight, we also present an efficient algorithm for computing the distance between two simple surfaces, even though they may in general be non-convex. Using an experimental implementation, the distance between two surfaces of revolution was computed in less than 0.5 msec on average.

We present an algorithm for computing the distance between two free-form surfaces. Using line geometry, the distance computation is reformulated as a simple instance of a surface-surface intersection problem, which leads to low-dimensional root finding in a system of equations. This approach produces an efficient algorithm for computing the distance between two ellipsoids, where the problem is reduced to finding a specific solution in a system of two equations in two variables. Similar algorithms can be designed for computing the distance between an ellipsoid and a simple surface (such as cylinder cone, or torus). In an experimental implementation (on a 500 MHz Windows PC), the distance between two ellipsoids was computed in less than 0.3 msec on average; and the distance between an ellipsoid and a simple convex surface was computed in less than 0.15 msec on average.

The paper proposes a simple approach to the affine motion interpolation problem, where an affine spline motion is generated that interpolates a given sequence of affine keyframes and approximately satisfies rigidity constraints and certain optimization criteria. An affine spline motion is first generated so as to interpolate the given keyframes; after that, it is progressively refined via knot insertion and degree elevation into an optimal affine motion by an iterative optimization procedure.

We present a simple method for constructing a physical 3D trackball that can input 3D rotation about an arbitrary axis. This input device is based on multiple sensors that can detect the tangential velocities at certain points on the boundary sphere of the trackball. A mathematical analysis is given for the optimal locations of multiple sensors. A prototype hardware device has been built for the 3D trackball. We demonstrate the effectiveness of this input device in 3D rotation by comparing its performance with the Magellan/SPACE MOUSE.

The bisector of two rational surfaces in R/sup 3/ is, in general, nonrational; and so is the bisector of a rational curve and a rational surface. Thus, bisector surfaces in these two cases must be approximated numerically. Unfortunately, they are algebraic surfaces of very high degree and numerical approximation is non-trivial. This paper suggests a new computational model for constructing curve-surface and surface-surface bisectors in R/sup 3/. The curve-surface bisector problem is reformulated as the search for a trivariate zero-set; and the surface-surface bisector problem is reduced to that of finding the common zero-set of two four-variate functions.

Presents an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep /spl cup//sub u/C/sup u/ generated by a moving circle. Given two ringed surfaces /spl cup//sub u/C/sub 1//sup u/ and /spl cup//sub v/C/sub 2//sup v/, we formulate the condition C/sub 1//sup u//spl cap/C/sub 2//sup v//spl ne/O (i.e. that the intersection of the two circles C/sub 1//sup u/ and C/sub 2//sup v/ is non-empty) as a bivariate equation /spl lambda/(u,v)= 0 of relatively low degree. Except for some redundant solutions and degenerate cases, there is a rational map from each solution of /spl lambda/(u,v)=0 to the intersection point C/sub 1//sup u//spl cap/C/sub 2//sup v/. Thus, it is trivial to construct the intersection curve once we have computed the zero-set of /spl lambda/(u,v)=0. We also analyze some exceptional cases and consider how to construct the corresponding intersection curves.

The paper reviews new approaches to computing the intersection curve of two surfaces of special types. We consider algorithms for intersecting a torus with a natural quadric or another torus. After that, we review a topological technique that computes the intersection curve of a sphere and a surface of revolution. Finally, we consider the intersection of two ruled surfaces.

Given a point and a rational curve in the plane, their bisector curve is rational [Farouki and Johnston 1994a]. However, in general, the bisector of two rational curves in the plane is not rational [Farouki and Johnstone 1994b]. Given a point and a rational space curve, this art icle shows that the bisector surface is a rational ruled surface. Moreover, given two rational space curves, we show that the bisector surface is rational (except for the degenerate case in which the two curves are coplanar).

