Generating Tool Paths for Free-Form Pocket Machining Using z-buffer-based Voronoi Diagrams

Size: px
Start display at page:

Download "Generating Tool Paths for Free-Form Pocket Machining Using z-buffer-based Voronoi Diagrams"

Transcription

1 Int J Adv Manuf Technol (1999) 15: Springer-Verlag London Limited Generating Tool Paths for Free-Form Pocket Machining Using z-buffer-based Voronoi Diagrams Jaehun Jeong and Kwangsoo Kim Department of Industrial Engineering, Pohang University of Science and Technology, South Korea Voronoi diagrams for closed shapes have many practical applications, ranging from numerical control machining to mesh generation. Curve offsetting based on Voronoi diagrams avoids the topological problems encountered in traditional offsetting algorithms. In this paper, we present a new procedure for generating tool paths using z-buffer-based Voronoi diagrams specially to deal with free-form shaped pockets. Using a z-buffer, a proposed algorithm effectively extracts the topological information on the Voronoi diagram, and generates the geometric information on the Voronoi edges approximately. These Voronoi edges are refined using a numerical algorithm. This method is independent of curve type and is applicable to any pockets with parametric curve boundaries that are twice differentiable. Keywords: Pocket machining; Voronoi diagram 1. Introduction One of the most important functions of CAD/CAM systems is generating tool paths for pocketing 2.-D parts on NC milling machines. A popular pocketing approach is to use successive offsets of the original boundary. These offsets are then chained together into a single spiralling tool path that follows the contours of the pocket. This approach is therefore called contour pocketing. The literature on contour pocketing algorithms can be roughly divided into two different approaches: 1. Voronoi diagrams. 2. Pairwise intersection [1]. In the first approach, the individual offset segments are trimmed to their intersections with the Voronoi diagram of the original boundary. The second approach for finding successive offsets uses pairwise intersection of individual offset segments to trim them into an offset boundary. The main problem of Correspondence and offprint requests to: Dr K. S. Kim, Pohang University of Science and Technology, Department of Industrial Engineering, IE2195, San 31, Hyoja-dong, Pohang , South Korea. kskim postech.ac.kr this approach is the need to eliminate all self-intersections after offsetting, leading to an O(n 2 ) calculation complexity where n denotes the number of boundary curve segments. Besides, the connection of these offset contours into one tool path, covering the entire pocket, remains a difficult problem [2]. Although many efficient procedures for pocketing have been published [1 12], most of them apply to a limited range of pocket shapes. Only a few publications are suitable for machining pockets bounded by free-form curves using Voronoi diagrams, but they do not give detailed descriptions of the pocketing algorithms. In this paper, we present a new procedure for generating tool paths using z-buffer-based Voronoi diagrams, specially to deal with free-form shaped pockets. This method is independent of curve type and is applicable to any pockets with parametric curve boundaries that are twice differentiable. The pocketing procedure consists of the following steps: 1. Segmenting each curve element in the pocket boundary into a set of curve segments. 2. Constructing the Voronoi diagram using a z-buffer. 3. Generating tool paths using the Voronoi diagram. The remainder of this paper will be presented in the following order. In Section 2, the segmentation procedure for pocket boundaries is presented. Section 3 presents the construction procedure of Voronoi diagrams. In Section 4, the procedure for generating tool paths based on Voronoi diagrams is described. The generated tool paths are verified through NC machining simulation in Section 5. The conclusion is discussed in Section Segmentation of Pocket Boundaries If the offset distance is larger than the radius of curvature at a point on a 2D curve element, topological problems such as loop and cusp occur around the point during offsetting. To avoid these topological problems, we divide each of the curve elements of the pocket boundaries into a set of curve segments that are monotonous in positive curvature. The segmentation procedure is summarised in the following steps:

2 Generating Tool Paths for Free-Form Pocket Machining Segmenting curve elements in the pocket boundaries at the points with positive maximum curvature. 2. Removing the non-effective points that do not cause the topological problems from the set of segmentation points. 2.1 Segmenting a Boundary Curve Element Because we are interested in only the portion of a Voronoi diagram that is internal to the pocket boundary C, we restrict our discussion to the interior of C. To distinguish the interior of C from the exterior, we assume that C is traversed in the counterclockwise direction so that the interior of C always lies to the left. A curve element in the pocket boundary C is divided into curve segments at the points where its curvature is positive and locally maximum. Thus, each curve segment is monotonously increasing or decreasing in its positive curvature. The extreme curvature points can be found by numerical methods. 2.2 Removing Non-Effective Segmentation Points At some of the segmentation points generated in the previous step, the segmentation operation is not necessary where the radius of curvature is larger than the minimum distance between the centre of curvature and the other curve segments, as shown in Fig. 1. If the centre of curvature at a point is closer to the other curve segments in the pocket boundary, geometrical degeneracy does not occur around this point during offsetting because the offset distance never exceeds the radius of curvature at this point. Thus, these segmentation points can be removed from the list of segmentation points. Figure 2(a) illustrates a pocket boundary that consists of Bezier curve segments and Fig. 2(b) shows the segmented pocket boundary with the effective segmentation points. 3. Construction of Voronoi Diagrams The Voronoi diagram of a pocket partitions the inside of the pocket into several regions. A region bounded by a Voronoi polygon contains the set of points that are closer to a curve segment than to the others in the pocket boundary. A Voronoi polygon is enclosed by the Voronoi edges, and each point on a Voronoi edge is equidistant from two or more boundary Fig. 2. A free-form pocket boundary and its segmentation. curve segments. In Fig. 3, the Voronoi diagram that consists of five Voronoi edges is represented. The Voronoi edges separate the pocket area into four Voronoi polygons. For example, the Voronoi polygon VP1 is enclosed by three Voronoi edges b 12, b 13, and b 14. In this paper, Voronoi diagrams are constructed using a new procedure based on z-buffer representation. The construction procedure is summarised in the following three steps: 1. First, each curve segment is assigned a unique colour. Then, a z-buffer is computed by rendering a circular cone with the colour assigned to each curve segment, while the cone apex is tracing along the curve segments. 2. The topological and geometric information needed for constructing Voronoi diagrams is extracted from the z-buffer representation. Using the extracted information, a planar graph is generated. 3. Then, the Voronoi diagram for the boundary curve segments is constructed by computing accurate Voronoi edge points using the geometric information extracted from the z- buffer representation. Fig. 1. Removing non-effective segmentation points. C 1, C 2, curve elements; C p, centre of curvature at P 0 ; D 1, curvature radius at P 0 ; D 2, distance between C p and C 2. Fig. 3. A simple Voronoi diagram.

