DOCSLIB.ORG

Models of the Cutting Edge Geometry of Medical Needles with Applications to Needle Design

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Models of the Cutting Edge Geometry of Medical Needles with Applications to Needle Design International Journal of Mechanical Sciences 65 (2012) 157–167 Contents lists available at SciVerse ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Models of the cutting edge geometry of medical needles with applications to needle design Peidong Han n, Demeng Che, Kumar Pallav, Kornel Ehmann Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, United States article info abstract Article history: Tissue cutting by a needle is greatly influenced by the geometry of its cutting edges. Yet, there are very Received 13 May 2012 few results in the technical literature that address the issues surrounding the relation between needle Received in revised form tip and cutting edge geometry and its cutting/insertion force. In the present paper, general mathema- 20 August 2012 tical models of the included and inclination angles of a needle’s cutting edge are formulated since they Accepted 25 September 2012 exert a profound influence on cutting behavior. Specific models for various tip geometries are Available online 3 October 2012 developed to investigate four types of needle tips including one-plane, asymmetric three-plane, Keywords: symmetric multi-plane, and symmetric three-curved-surface needles. The results show that the cutting Tissue cutting edge of the one-plane needle has a very undesirable configuration. Asymmetric three-plane needles, Medical needle design such as the lancet and back bevel tip needles, can significantly improve the configuration of the cutting Cutting edge geometry edge. Back bevel tip needles, in turn, offer even more desirable cutting edge geometry than lancet tip Needle insertion needles. In evaluating the cutting edge angles of multi-plane needles, it was shown that the included angle is limited by its tip geometry. To resolve this problem a needle formed by three curved surfaces was proposed. It was found that this new design has significant potential for reducing patient trauma as well as target accuracy during needle insertion. The proposed general approach provides the foundation for designing medical needles with specific cutting edge properties. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction critical to the success of the procedure. Additionally, the studies by Egekvist et al. [5,6] and Arendt-Nielsen et al. [7] have shown that the Medical needles are the most commonly used medical devices needle insertion force is positively correlated with the frequency and and are widely used in minimally invasive percutaneous proce- intensity of pain and trauma. For example, the insertion force is dures such as injection, regional anesthesia, blood sampling, significantly higher for 27G needle insertions compared to 30G biopsy and brachytherapy. These procedures require inserting a needle insertions. Correspondingly, 53% of insertions with the 27G needle to a target inside the body for either drug delivery or tissue needles caused pain and 39% of insertions with 30G needles caused sample removal. Brachytherapy is a medical procedure to place pain. In [8], the bluntness of the needle tip has been found to be radioactive seeds inside or near the tumor using a needle for related to the increased pain in subcutaneous injections. Thus, cancer treatment. The success of this procedure depends on the needles with lower insertion forces are important to reduce the pain accurate placement of the seeds. However, inserting a needle and trauma experienced by the patients during the procedure. deforms the tissue and displaces the target causing placement Needle insertion is essentially a tissue cutting process. The errors [1,2]. Substantial practice is needed to avoid placement force of tissue cutting by a needle greatly depends on its cutting errors by compensating for tissue deformation and target edge geometry. Prior to studying the cutting edge geometry of a displacement caused by the needle tip force and friction. It has been needle, one may begin by examining a blade with a straight shown by Alterovitz [2–4] that without preoperative planning cutting edge that is defined by its included angle y and the cutting placement errors may be as large as 26% of the prostate diameter edge radius also referred to as the ‘‘roundness’’ of the cutting edge, resulting in damage to healthy tissue and failure to kill cancerous as illustrated in Fig. 1(a). Maximal knife sharpness is obtained with a cells. A practical approach to reducing the placement error is to small included angle and a small radius for the cutting edge [9]. minimize the needle insertion force. Thus, a needle that creates The knives with a small included angle, such as surgical scalpels and lower insertion forces and smaller displacements of the target is