Article The bisector surface of freeform rational space curves Share on Authors: Gershon Elber Department of Computer Science Technion, Israel Institute of Technology, Haifa 32000, Israel Department of Computer Science Technion, Israel Institute of Technology, Haifa 32000, IsraelView Profile , Myung-Soo Kim Department of Computer Science, POSTECH, Pohang 790-784, South Korea Department of Computer Science, POSTECH, Pohang 790-784, South KoreaView Profile Authors Info & Claims SCG '97: Proceedings of the thirteenth annual symposium on Computational geometryAugust 1997 Pages 473–474https://doi.org/10.1145/262839.263091Online:01 August 1997Publication History 3citation171DownloadsMetricsTotal Citations3Total Downloads171Last 12 Months1Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Pacific Graphics '95, the third international conference on Computer Graphics and its Applications, was held on 21-24 August 1995, in Seoul, Korea.Following on the success of the two previous conferences: Pacific Graphics '93 (Seoul, Korea) and Pacific Graphics '94 (Beijing, China), we received 62 submissions from 22 different countries all over the world and, after peer review, selected 24 papers for publication in the conference proceedings: Computer Graphics and Applications, published by World-Scientific, Singapore.(The proceedings also include 10 invited papers.)The four papers presented in this special issue were judged by the programme committee members to be the best of the 24 selected papers presented at the conference.Each of the four papers was further reviewed by experts in the respective research area and the papers were further improved on the basis of comments received.The first paper, by Kim and Elber, presents a symbolic approach to generate blending surfaces between two polynomial and/or rational surfaces.Using a purely symbolic technique, the authors demonstrate how to compute various cross boundary tangent vector fields on each rail curve and represent the blending surfaces as polynomial or rational surfaces.The next paper, by Sun, Wang and Chin, presents a 3D morphing algorithm which transforms one polyhedral model into another.Considering each polyhedron as a graph representing vertex adjacencies, the authors develop a simple and elegant method which generalizes the 2D polygonal morphing algorithms of Sederberg to a 3D polyhedral morphing algorithm.The third paper, by Lee and Shin, presents an efficient ray casting algorithm for terrain rendering from DTMs (digital terrain models).The authors introduce a formal analysis to vertical ray coherence.Based on this analysis, they propose a new technique which covers the entire image plane with non-parallel ray lines.This

This paper presents a compact differential formula for the first derivative of a unit quaternion curve defined on SO(3) or S3. The formula provides a convenient way to compute the angular velocity of a rotating 3D solid. We demonstrate the effectiveness of this formula by deriving the differential properties of various unit quaternion curves1–5 at the curve end points.

Construction methods are presented that generate Hermite interpolation quaternion curves on SO(3). Two circular curves C1(t) and C2(t), 0 ≤ t ≤ 1, are generated that interpolate two orientations q1 and q2, and have boundary angular velocities: C1′(0) = ω1 and C2′(1) = ω2, respectively. They are smoothly blended together on SO(3) to generate a Hermite quaternion curve Q(t) ∈ SO(3), 0 ≤ t ≤ 1, which satisfies the boundary conditions: Q(0) = q1, Q(1) = 2, Q′(0) = ω1, and Q′ (1) = ω2.

An algorithm is presented that constructs a C/sup 2/-continuous B-spline quaternion curve which interpolates a given sequence of unit quaternions on the rotation group SO(3). The de Casteljau type construction method of B-spline curves can be extended to generate B-spline quaternion curves; however, the B-spline quaternion curves do not have C/sup 2/-continuity in SO(3). The authors recently suggested a new construction method that can extend a B-spline curve to a similar one in SO(3) while preserving the C/sup k/-continuity of the B-spline curve. We adapt this method for the construction of a B-spline quaternion interpolation curve. Thus, the problem essentially reduces to the problem of finding the control points for the B-spline interpolation curve. However, due to the non-linearity of the associated constraint equations, it is non-trivial to compute the B-spline control points. We provide an efficient iterative refinement solution which can approximate the control points very precisely.>