3 184 J. Jeong and K. Kim Fig. 4. A z-buffer representation. (a) A 3D view. (b) A 2D view projected onto the xy-plane. Fig. 6. Constructing a planar graph. 3.1 Computing a z-buffer After assigning a unique colour to each curve segment, we compute a z-buffer by rendering a right circular cone with the colour assigned to each curve segment while moving its apex along the curve segments, as shown in Fig. 4. The cone axis is selected to be parallel to the z-axis. The z-buffer stores information on the z-value and colour of the visible cone. If the difference in z-value is smaller than the size of a pixel during the rendering process, both colour codes are stored in the z-buffer. In the 2D view of the z-buffer representation projected onto the xy-plane, a region with a common colour represents the Voronoi region of the curve segment associated with the colour. A sequence of pixels having two or more colours represents Voronoi edges. 3.2 Constructing a Planar Graph Fig. 5. Characteristic points. We first find two types of the characteristic points for constructing Voronoi diagrams: 1. Terminal points. 2. Branch points. A terminal point is the point on a Voronoi edge that is also on the boundary curve. The segmentation points, shown in

4 Generating Tool Paths for Free-Form Pocket Machining Generating a Voronoi Diagram Fig. 7. Computing bisector points. Fig. 8. A Voronoi diagram. Fig. 5(a), are the terminal points. A branch point is the intersection point of two Voronoi edges. These points are obtained by finding the pixels in the z-buffer that are associated with three or more colours, as shown in Fig. 5(b). Using these characteristic points, we can construct a planar graph that represents the complete topological structure of a Voronoi diagram as follows. First, every pair of two branch points that shares two common colours with each other is connected by a line as shown in Fig. 6(a). For example, the branch points, a and b, connected by a line in Fig. 6(a) have two common colours (3,5). Then, each pair of a terminal point and a branch point that share two common colours are connected by a line as shown in Fig. 6(b). Finally, the counterclockwise order of Voronoi edges connected to each branch point is determined by checking the tangent vector of each Voronoi edge. The topological information extracted in this step is used in offsetting the boundary curve segments for tool path generation. This completes the construction of the planar graph for a Voronoi diagram. We can complete the construction of a Voronoi diagram by computing the accurate Voronoi edges. The Voronoi edge or bisector of two free-form curve segments is not algebraically represented. Thus, we approximate the bisector with a sequence of discrete points. A point B i on a bisector of free-from curves satisfies the following conditions: 1. B i is equidistant to the two foot points on the curve segments, where the foot point is the closest point on the boundary curve segment to the given bisector point. 2. B i lies on the normal lines of the foot points. 3. B i is on the left of both foot points. 4. The distance from B i to a foot point is smaller than the radius of curvature. The first three conditions state that B i is the centre of the circle touching the boundary at the foot point. Each bisector point consists of 5 parameter values: 1. Two coordinate values (x,y). 2. Two parameter values (u,v) of two foot points on the two curve segments. 3. The distance d f from the bisector point to the foot points. This foot distance d f has an important role in offsetting pocket boundaries. Each bisector included in the Voronoi diagram has a monotonous foot-distance function. As shown in Fig. 7, the bisector points are traced out by stepping point by point. We use the pixels in the z-buffer representing the Voronoi edges as initial bisector points in computing accurate bisector points. For a foot-distance, we refine the initial bisector point using Algorithm 1. After increasing the foot-distance, we repeat this procedure to compute the next bisector points. Figure 8 shows the Voronoi diagram of the pocket shown in Fig. 2. Algorithm 1. Computing a bisector point, B i 0. The foot-distance d f and an initial bisector point B i are given. 1. For the bisector point B i, obtain two foot points F 1, F 2 on the curve segments. 2. Compute the distances d 1, d 2 between the bisector point B i and the foot points F 1, F 2, respectively. Fig. 9. Offset pocket boundary curves with and without Uoronoi edges.

5 186 J. Jeong and K. Kim connecting the offset profiles, starting from the innermost offset profile. The new procedure for offsetting a free-form curve segment is described below. The offset boundary curve segments of the pocket in Fig. 2 are shown in Fig. 9. Fig. 10. Offsetting a curve segment. o i, curve segments; b i, bisectors enclosing the curve segment o i ; F i, foot points; u i, parameter values at foot points; d, offset distance; B i, bisector points with foot-distance d. 3. Compute the differences e 1, e 2 between the given foot distance d f and the distances d 1, d 2, respectively. 4. If e 1 e 2, update the bisector point B i by B i B i +. (F 1 B i ) e 1 d 1 Otherwise, update the bisector point B i by B i B i +. (F 2 B i ) e 2 d 2 5. If the maximum of (e 1, e 2 ) is greater than the given error tolerance, repeat steps Generation of Tool Paths Once the Voronoi diagram of a pocket and an offset distance are given, the tool path can be constructed by following the procedure described below. First, each curve segment is offset within its associated Voronoi polygon to avoid the degeneracy problem during offsetting. Secondly, the offset curve segments are connected to form a closed offset profile by exploring every Voronoi polygon in the Voronoi diagram. Thirdly, these offsetting and connecting procedures are repeated to obtain the next offset profiles after increasing the offset distance by the specified amount. Fourthly, the tool path is constructed by Algorithm 2. Offsetting a curve segment 0. An offset distance d is given. 1. Among the bisectors enclosing the associated Voronoi polygon, find the bisectors of which the foot-distance intervals include the offset distance d. Since each bisector has its upper and lower limits of foot-distance, we find these bisectors by comparing the given offset distance with the limit values of each bisector foot-distance. (The bisectors b 1, b 3 are selected in Fig. 10.) 2. For each selected bisector, determine a point B i on the bisector that has the foot-distance d to the associated curve segment o i. 2.1 As an initial bisector point, select the point from the set of bisector points that has the nearest foot-distance value to the given offset distance. 2.2 Refine the selected bisector point B i using the modified version of the algorithm Find the parameter interval(s) [u i, u j ] in which the offset curve is defined, where u i, u j are the parameter values at the foot points F i, F j associated with B i, B j, respectively. (In Fig. 10, the interval is [u 1, u 2 ]). 4. Obtain the offset curve segment(s) by offsetting a set of points on the curve segment o i within the interval(s) [u i, u j ]. 5. Implementation The tool path generation procedure using z buffer-based Voronoi diagrams is implemented in C on a SUN workstation. Figure 11 shows two examples of tool paths generated for machining pockets with free-form curve boundaries. In order to check whether the generated tool path is covering the entire pocket with the chosen tool radius and overlap, a machining simulation is carried out. Figure 12 illustrates a few steps of the machining simulation for the pocket shown in Fig. 2. Fig. 11. Examples of tool paths for pocketing.