straight razors, are extremely sharp but fragile. On the contrary, knives that require a tough edge usually have a large included angle. Typical values of the included angle are about 151 for razor blades n Corresponding author. Tel.: þ1 84 7467 0712. and veneer cutting knives, 20–301 for microtome knives and 30–401 E-mail address: [email protected] (P. Han). for kitchen knives [9]. Typical radii of cutting edges are 5 mmfora 0020-7403/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2012.09.014 158 P. Han et al. / International Journal of Mechanical Sciences 65 (2012) 157–167 Bevel tip needle stylet Edge bevel Bevel tip needle cannula Bevel cutting edge Lancet point needle stylet Back bevel needle stylet Edge bevel Motion Back bevel needle cannula Work Cutting edge radius Trocar tip needle stylet Franseen tip needle cannula Conical tip needle stylet 2 1 Fig. 2. Examples of needle tips. Motion Motion Surprisingly little research has been carried out related to the Work Work modeling of the needle cutting edge geometry and to investigate how the cutting edge geometry influences the penetration force of a Fig. 1. (a) cross-section of a blade showing the included angle y and cutting edge needle. Studies have been conducted to investigate the effects of radius; (b) plan view of cutting with inclination angle l; (c) illustration of the cutting process (a¼rake angle, b¼setting angle, z¼relief/clearance angle); and needle tip geometry on the penetration force and needle sharpness (d) the cutting process at b¼01. [14–20]. In these works needle tip geometry was defined only by the bevel angle, the angle between the bevel ground surface and the needle longitudinal axis, which does not directly reveal any specific scalpel and 17 mm for a new safety razor blade (34 mmwhen information about the cutting edges. The needle sharpness was worn) [9]. Another important factor in cutting is the inclination judged by the penetration force encountered during passing through angle l, which describes the angle of the cutting edge relative to the tissue. However, little is known about how sharpness and penetra- cutting direction, as illustrated in Fig. 1(b). It has been shown by tion force are directly influenced by tip geometry. Moore et al. Atkins [10–12] that the cutting force exerted by a sharp object [21–23] have used the customary cutting tool geometry approach depends on the slice/push ratio k given by k¼(speed parallel to widely used in metal cutting theory to model the hollow biopsy cutting edge/speed perpendicular to the cutting edge). An increase needle tip. The emphasis of their work was on the determination of in the slice/push ratio reduces the cutting force. When the cutting the variation of the inclination and rake angles along the needle edge is inclined at an inclination angle l, the slice/push ratio is given cutting edge. For a biopsy cannula, where the cutting edge is by k¼tan l. An increase in the inclination angle increases k, leading generated by the intersection of the needle side surface and the to a lower cutting force. For a general cutting process, three other bevel plane (rake face), one rake angle together with the inclination basic angles are important: the rake angle (a), the clearance/relief angle and edge radius properly describes the cutting edge. However, angle (z), and the setting angle (b), as shown in Fig. 1(c). The cutting in case of a multi-plane needle stylet where the needle cutting edge surface of the blade is the rake face. The rake angle is the angle of is formed by the intersection of two bevel surfaces, one rake angle the rake face with respect to the perpendicular to the cut surface. together with the inclination angle and edge radius does not The clearance/relief angle is the angle between the second face of adequately represent the cutting edge geometry. Until now, no work the blade and the cut surface. Hence, yþaþz¼901. It can be seen has been reported on modeling the cutting edge of a needle stylet. that the rake angle and clearance/relief angle depend on the setting In this paper, the cutting edge geometry of a needle is defined angle. Tissue cutting by a scalpel or a needle is a process where the by the included angle, the inclination angle, and the cutting edge tool cuts by penetration into the tissue. This type of cutting has a radius, as is defined for a blade. The radius of a cutting edge setting angle b¼01,asshowninFig. 1(d), where both the edge depends on the manufacturing processes including the machin- bevels act as rake faces. Hence, yþa1þa2¼1801.Asymmetricblade ing, deburring, and coating processes, while the included angle has a1¼a2. For a blade with only one side beveled and the other side and inclination angle of a needle’s cutting edge are determined by remaining flat, a2 is equal to 901, hence yþa1¼901.
Load more

Recommended publications
  • 1 Lifts of Polytopes
    Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe.