This paper proposesa new class of unit quaternion curves in SO(3). A general method is developed that transforms a curve in R 3 (defined as a weighted sum of basis functions) into its unit quaternion analogue in SO(3). Applying the method to well-known spline curves (such as Bezier, Hermite, and B-spline curves), we are able to construct various unit quaternion curves which share many important differential properties with their original curves. Many of our naive common beliefs in geometry break down even in the simple non-Euclidean space S 3 or SO(3). For example, the de Casteljau type construction of cubic B-spline quaternion curves does not preserve C 2 -continuity [10]. Through the use of decomposition into simple primitive quaternion curves, our quaternion curves preserve most of the algebraic and differential properties of the original spline curves. CR Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling -- Curve, surface, solid, and object representation, -- Geometric algorithms. Keywords: Quaternion, rotation, orientation, SO(3), Bezier, Hermite, B-spline 1

Abstract A method is presented for the modelling and animation of generalized cylinders with variable radius offset space curves. The boundary surface of a generalized cylinder is constructed: either as a translational sweep of cross‐sectional curves along the skeleton curve, or as a rotational sweep of profile curves around the skeleton curve. The cross‐sectional curves are computed as the variable radius offset curves of a circle in the normal plane, and the profile curves are computed as the variable radius offset space curves of the skeleton curve. The offset curves are approximated by spline curves, and the boundary surface of a generalized cylinder is approximated by the tensor product surface patches of the offset spline curves.

Algebraic algorithms for collision-avoidance robot motion planning problems with planar geometric models are presented. By decomposing the collision-free space into horizontal vertex visibility cells and connecting these cells into a connectivity graph, the global topological structure of collision-free space is represented. Using the C-space obstacle boundaries and this connectivity graph, exact (nonheuristic) compliant and gross motion paths of planar curved objects moving with a fixed orientation amidst similar obstacles are generated. The gross motion planning algorithm is further, extended (using approximations) to the case of objects moving with both translations and rotations by using the conventional slicing method.>

How to compute and represent the Gaussian approximations of planar curved objects is described. Also considered are various applications of the Gaussian approximation to various primitive geometric operations on monotone curve segments. The exact solutions for these problems can be computed by solving simultaneous polynomial equations, however, this required an intensive computation time. Efficient heuristic approximation algorithms using simple binary subdivisions on the original geometric components are suggested. It is shown that simple data structures such as arrays and circular lists can be used to represent the Gaussian approximations of planar curved objects.>

An algebraic algorithm is presented for generating the purely rotational sweeping volumes of planar objects bounded by algebraic curves. The boundaries of purely rotational sweep volumes of planar objects are characterized. The sweep volume boundary is related to convolutions consisting of the planar object boundaries at its start and final angles and the circular sweep arcs of radial extreme points. Algebraic algorithms are given to detect radial extreme points and ineffective radial extreme points. Algorithms are also given for the internal representation of algebraic curves, that is, whether they are parametrically or implicitly defined. Redundancies that arise in the effective convolution are treated, and plane sweep methods by which to remove these redundancies and thereby obtain the boundary of the rotational sweep volumes are given.>

Algebraic algorithms are presented for generating the boundary of configuration space obstacles arising from the motion of a sphere among obstacles. The boundaries of the obstacles are given by patches of algebraic surfaces. Algorithms are given for both implicit and parametric surface patches. Both convex and nonconvex obstacles are considered. In the case of convex obstacles, the topology of convolution faces is the same as the adjacency graph of faces, edges, and vertices of the obstacle. Further, there are no redundancies in the convolution faces. Redundancies on the convolution can occur in the case of nonconvex obstacles. It is possible to detect these redundancies from the the intersections and self-intersections of convolution faces. Simple solids are also considered.>

We present algebraic algorithms to generate the boundary of configuration space obstacles arising from the translatory motion of objects amongst obstacles. In particular we consider obtaining compliant motion paths where a curved convex object with fixed orientation moves in continuous contact with the boundary of curved convex obstacles in three Dimensions. Both the boundaries of the objects and obstacles are given by patches of algebraic surfaces. We also give a method to obtain approximate geodesic paths on convex C-space obstacles with algebraic boundary surfaces.

We present algebraic algorithms to generate the boundary of planar configuration space obstacles arising from the translatory motion of objects amongst obstacles. Both the boundaries of the objects and obstacles are given by segments of algebraic curves.