6 Generating Tool Paths for Free-Form Pocket Machining 187 a proposed procedure. Later, these tool paths are easily translated into the appropriate NC movement commands to machine the pocket on a milling machine. Acknowledgements The authors would like to thank Dr M. S. Kim of POSTECH for his valuable suggestions. This research is supported in part by the KOSEF. References 6. Conclusion Fig. 12. NC machining simulation. Tool path generation based on Voronoi diagrams avoids the topological problems encountered in traditional offsetting algorithms. In this paper, we have presented an efficient method for generating contour-parallel tool paths that uses a new z- buffer-based algorithm for constructing Voronoi diagrams. Using a z-buffer, this algorithm effectively extracts the topological information for the Voronoi diagram, and generates the geometric information for the Voronoi edges approximately. These Voronoi edges are refined in the following procedure using a numerical algorithm. Using the Voronoi diagram, the tool paths without the topological problems are computed by 1. K. Preiss, Automated mill pocketing computations, in Proceedings of the International Symposium on Advanced Geometric Modeling for Engineering Applications, Berlin, Germany, C. Lambregts, F. Delbrssine, W. de Vries and A. van der Wolf, An efficient automatic tool path generator for 2D free-form pockets, Computers in Industry, 29(3), pp , H. Persson, NC machining of arbitrarily shaped pockets, Computer-Aided Design, 10(3), pp , S. Chuang and W. Lin, Tool path generation for pockets with free-form curves using Bezier convex hulls, International Journal of Advanced Manufacturing Technology, 13, pp , M. Held, GEOPOCKET: A sophisticated computational geometry solution of geometrical and technical problems arising from pocket machining, F. Kimura and A. Rolstadas (eds.), In Computer Applications in Production and Engineering, Elsevier, P. Prabhu, A. Gramopadhye and H. Wang, A general mathematical model for optimizing NC tool path for face milling of flat convex polygon surfaces, International Journal of Production Research, 28(1), pp , M. Guyder, Automating the optimization of 2.5 axis milling, Computers in Industry, 15, pp , Y. Suh and K. Lee, NC milling tool path generation for arbitrary pockets defined by sculptured surfaces, Computer-Aided Design, 22(5), pp , L. Zeidner, Automatic process generation and the SURROUND problem: solution and applications, Manufacturing Review, 4(1), pp , T. Kramer, Pocket milling with tool engagement detection, Journal of Manufacturing Systems, 11(2), pp , M. Held, G. Lukacs and L. Andor, Pocket machining based on contour-parallel tool paths generated by means of proximity maps, Computer-Aided Design, 26(3), pp , A. Hansen and F. Arbab, An algorithm for generating NC tool paths for arbitrarily shaped pockets with islands, ACM Transactions on Graphics, 11(2), pp , 1992.

A new offset algorithm for closed 2D lines with Islands

A new offset algorithm for closed 2D lines with Islands Int J Adv Manuf Technol (2006) 29: 1169 1177 DOI 10.1007/s00170-005-0013-1 ORIGINAL ARTICLE Hyun-Chul Kim. Sung-Gun Lee. Min-Yang Yang A new offset algorithm for closed 2D lines with Islands Received:

More information

Multipatched B-Spline Surfaces and Automatic Rough Cut Path Generation

Multipatched B-Spline Surfaces and Automatic Rough Cut Path Generation Int J Adv Manuf Technol (2000) 16:100 106 2000 Springer-Verlag London Limited Multipatched B-Spline Surfaces and Automatic Rough Cut Path Generation S. H. F. Chuang and I. Z. Wang Department of Mechanical

More information

and Molds 1. INTRODUCTION

and Molds 1. INTRODUCTION Optimal Tool Path Generation for 2 and Molds D Milling of Dies HuiLi Automotive Components Division Ford Motor Company, Dearborn, MI, USA Zuomin Dong (zdong@me.uvic.ca) and Geoffrey W Vickers Department

More information

Offset Triangular Mesh Using the Multiple Normal Vectors of a Vertex

Offset Triangular Mesh Using the Multiple Normal Vectors of a Vertex 285 Offset Triangular Mesh Using the Multiple Normal Vectors of a Vertex Su-Jin Kim 1, Dong-Yoon Lee 2 and Min-Yang Yang 3 1 Korea Advanced Institute of Science and Technology, sujinkim@kaist.ac.kr 2 Korea

More information

Incomplete two-manifold mesh-based tool path generation

Incomplete two-manifold mesh-based tool path generation Int J Adv Manuf Technol (2006) 27: 797 803 DOI 10.1007/s00170-004-2239-8 ORIGINAL ARTICLE Dong-Yoon Lee Su-Jin Kim Hyun-Chul Kim Sung-Gun Lee Min-Yang Yang Incomplete two-manifold mesh-based tool path

More information

Computing Offsets and Tool Paths With Voronoi Diagrams l UUCS

Computing Offsets and Tool Paths With Voronoi Diagrams l UUCS Computing Offsets and Tool Paths With Voronoi Diagrams l Jin Jacob Chou and Elaine Cohen UUCS-89-017 Department of Computer Science University of Utah Salt Lake City, UT 84112 USA August 29, 1990 1 This

More information

Divided-and-Conquer for Voronoi Diagrams Revisited. Supervisor: Ben Galehouse Presenter: Xiaoqi Cao

Divided-and-Conquer for Voronoi Diagrams Revisited. Supervisor: Ben Galehouse Presenter: Xiaoqi Cao Divided-and-Conquer for Voronoi Diagrams Revisited Supervisor: Ben Galehouse Presenter: Xiaoqi Cao Outline Introduction Generalized Voronoi Diagram Algorithm for building generalized Voronoi Diagram Applications

More information

Solving 3D Geometric Constraints for Assembly Modelling

Solving 3D Geometric Constraints for Assembly Modelling Int J Adv Manuf Technol () 6:843 849 Springer-Verlag London Limited Solving 3D Geometric Constraints for Assembly Modelling J. Kim, K. Kim, K. Choi and J. Y. Lee 3 School of Mechanical and Industrial Engineering,

More information

Keyword: Quadratic Bézier Curve, Bisection Algorithm, Biarc, Biarc Method, Hausdorff Distances, Tolerance Band.