    [Show full text]
  • Geometry © 2000 Springer-Verlag New York Inc
    Discrete Comput Geom OF1–OF15 (2000) Discrete & Computational DOI: 10.1007/s004540010046 Geometry © 2000 Springer-Verlag New York Inc. Convex and Linear Orientations of Polytopal Graphs J. Mihalisin and V. Klee Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA {mihalisi,klee}@math.washington.edu Abstract. This paper examines directed graphs related to convex polytopes. For each fixed d-polytope and any acyclic orientation of its graph, we prove there exist both convex and concave functions that induce the given orientation. For each combinatorial class of 3-polytopes, we provide a good characterization of the orientations that are induced by an affine function acting on some member of the class. Introduction A graph is d-polytopal if it is isomorphic to the graph G(P) formed by the vertices and edges of some (convex) d-polytope P. As the term is used here, a digraph is d-polytopal if it is isomorphic to a digraph that results when the graph G(P) of some d-polytope P is oriented by means of some affine function on P. 3-Polytopes and their graphs have been objects of research since the time of Euler. The most important result concerning 3-polytopal graphs is the theorem of Steinitz [SR], [Gr1], asserting that a graph is 3-polytopal if and only if it is planar and 3-connected. Also important is the related fact that the combinatorial type (i.e., the entire face-lattice) of a 3-polytope P is determined by the graph G(P). Steinitz’s theorem has been very useful in studying the combinatorial structure of 3-polytopes because it makes it easy to recognize the 3-polytopality of a graph and to construct graphs that represent 3-polytopes without producing an explicit geometric realization.
    [Show full text]
  • Archimedean Solids
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat).
    [Show full text]
  • Steinitz's Theorem Project Report §1 Introduction §2 Basic Definitions
    Steinitz's Theorem Project Report Jon Hillery May 17, 2019 §1 Introduction Looking at the vertices and edges of polyhedra gives a family of graphs that we might expect has nice properties. It turns out that there is actually a very nice characterization of these graphs! We can use this characterization to find useful representations of certain graphs. §2 Basic Definitions We define a space to be convex if the line segment connecting any two points in the space remains entirely inside the space. This works for two-dimensional sets: and three-dimensional sets: Given a polyhedron, we define its 1-skeleton to be the graph formed from the vertices and edges of the polyhedron. For example, the 1-skeleton of a tetrahedron is K4: 1 Jon Hillery (May 17, 2019) Steinitz's Theorem Project Report Here are some further examples of the 1-skeleton of an icosahedron and a dodecahedron: §3 Properties of 1-Skeletons What properties do we know the 1-skeleton of a convex polyhedron must have? First, it must be planar. To see this, imagine moving your eye towards one of the faces until you are close enough that all of the other faces appear \inside" the face you are looking through, as shown here: This is always possible because the polyedron is convex, meaning intuitively it doesn't have any parts that \jut out". The graph formed from viewing in this way will have no intersections because the polyhedron is convex, so the straight-line rays our eyes see are not allowed to leave via an edge on the boundary of the polyhedron and then go back inside.
    [Show full text]
  • Edge Energies and S S of Nan O Pred P I Tatt
    SAND2007-0238 .. j c..* Edge Energies and S s of Nan o pred p i tatt dexim 87185 and Llverm ,ultIprogramlabora lartln Company, for Y'S 11 Nuclear Security Admini 85000. Approved for Inallon unlimited. Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government, nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, make any warranty, express or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefidness of any information, apparatus, product, or process disclosed, or represent that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government, any agency thereof, or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof, or any of their contractors. Printed in the United States of America. This report has been reproduced directly from the best available copy. Available to DOE and DOE contractors from U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN 3783 1 Telephone: (865) 576-8401 Facsimile: (865) 576-5728 E-Mail: reDorts(cL.adonls.os(I.oov Online ordering: htto://www.osti.ao\ /bridge Available to the public from U.S.
    [Show full text]
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
    [Show full text]
  • What Is a Polyhedron?
    3. POLYHEDRA, GRAPHS AND SURFACES 3.1. From Polyhedra to Graphs What is a Polyhedron? Now that we’ve covered lots of geometry in two dimensions, let’s make things just a little more difficult. We’re going to consider geometric objects in three dimensions which can be made from two-dimensional pieces. For example, you can take six squares all the same size and glue them together to produce the shape which we call a cube. More generally, if you take a bunch of polygons and glue them together so that no side gets left unglued, then the resulting object is usually called a polyhedron.1 The corners of the polygons are called vertices, the sides of the polygons are called edges and the polygons themselves are called faces. So, for example, the cube has 8 vertices, 12 edges and 6 faces. Different people seem to define polyhedra in very slightly different ways. For our purposes, we will need to add one little extra condition — that the volume bound by a polyhedron “has no holes”. For example, consider the shape obtained by drilling a square hole straight through the centre of a cube. Even though the surface of such a shape can be constructed by gluing together polygons, we don’t consider this shape to be a polyhedron, because of the hole. We say that a polyhedron is convex if, for each plane which lies along a face, the polyhedron lies on one side of that plane. So, for example, the cube is a convex polyhedron while the more complicated spec- imen of a polyhedron pictured on the right is certainly not convex.
    [Show full text]
  • Graphs of Polyhedra and the Theorem of Steinitz by António Guedes De Oliveira*
    Graphs of polyhedra and the theorem of Steinitz by António Guedes de Oliveira* The theorem of Steinitz characterizes in simple terms the graphs of the polyhedra. In fact, the characteristic properties of such graphs, according to the theorem, are not only simple but “very natural”, in that they occur in various different contexts. As a consequence, for exam- ple, polyhedra and typical polyhedral constructions can be used for finding rectangles that can be decomposed in non-congruent squares (see Figure 1). The extraordi- nary theorem behind this relation is due to Steinitz and is the main topic of the present paper. Steinitz’s theorem was first published in a scientific encyclopaedia, in 1922 [16], and later, in 1934, in a book [17], after Steinitz’s death. It was ignored for a long time, but after “being discovered” its importance never ceased to increase and it is the starting point for active research even to our days. In the middle of the last century, two very important books were published in Polytope The- ory. The first one, by Alexander D. Alexandrov, which was published in Russian in 1950 and in German, under the title “Konvexe Polyeder” [1], in 1958, does not men- Figure 1.— Decomposition of a rectangle in tion this theorem. The second one, by Branko Grünbaum, non-congruent squares “Convex Polytopes”, published for the first time in 1967 (and dedicated exactly to the “memory of the extraordi- nary geometer Ernst Steinitz”), considers this theorem as the “most important and the deepest of the known results about polyhedra” [9, p.
    [Show full text]
  • Steinitz's Theorem, I
    Steinitz’s Theorem, I Realization Spaces of Polytopes Winter Semester 2020/2021 Marta Panizzut November 24, 2020 1 / 20 Steinitz’s Theorem Theorem 8 [Steinitz ’22] A finite graph is the edge-graph of a 3-polytope if and only if it is simple, planar and 3-connected. “only if”: radial projection to the sphere from an interior point or Schlegel diagram + Balinski’s theorem. 2 / 20 Theorem 9 Let G be a graph. (i) [Menger’s Theorem] If G is 3-connected, then between any pair of vertices there are three paths G1, G2 and G3 that are disjoint except for the endpoints. (ii) If there are three vertices u,v,w that are connected by three different Y -graphs Y1, Y2 and Y3 that are disjoint except for u, v and w, then G is not planar. (iii) [Whitney’s Theorem] If G is planar and 3-connected then the set cells(G ,D) is independent on the particular choice of a drawing D of G. (iv) [Euler’s Theorem] If G is planar and 2-connected and D is a drawing of G, then the numbers of cells c cells(G ,D) , Æ j j vertices v V , and edges e E are related by Æ j j Æ j j v c e 2. Å Æ Å 3 / 20 Idea of the proof Draw a 3-connected planar graph such that its edges are Ï represented by line segments, and the cells are convex polygons. If the boundary of the drawing of G is a triangle, then the Ï figure is a Schlegel diagram of a 3-polytope with edge-graph G.