Keyword: Quadratic Bézier Curve, Bisection Algorithm, Biarc, Biarc Method, Hausdorff Distances, Tolerance Band. Department of Computer Science Approximation Methods for Quadratic Bézier Curve, by Circular Arcs within a Tolerance Band Seminar aus Informatik Univ.-Prof. Dr. Wolfgang Pree Seyed Amir Hossein Siahposhha

More information

Smallest Intersecting Circle for a Set of Polygons

Smallest Intersecting Circle for a Set of Polygons Smallest Intersecting Circle for a Set of Polygons Peter Otfried Joachim Christian Marc Esther René Michiel Antoine Alexander 31st August 2005 1 Introduction Motivated by automated label placement of groups

More information

Curve and Surface Basics

Curve and Surface Basics Curve and Surface Basics Implicit and parametric forms Power basis form Bezier curves Rational Bezier Curves Tensor Product Surfaces ME525x NURBS Curve and Surface Modeling Page 1 Implicit and Parametric

More information

Flank Millable Surface Design with Conical and Barrel Tools

Flank Millable Surface Design with Conical and Barrel Tools 461 Computer-Aided Design and Applications 2008 CAD Solutions, LLC http://www.cadanda.com Flank Millable Surface Design with Conical and Barrel Tools Chenggang Li 1, Sanjeev Bedi 2 and Stephen Mann 3 1

More information

Packing Two Disks into a Polygonal Environment

Packing Two Disks into a Polygonal Environment Packing Two Disks into a Polygonal Environment Prosenjit Bose, School of Computer Science, Carleton University. E-mail: jit@cs.carleton.ca Pat Morin, School of Computer Science, Carleton University. E-mail:

More information

Voronoi Diagrams in the Plane. Chapter 5 of O Rourke text Chapter 7 and 9 of course text

Voronoi Diagrams in the Plane. Chapter 5 of O Rourke text Chapter 7 and 9 of course text Voronoi Diagrams in the Plane Chapter 5 of O Rourke text Chapter 7 and 9 of course text Voronoi Diagrams As important as convex hulls Captures the neighborhood (proximity) information of geometric objects

More information

Smart Strategies for Steep/Shallow Milling

Smart Strategies for Steep/Shallow Milling Smart Strategies for Steep/Shallow Milling A Technical Overview contents Smoothing the Ups and Downs of Surface Machining..... 2 Traditional Finishing Strategies....... 2 Planar Toolpath.... 2 Z-Level

More information

Other Voronoi/Delaunay Structures

Other Voronoi/Delaunay Structures Other Voronoi/Delaunay Structures Overview Alpha hulls (a subset of Delaunay graph) Extension of Voronoi Diagrams Convex Hull What is it good for? The bounding region of a point set Not so good for describing

More information

VORONOI DIAGRAM PETR FELKEL. FEL CTU PRAGUE Based on [Berg] and [Mount]

VORONOI DIAGRAM PETR FELKEL. FEL CTU PRAGUE   Based on [Berg] and [Mount] VORONOI DIAGRAM PETR FELKEL FEL CTU PRAGUE felkel@fel.cvut.cz https://cw.felk.cvut.cz/doku.php/courses/a4m39vg/start Based on [Berg] and [Mount] Version from 9.11.2017 Talk overview Definition and examples

More information

An efficient implementation of the greedy forwarding strategy

An efficient implementation of the greedy forwarding strategy An efficient implementation of the greedy forwarding strategy Hannes Stratil Embedded Computing Systems Group E182/2 Technische Universität Wien Treitlstraße 3 A-1040 Vienna Email: hannes@ecs.tuwien.ac.at

More information

3. Voronoi Diagrams. 3.1 Definitions & Basic Properties. Examples :

3. Voronoi Diagrams. 3.1 Definitions & Basic Properties. Examples : 3. Voronoi Diagrams Examples : 1. Fire Observation Towers Imagine a vast forest containing a number of fire observation towers. Each ranger is responsible for extinguishing any fire closer to her tower

More information

BISECTOR CURVES OF PLANAR RATIONAL CURVES IN LORENTZIAN PLANE

BISECTOR CURVES OF PLANAR RATIONAL CURVES IN LORENTZIAN PLANE INTERNATIONAL JOURNAL OF GEOMETRY Vol. (03), No., 47-53 BISECTOR CURVES OF PLANAR RATIONAL CURVES IN LORENTZIAN PLANE MUSTAFA DEDE, YASIN UNLUTURK AND CUMALI EKICI Abstract. In this paper, the bisector

More information

Week 8 Voronoi Diagrams

Week 8 Voronoi Diagrams 1 Week 8 Voronoi Diagrams 2 Voronoi Diagram Very important problem in Comp. Geo. Discussed back in 1850 by Dirichlet Published in a paper by Voronoi in 1908 3 Voronoi Diagram Fire observation towers: an

More information

A fast approximation of the Voronoi diagram for a set of pairwise disjoint arcs

A fast approximation of the Voronoi diagram for a set of pairwise disjoint arcs A fast approximation of the Voronoi diagram for a set of pairwise disjoint arcs Dmytro Kotsur Taras Shevchenko National University of Kyiv 64/13, Volodymyrska, st., Kyiv, Ukraine dkotsur@gmail.com Vasyl

More information

Chapter 3. Sukhwinder Singh

Chapter 3. Sukhwinder Singh Chapter 3 Sukhwinder Singh PIXEL ADDRESSING AND OBJECT GEOMETRY Object descriptions are given in a world reference frame, chosen to suit a particular application, and input world coordinates are ultimately

More information

In what follows, we will focus on Voronoi diagrams in Euclidean space. Later, we will generalize to other distance spaces.