    [Show full text]
  • Quantitative Steinitz's Theorems with Applications to Multi Ngered
    Quantitative Steinitzs Theorems with Applications to Multingered Grasping David Kirkpatrick Bhubaneswar Mishra and CheeKeng Yap Courant Institute of Mathematical Sciences New York University New York NY Abstract We prove the following quantitative form of a classical theorem of Steinitz Let m b e suciently large If the convex hull of a subset S of Euclidean dspace contains a unit ball centered on the origin then there is a subset of S with at most m p oints whose convex hull contains a solid ball also centered on the origin and having residual radius 2 d1 d d m The case m d was rst considered by Barany Katchalski and Pach We also show an upp er b ound on the achievable radius the residual radius must b e less than 2 d1 d m These results have applications in the problem of computing the socalled closure grasps by an mngered rob ot hand The ab ove quantitative form of Steinitzs the orem gives a notion of eciency for closure grasps The theorem also gives rise to some new problems in computational geometry We present some ecient algorithms for these problems esp ecially in the two dimensional case Septemb er 1 Supp orted by NSF Grants DCR and DCR ONR Grant NJ NSF Grant Sub contract CMU Chee Yap is supp orted in part by the German Research Foundation DFG and by the ESPRIT II Basic Research Actions Program of the EC under contract No pro ject ALCOM 2 Department of Computer Science University of British Columbia Vancouver Canada Section Intro duction d Caratheo dorys theorem states that given a subset S of Euclidean
    [Show full text]
  • REGULAR POLYTOPES in Zn Contents 1. Introduction 1 2. Some
    REGULAR POLYTOPES IN Zn ANDREI MARKOV Abstract. In [3], all embeddings of regular polyhedra in the three dimen- sional integer lattice were characterized. Here, we prove some results toward solving this problem for all higher dimensions. Similarly to [3], we consider a few special polytopes in dimension 4 that do not have analogues in higher dimensions. We then begin a classification of hypercubes, and consequently regular cross polytopes in terms of the generalized duality between the two. Fi- nally, we investigate lattice embeddings of regular simplices, specifically when such simplices can be inscribed in hypercubes of the same dimension. Contents 1. Introduction 1 2. Some special cases, and a word on convexity. 2 3. Hypercubes and Cross Polytopes 4 4. Simplices 4 5. Corrections 6 Acknowledgments 7 References 7 1. Introduction Definition 1.1. A regular polytope is a polytope such that its symmetry group action is transitive on its flags. This definition is not all that important to what we wish to accomplish, as the classification of all such polytopes is well known, and can be found, for example, in [2]. The particulars will be introduced as each case comes up. Definition 1.2. The dual of a polytope of dimension n is the polytope formed by taking the centroids of the n − 1 cells to be its vertices. The dual of the dual of a regular polytope is homothetic to the original with respect to their mutual center. We will say that a polytope is \in Zn" if one can choose a set of points in the lattice of integer points in Rn such that the points can form the vertices of said polytope.
    [Show full text]
  • Basic Concepts Today
    Basic Concepts Today • Mesh basics – Zoo – Definitions – Important properties • Mesh data structures • HW1 Polygonal Meshes • Piecewise linear approximation – Error is O(h2) 3 6 12 24 25% 6.5% 1.7% 0.4% Polygonal Meshes • Polygonal meshes are a good representation – approximation O(h2) – arbitrary topology – piecewise smooth surfaces – adtidaptive refinemen t – efficient rendering Triangle Meshes • Connectivity: vertices, edges, triangles • Geometry: vertex positions Mesh Zoo Single component, With boundaries Not orientable closed, triangular, orientable manifold Multiple components Not only triangles Non manifold Mesh Definitions • Need vocabulary to describe zoo meshes • The connectivity of a mesh is just a graph • We’ll start with some graph theory Graph Definitions B A G = graph = <VEV, E> E F C D V = vertices = {A, B, C, …, K} E = edges = {(AB), (AE), (CD), …} H I F ABE DHJG G = faces = {( ), ( ), …} J K Graph Definitions B A Vertex degree or valence = number of incident edges E deg(A) = 4 F C D deg(E) = 5 H I G Regular mesh = all vertex degrees are equal J K Connectivity Connected = B A path of edges connecting every two vertices E F C D H I G J K Connectivity Connected = B A path of edges connecting every two vertices E F Subgraph = C D G’=<VV,E’,E’> is a subgraph of G=<V,E> if I H V’ is a subset of V and G E’ is the subset of E incident on V J K Connectivity Connected = B A path of edges connecting every two vertices E F Subgraph = C D G’=<VV,E’,E’> is a subgraph of G=<V,E> if I H V’ is a subset of V and G E’ is a subset of E incident
    [Show full text]