In what follows, we will focus on Voronoi diagrams in Euclidean space. Later, we will generalize to other distance spaces. Voronoi Diagrams 4 A city builds a set of post offices, and now needs to determine which houses will be served by which office. It would be wasteful for a postman to go out of their way to make a delivery

More information

Cutter path generation for 2.5D milling by combining multiple different cutter path patterns

Cutter path generation for 2.5D milling by combining multiple different cutter path patterns int. j. prod. res., 01 June 2004, vol. 42, no. 11, 2141 2161 Cutter path generation for 2.5D milling by combining multiple different cutter path patterns ZHIYANG YAOy and SATYANDRA K. GUPTAy* Different

More information

Simultaneously flippable edges in triangulations

Simultaneously flippable edges in triangulations Simultaneously flippable edges in triangulations Diane L. Souvaine 1, Csaba D. Tóth 2, and Andrew Winslow 1 1 Tufts University, Medford MA 02155, USA, {dls,awinslow}@cs.tufts.edu 2 University of Calgary,

More information

Practical Linear Algebra: A Geometry Toolbox

Practical Linear Algebra: A Geometry Toolbox Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 18: Putting Lines Together: Polylines and Polygons Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book

More information

6.854J / J Advanced Algorithms Fall 2008

6.854J / J Advanced Algorithms Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 6.854J / 18.415J Advanced Algorithms Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.415/6.854 Advanced

More information

CMPS 3130/6130 Computational Geometry Spring Voronoi Diagrams. Carola Wenk. Based on: Computational Geometry: Algorithms and Applications

CMPS 3130/6130 Computational Geometry Spring Voronoi Diagrams. Carola Wenk. Based on: Computational Geometry: Algorithms and Applications CMPS 3130/6130 Computational Geometry Spring 2015 Voronoi Diagrams Carola Wenk Based on: Computational Geometry: Algorithms and Applications 2/19/15 CMPS 3130/6130 Computational Geometry 1 Voronoi Diagram

More information

Technical Report. Removing polar rendering artifacts in subdivision surfaces. Ursula H. Augsdörfer, Neil A. Dodgson, Malcolm A. Sabin.

Technical Report. Removing polar rendering artifacts in subdivision surfaces. Ursula H. Augsdörfer, Neil A. Dodgson, Malcolm A. Sabin. Technical Report UCAM-CL-TR-689 ISSN 1476-2986 Number 689 Computer Laboratory Removing polar rendering artifacts in subdivision surfaces Ursula H. Augsdörfer, Neil A. Dodgson, Malcolm A. Sabin June 2007

More information

The Convex Hull of Rational Plane Curves

The Convex Hull of Rational Plane Curves Graphical Models 63, 151 162 (2001) doi:10.1006/gmod.2001.0546, available online at http://www.idealibrary.com on The Convex Hull of Rational Plane Curves Gershon Elber Department of Computer Science,

More information

Lecture 16: Voronoi Diagrams and Fortune s Algorithm

Lecture 16: Voronoi Diagrams and Fortune s Algorithm contains q changes as a result of the ith insertion. Let P i denote this probability (where the probability is taken over random insertion orders, irrespective of the choice of q). Since q could fall through

More information

Trimming Local and Global Self-intersections in Offset Curves/Surfaces using Distance Maps

Trimming Local and Global Self-intersections in Offset Curves/Surfaces using Distance Maps Trimming Local and Global Self-intersections in Offset Curves/Surfaces using Distance Maps Joon-Kyung Seong a Gershon Elber b, Myung-Soo Kim a,c a School of Computer Science and Engineering, Seoul National

More information

Path-planning by Tessellation of Obstacles

Path-planning by Tessellation of Obstacles Path-planning by Tessellation of Obstacles Tane Pendragon and Lyndon While School of Computer Science & Software Engineering, The University of Western Australia, Western Australia 6009 email: {pendrt01,

More information

Lectures 19: The Gauss-Bonnet Theorem I. Table of contents

Lectures 19: The Gauss-Bonnet Theorem I. Table of contents Math 348 Fall 07 Lectures 9: The Gauss-Bonnet Theorem I Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In

More information

Curves and Surfaces. Chapter 7. Curves. ACIS supports these general types of curves:

Curves and Surfaces. Chapter 7. Curves. ACIS supports these general types of curves: Chapter 7. Curves and Surfaces This chapter discusses the types of curves and surfaces supported in ACIS and the classes used to implement them. Curves ACIS supports these general types of curves: Analytic

More information

An approach to 3D surface curvature analysis

An approach to 3D surface curvature analysis An approach to 3D surface curvature analysis Dr. Laith A. Mohammed* Dr. Ghasan A. Al-Kindi** Published in J. of Engineering and Technology, University of Technology, Baghdad, Iraq, Vol.24, No.7, 2005,

More information

Subset Warping: Rubber Sheeting with Cuts

Subset Warping: Rubber Sheeting with Cuts Subset Warping: Rubber Sheeting with Cuts Pierre Landau and Eric Schwartz February 14, 1994 Correspondence should be sent to: Eric Schwartz Department of Cognitive and Neural Systems Boston University

More information

L1 - Introduction. Contents. Introduction of CAD/CAM system Components of CAD/CAM systems Basic concepts of graphics programming

L1 - Introduction. Contents. Introduction of CAD/CAM system Components of CAD/CAM systems Basic concepts of graphics programming L1 - Introduction Contents Introduction of CAD/CAM system Components of CAD/CAM systems Basic concepts of graphics programming 1 Definitions Computer-Aided Design (CAD) The technology concerned with the

More information

Outline of the presentation

Outline of the presentation Surface Reconstruction Petra Surynková Charles University in Prague Faculty of Mathematics and Physics petra.surynkova@mff.cuni.cz Outline of the presentation My work up to now Surfaces of Building Practice

More information

Computational Geometry

Computational Geometry Lecture 1: Introduction and convex hulls Geometry: points, lines,... Geometric objects Geometric relations Combinatorial complexity Computational geometry Plane (two-dimensional), R 2 Space (three-dimensional),

More information

A Constrained Delaunay Triangle Mesh Method for Three-Dimensional Unstructured Boundary Point Cloud

A Constrained Delaunay Triangle Mesh Method for Three-Dimensional Unstructured Boundary Point Cloud International Journal of Computer Systems (ISSN: 2394-1065), Volume 03 Issue 02, February, 2016 Available at http://www.ijcsonline.com/ A Constrained Delaunay Triangle Mesh Method for Three-Dimensional

More information

Contours & Implicit Modelling 4

Contours & Implicit Modelling 4 Brief Recap Contouring & Implicit Modelling Contouring Implicit Functions Visualisation Lecture 8 lecture 6 Marching Cubes lecture 3 visualisation of a Quadric toby.breckon@ed.ac.uk Computer Vision Lab.

More information

pine cone Ratio = 13:8 or 8:5

pine cone Ratio = 13:8 or 8:5 Chapter 10: Introducing Geometry 10.1 Basic Ideas of Geometry Geometry is everywhere o Road signs o Carpentry o Architecture o Interior design o Advertising o Art o Science Understanding and appreciating

More information

Parameterization of Triangular Meshes with Virtual Boundaries

Parameterization of Triangular Meshes with Virtual Boundaries Parameterization of Triangular Meshes with Virtual Boundaries Yunjin Lee 1;Λ Hyoung Seok Kim 2;y Seungyong Lee 1;z 1 Department of Computer Science and Engineering Pohang University of Science and Technology

More information

References. Additional lecture notes for 2/18/02.

References. Additional lecture notes for 2/18/02. References Additional lecture notes for 2/18/02. I-COLLIDE: Interactive and Exact Collision Detection for Large-Scale Environments, by Cohen, Lin, Manocha & Ponamgi, Proc. of ACM Symposium on Interactive

More information

The Topology of Skeletons and Offsets

The Topology of Skeletons and Offsets The Topology of Skeletons and Offsets Stefan Huber 1 1 B&R Industrial Automation stefan.huber@br-automation.com Abstract Given a polygonal shape with holes, we investigate the topology of two types of

More information

Polynomial/Rational Approximation of Minkowski Sum Boundary Curves 1

Polynomial/Rational Approximation of Minkowski Sum Boundary Curves 1 GRAPHICAL MODELS AND IMAGE PROCESSING Vol. 60, No. 2, March, pp. 136 165, 1998 ARTICLE NO. IP970464 Polynomial/Rational Approximation of Minkowski Sum Boundary Curves 1 In-Kwon Lee and Myung-Soo Kim Department

More information

Ray Casting of Trimmed NURBS Surfaces on the GPU

Ray Casting of Trimmed NURBS Surfaces on the GPU Ray Casting of Trimmed NURBS Surfaces on the GPU Hans-Friedrich Pabst Jan P. Springer André Schollmeyer Robert Lenhardt Christian Lessig Bernd Fröhlich Bauhaus University Weimar Faculty of Media Virtual

More information

LASER ADDITIVE MANUFACTURING PROCESS PLANNING AND AUTOMATION

LASER ADDITIVE MANUFACTURING PROCESS PLANNING AND AUTOMATION LASER ADDITIVE MANUFACTURING PROCESS PLANNING AND AUTOMATION Jun Zhang, Jianzhong Ruan, Frank Liou Department of Mechanical and Aerospace Engineering and Engineering Mechanics Intelligent Systems Center

More information

CSCI 4620/8626. Coordinate Reference Frames

CSCI 4620/8626. Coordinate Reference Frames CSCI 4620/8626 Computer Graphics Graphics Output Primitives Last update: 2014-02-03 Coordinate Reference Frames To describe a picture, the world-coordinate reference frame (2D or 3D) must be selected.

More information

Geometric Computations for Simulation

Geometric Computations for Simulation 1 Geometric Computations for Simulation David E. Johnson I. INTRODUCTION A static virtual world would be boring and unlikely to draw in a user enough to create a sense of immersion. Simulation allows things

More information

Robot Motion Planning Using Generalised Voronoi Diagrams

Robot Motion Planning Using Generalised Voronoi Diagrams Robot Motion Planning Using Generalised Voronoi Diagrams MILOŠ ŠEDA, VÁCLAV PICH Institute of Automation and Computer Science Brno University of Technology Technická 2, 616 69 Brno CZECH REPUBLIC Abstract:

More information

The Geometry of Carpentry and Joinery

The Geometry of Carpentry and Joinery The Geometry of Carpentry and Joinery Pat Morin and Jason Morrison School of Computer Science, Carleton University, 115 Colonel By Drive Ottawa, Ontario, CANADA K1S 5B6 Abstract In this paper we propose

More information

Shape Control of Cubic H-Bézier Curve by Moving Control Point

Shape Control of Cubic H-Bézier Curve by Moving Control Point Journal of Information & Computational Science 4: 2 (2007) 871 878 Available at http://www.joics.com Shape Control of Cubic H-Bézier Curve by Moving Control Point Hongyan Zhao a,b, Guojin Wang a,b, a Department

More information

Triangle Graphs and Simple Trapezoid Graphs

Triangle Graphs and Simple Trapezoid Graphs JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 18, 467-473 (2002) Short Paper Triangle Graphs and Simple Trapezoid Graphs Department of Computer Science and Information Management Providence University

More information

Coarse-to-Fine Search Technique to Detect Circles in Images

Coarse-to-Fine Search Technique to Detect Circles in Images Int J Adv Manuf Technol (1999) 15:96 102 1999 Springer-Verlag London Limited Coarse-to-Fine Search Technique to Detect Circles in Images M. Atiquzzaman Department of Electrical and Computer Engineering,

More information

(Master Course) Mohammad Farshi Department of Computer Science, Yazd University. Yazd Univ. Computational Geometry.

(Master Course) Mohammad Farshi Department of Computer Science, Yazd University. Yazd Univ. Computational Geometry. 1 / 17 (Master Course) Mohammad Farshi Department of Computer Science, Yazd University 1392-1 2 / 17 : Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars, Algorithms and Applications, 3rd Edition,

More information

VALLIAMMAI ENGINEERING COLLEGE

VALLIAMMAI ENGINEERING COLLEGE VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203 DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK M.E: CAD/CAM I SEMESTER ED5151 COMPUTER APPLICATIONS IN DESIGN Regulation 2017 Academic

More information

Central issues in modelling

Central issues in modelling Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction includes: manual modelling; fitting to

More information

COMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg

COMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg COMPUTER AIDED GEOMETRIC DESIGN Thomas W. Sederberg January 31, 2011 ii T. W. Sederberg iii Preface This semester is the 24 th time I have taught a course at Brigham Young University titled, Computer Aided

More information

Preferred directions for resolving the non-uniqueness of Delaunay triangulations

Preferred directions for resolving the non-uniqueness of Delaunay triangulations Preferred directions for resolving the non-uniqueness of Delaunay triangulations Christopher Dyken and Michael S. Floater Abstract: This note proposes a simple rule to determine a unique triangulation

More information

A New Slicing Procedure for Rapid Prototyping Systems

A New Slicing Procedure for Rapid Prototyping Systems Int J Adv Manuf Technol (2001) 18:579 585 2001 Springer-Verlag London Limited A New Slicing Procedure for Rapid Prototyping Systems Y.-S. Liao 1 and Y.-Y. Chiu 2 1 Department of Mechanical Engineering,

More information

2D Grey-Level Convex Hull Computation: A Discrete 3D Approach

2D Grey-Level Convex Hull Computation: A Discrete 3D Approach 2D Grey-Level Convex Hull Computation: A Discrete 3D Approach Ingela Nyström 1, Gunilla Borgefors 2, and Gabriella Sanniti di Baja 3 1 Centre for Image Analysis, Uppsala University Uppsala, Sweden ingela@cb.uu.se

More information

X i p. p = q + g. p=q

X i p. p = q + g. p=q Geometric Contributions to 3-Axis Milling of Sculptured Surfaces Johannes Wallner, Georg Glaeser y, Helmut Pottmann Abstract: When we are trying to shape a surface X by 3-axis milling, we encounter a list

More information

Voronoi Diagram. Xiao-Ming Fu

Voronoi Diagram. Xiao-Ming Fu Voronoi Diagram Xiao-Ming Fu Outlines Introduction Post Office Problem Voronoi Diagram Duality: Delaunay triangulation Centroidal Voronoi tessellations (CVT) Definition Applications Algorithms Outlines

More information

Lecture 13 Geometry and Computational Geometry

Lecture 13 Geometry and Computational Geometry Lecture 13 Geometry and Computational Geometry Euiseong Seo (euiseong@skku.edu) 1 Geometry a Branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties

More information

Chapter 2: Rhino Objects

Chapter 2: Rhino Objects The fundamental geometric objects in Rhino are points, curves, surfaces, polysurfaces, extrusion objects, and polygon mesh objects. Why NURBS modeling NURBS (non-uniform rational B-splines) are mathematical

More information

MISCELLANEOUS SHAPES

MISCELLANEOUS SHAPES MISCELLANEOUS SHAPES 4.1. INTRODUCTION Five generic shapes of polygons have been usefully distinguished in the literature: convex, orthogonal, star, spiral, and monotone. 1 Convex polygons obviously do

More information

Shape fitting and non convex data analysis

Shape fitting and non convex data analysis Shape fitting and non convex data analysis Petra Surynková, Zbyněk Šír Faculty of Mathematics and Physics, Charles University in Prague Sokolovská 83, 186 7 Praha 8, Czech Republic email: petra.surynkova@mff.cuni.cz,

More information

G 2 Interpolation for Polar Surfaces

G 2 Interpolation for Polar Surfaces 1 G 2 Interpolation for Polar Surfaces Jianzhong Wang 1, Fuhua Cheng 2,3 1 University of Kentucky, jwangf@uky.edu 2 University of Kentucky, cheng@cs.uky.edu 3 National Tsinhua University ABSTRACT In this

More information

CATIA V5 Parametric Surface Modeling

CATIA V5 Parametric Surface Modeling CATIA V5 Parametric Surface Modeling Version 5 Release 16 A- 1 Toolbars in A B A. Wireframe: Create 3D curves / lines/ points/ plane B. Surfaces: Create surfaces C. Operations: Join surfaces, Split & Trim

More information

Voronoi Diagram and Convex Hull

Voronoi Diagram and Convex Hull Voronoi Diagram and Convex Hull The basic concept of Voronoi Diagram and Convex Hull along with their properties and applications are briefly explained in this chapter. A few common algorithms for generating

More information

Measuring Lengths The First Fundamental Form

Measuring Lengths The First Fundamental Form Differential Geometry Lia Vas Measuring Lengths The First Fundamental Form Patching up the Coordinate Patches. Recall that a proper coordinate patch of a surface is given by parametric equations x = (x(u,

More information

Geometric Modeling Mortenson Chapter 11. Complex Model Construction

Geometric Modeling Mortenson Chapter 11. Complex Model Construction Geometric Modeling 91.580.201 Mortenson Chapter 11 Complex Model Construction Topics Topology of Models Connectivity and other intrinsic properties Graph-Based Models Emphasize topological structure Boolean

More information

Chapter 8. Voronoi Diagrams. 8.1 Post Oce Problem

Chapter 8. Voronoi Diagrams. 8.1 Post Oce Problem Chapter 8 Voronoi Diagrams 8.1 Post Oce Problem Suppose there are n post oces p 1,... p n in a city. Someone who is located at a position q within the city would like to know which post oce is closest

More information

Distance Trisector Curves in Regular Convex Distance Metrics

Distance Trisector Curves in Regular Convex Distance Metrics Distance Trisector Curves in Regular Convex Distance Metrics Tetsuo sano School of Information Science JIST 1-1 sahidai, Nomi, Ishikawa, 923-1292 Japan t-asano@jaist.ac.jp David Kirkpatrick Department

More information

Construction of Voronoi Diagrams

Construction of Voronoi Diagrams Construction of Voronoi Diagrams David Pritchard April 9, 2002 1 Introduction Voronoi diagrams have been extensively studied in a number of domains. Recent advances in graphics hardware have made construction

More information

Flavor of Computational Geometry. Voronoi Diagrams. Shireen Y. Elhabian Aly A. Farag University of Louisville

Flavor of Computational Geometry. Voronoi Diagrams. Shireen Y. Elhabian Aly A. Farag University of Louisville Flavor of Computational Geometry Voronoi Diagrams Shireen Y. Elhabian Aly A. Farag University of Louisville March 2010 Pepperoni Sparse Pizzas Olive Sparse Pizzas Just Two Pepperonis A person gets the

More information

Aspect-Ratio Voronoi Diagram with Applications

Aspect-Ratio Voronoi Diagram with Applications Aspect-Ratio Voronoi Diagram with Applications Tetsuo Asano School of Information Science, JAIST (Japan Advanced Institute of Science and Technology), Japan 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

More information

Three dimensional biarc approximation of freeform surfaces for machining tool path generation

Three dimensional biarc approximation of freeform surfaces for machining tool path generation INT. J. PROD. RES., 000, VOL. 38, NO. 4, 739± 763 Three dimensional biarc approximation of freeform surfaces for machining tool path generation YUAN-JYE TSENGy * and YII-DER CHEN y In typical methods for

More information

CATIA Surface Design

CATIA Surface Design CATIA V5 Training Exercises CATIA Surface Design Version 5 Release 19 September 2008 EDU_CAT_EN_GS1_FX_V5R19 Table of Contents (1/2) Creating Wireframe Geometry: Recap Exercises 4 Creating Wireframe Geometry:

More information

A Simple Method of the TEX Surface Drawing Suitable for Teaching Materials with the Aid of CAS

A Simple Method of the TEX Surface Drawing Suitable for Teaching Materials with the Aid of CAS A Simple Method of the TEX Surface Drawing Suitable for Teaching Materials with the Aid of CAS Masataka Kaneko, Hajime Izumi, Kiyoshi Kitahara 1, Takayuki Abe, Kenji Fukazawa 2, Masayoshi Sekiguchi, Yuuki

More information

CS S Lecture February 13, 2017

CS S Lecture February 13, 2017 CS 6301.008.18S Lecture February 13, 2017 Main topics are #Voronoi-diagrams, #Fortune. Quick Note about Planar Point Location Last week, I started giving a difficult analysis of the planar point location

More information

Voronoi diagrams Delaunay Triangulations. Pierre Alliez Inria

Voronoi diagrams Delaunay Triangulations. Pierre Alliez Inria Voronoi diagrams Delaunay Triangulations Pierre Alliez Inria Voronoi Diagram Voronoi Diagram Voronoi Diagram The collection of the non-empty Voronoi regions and their faces, together with their incidence

More information

6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.

6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z. Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct

More information

A second order algorithm for orthogonal projection onto curves and surfaces

A second order algorithm for orthogonal projection onto curves and surfaces A second order algorithm for orthogonal projection onto curves and surfaces Shi-min Hu and Johannes Wallner Dept. of Computer Science and Technology, Tsinghua University, Beijing, China shimin@tsinghua.edu.cn;

More information

Polygonal Skeletons. Tutorial 2 Computational Geometry

Polygonal Skeletons. Tutorial 2 Computational Geometry Polygonal Skeletons Tutorial 2 Computational Geometry The Skeleton of a Simple Polygon A polygon is a closed contour in the plane, which might contain holes (which are simple polygons as well). A skeleton

More information

software isy-cam 2.8 and 3.6 CAD/CAM software Features isy-cam 2.8 Features isy-cam 3.6 D-4 CAD functionality (without volume modeller)

software isy-cam 2.8 and 3.6 CAD/CAM software Features isy-cam 2.8 Features isy-cam 3.6 D-4 CAD functionality (without volume modeller) CAD/CAM isy-cam 2.8 and 3.6 isy-cam 2.8 CAD functionality (without volume modeller) works with Win XP, Windows 7 and 8, 32-/64-bit version Import: DXF / EPS / AI / 3D STL data Export: NCP format proven

More information

We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.

We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the

More information

Research Article Polygon Morphing and Its Application in Orebody Modeling

Research Article Polygon Morphing and Its Application in Orebody Modeling Mathematical Problems in Engineering Volume 212, Article ID 732365, 9 pages doi:1.1155/212/732365 Research Article Polygon Morphing and Its Application in Orebody Modeling Hacer İlhan and Haşmet Gürçay

More information

Direct Rendering of Trimmed NURBS Surfaces

Direct Rendering of Trimmed NURBS Surfaces Direct Rendering of Trimmed NURBS Surfaces Hardware Graphics Pipeline 2/ 81 Hardware Graphics Pipeline GPU Video Memory CPU Vertex Processor Raster Unit Fragment Processor Render Target Screen Extended

More information

DiFi: Distance Fields - Fast Computation Using Graphics Hardware

DiFi: Distance Fields - Fast Computation Using Graphics Hardware DiFi: Distance Fields - Fast Computation Using Graphics Hardware Avneesh Sud Dinesh Manocha UNC-Chapel Hill http://gamma.cs.unc.edu/difi Distance Fields Distance Function For a site a scalar function f:r

More information

Precise Continuous Contact Motion Analysis for Freeform Geometry. Yong-Joon Kim Department of Computer Science Technion, Israel

Precise Continuous Contact Motion Analysis for Freeform Geometry. Yong-Joon Kim Department of Computer Science Technion, Israel Precise Continuous Contact Motion Analysis for Freeform Geometry Yong-Joon Kim Department of Computer Science Technion, Israel Two Main Parts Precise contact motion - planar curves. Joint Work with Prof.

More information

Normals of subdivision surfaces and their control polyhedra

Normals of subdivision surfaces and their control polyhedra Computer Aided Geometric Design 24 (27 112 116 www.elsevier.com/locate/cagd Normals of subdivision surfaces and their control polyhedra I. Ginkel a,j.peters b,,g.umlauf a a University of Kaiserslautern,

More information

Elementary Planar Geometry

Elementary Planar Geometry Elementary Planar Geometry What is a geometric solid? It is the part of space occupied by a physical object. A geometric solid is separated from the surrounding space by a surface. A part of the surface

More information

Basic tool: orientation tests

Basic tool: orientation tests Basic tool: orientation tests Vera Sacristán Computational Geometry Facultat d Informàtica de Barcelona Universitat Politècnica de Catalunya Orientation test in R 2 Given 3 points p,, r in the plane, efficiently

More information

Contours & Implicit Modelling 1

Contours & Implicit Modelling 1 Contouring & Implicit Modelling Visualisation Lecture 8 Institute for Perception, Action & Behaviour School of Informatics Contours & Implicit Modelling 1 Brief Recap Contouring Implicit Functions lecture

More information

ACCELERATING TOOL PATH COMPUTING IN TURNING LATHE MACHINING. Antonio Jimeno, Sergio Cuenca, Antonio Martínez, Jose Luis Sánchez Romero

ACCELERATING TOOL PATH COMPUTING IN TURNING LATHE MACHINING. Antonio Jimeno, Sergio Cuenca, Antonio Martínez, Jose Luis Sánchez Romero ACCELERATING TOOL PATH COMPUTING IN TURNING LATHE MACHINING Antonio Jimeno, Sergio Cuenca, Antonio Martínez, Jose Luis Sánchez Romero Computer Science Technology and Computation Department University of

More information

Planar union of rectangles with sides parallel to the coordinate axis

Planar union of rectangles with sides parallel to the coordinate axis 6 th International Conference on Applied Informatics Eger, Hungary, January 27 31, 2004. Planar union of rectangles with sides parallel to the coordinate axis Daniel Schmid, Hanspeter Bopp Department of

More information