Fathi El-Yafi Project and Software Development Manager Engineering Simulation
|
|
- Scarlett Golden
- 5 years ago
- Views:
Transcription
1 An Introduction to Geometry Design Algorithms Fathi El-Yafi Project and Software Development Manager Engineering Simulation 1
2 Geometry: Overview Geometry Basics Definitions Data Semantic Topology Mathematics Hierarchy CSG Approach BREP Approach Curves Surfaces 2
3 Geometry: Concept Basics vertices: x,y,z location 3
4 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices 4
5 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices surfaces: closed set of curves 5
6 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices surfaces: closed set of curves volumes: closed set of surfaces 6
7 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices surfaces: closed set of curves volumes: closed set of surfaces body: collection of volumes 7
8 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices surfaces: closed set of curves loops: ordered set of curves on surface volumes: closed set of surfaces body: collection of volumes 8
9 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices coedges: orientation of curve w.r.t. loop surfaces: closed set of curves (loops) loops: ordered set of curves on surface volumes: closed set of surfaces body: collection of volumes 9
10 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices coedges: orientation of curve w.r.t. loop surfaces: closed set of curves (loops) shell: oriented set of surfaces comprising a volume volumes: closed set of surfaces (shells) loops: ordered set of curves on surface body: collection of volumes 10
11 Geometry: Concept Basics vertices: x,y,z location curves: bounded by two vertices coedges: orientation of curve w.r.t. loop surfaces: closed set of curves (loops) shell: oriented set of surfaces comprising a volume volumes: closed set of surfaces (shells) loops: ordered set of curves on surface body: collection of volumes 11
12 Geometry: Concept Basics vertices: x,y,z location coface: oriented surface w.r.t. shell shell: oriented set of surfaces comprising a volume curves: bounded by two vertices volumes: closed set of surfaces (shells) coedges: orientation of curve w.r.t. loop loops: ordered set of curves on surface surfaces: closed set of curves (loops) body: collection of volumes 12
13 Geometry: Concept Basics Surface 11 Volume 2 Volume 1 Manifold Geometry: Each volume maintains its own set of unique surfaces Volume 2 Surface 7 Surface 8 Surface 9 Surface 10 Surface 11 Volume 1 Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6 Surface 7 13
14 Geometry: Concept Basics Volume 1 Non-Manifold Geometry: Volumes share matching surfaces Volume 2 Volume 2 Surface 7 Surface 8 Surface 9 Surface 10 Volume 1 Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6 Surface 7 14
15 Geometry: Data Model Wireframe Surface Volume Geometrical Data Separated from Attributes Attributes Colors Parameters Etc. Graphical Objects Visible Parts 15
16 Semantic: Surface-Volume-Deflection-Defects 16
17 Semantic: Surfaces and Features 17
18 Semantic: Detail 18
19 Semantic: Detail-Mesh Bounding Box 19
20 Semantic: Decomposition Curvature Mesh STL-Mesh FEM 20
21 Wireframe Model: Limits 21
22 Basics of Topology Topology? Concept of interior and exterior Orientation Reliable calculation of basic parameters: volume, center of gravity, axis of inertia Contour = oriented surface and limited area Tree construction 22
23 Relations: (Geometry, Topology) Geometry Topology Volume Solid Surface face Contour Curve Vertex Edge Point 23
24 Topology: Mathematics A contour grips" to its interior Either S is a Set of R 3 Adhesion : A(S) P A( S) : any neighborhood of P contains a point of S Interior : I(S) P I( S) if V(P) S Boundary : B(S) P B( S) if P A( S) and P A( C(S)) where C(S) refers to the complementary of S in R 3 24
25 Topology: Mathematics Open : S = A(S) Closed : S = I(S) Regular Solid S = A(I(S)) : adhesion of its interior = R(S) 25
26 Topology: Mathematics Operator of Regularization A B = R(A B) A B = R(A B) A - B = R(A-B) C(A) = R(C(A)) A B A A B R(A B) A - B R(A B) 26
27 Topology: Mathematics Operator of Regularization A B A B A B C(A)/B R(C(A)/B) 27
28 Topology: Mathematics Euler Formula: V+F = E+2 Polyhedron Type of Faces F V E Tetrahedron Equilateral Triangles Octahedron Equilateral Triangles Cube Squares Dodecahedron Pentagons Icosahedrons Equilateral Triangles F = number of faces, V = number of vertices E = number of edges 28
29 Topology: Mathematics Cauchy Proof ( ) V+F=E+2 V+F=E+1 V+F+1=(E+1)+1 V+F=E+1 V+F=E+1 V+F=E+1 (V-2)+F=(E-2)+1 V+F=E+1 3+1=3+1!! 29
30 Topology: Mathematics Mesh Examples F = 3844 V = 1924 E = V + F 2 = 5766 F = V = E = S + F 2 =
31 Topology : Hierarchy 31
32 CSG : Constructive Solid Geometry Constructive Representation Parametric Volume Primitives Transformations Boolean Operators: Union, common, subtract Advantage : Concept of Graph Simple Description Simulation of Object «Manufacturing» 32
33 CSG: Primitive Components 33
34 CSG: Boolean Operators Union U 34
35 CSG: Boolean Operators Union U 35
36 CSG: Boolean Operators Subtract - U 36
37 CSG: Boolean Operators Common 37
38 CSG: Boolean Operators Fillet 38
39 Geometry: Curves x, y, z function of u, continuous first order Lap back point s(u) u Multiple point Arc length, abscissa curvilinear ds = dx + dy + dz τ(s) τ(s+ds) s( u) u u dx dy dz = + + du du du du u=u 0 39
40 Geometry: Curvature 40
41 Geometry: Curvature 41
42 Geometry: Curvature O n Ray of Osculator Circle = Curvature Ray OM = n ρ M For the curve with the equation y = f(x): Parametric curve: 42
43 Geometry: Vector and Tangential Frenet Reference 2 dx d x 2 ds ds 2 2 dom dy d OM d y τ ( u) = et n( u) = 2 2 ds ds ds ds dz 2 d z ds 2 ds n = Principal normal t OM(s+dh2) OM(s) OM(s+dh1) 43
44 Geometry: Torsion M(s+ds) M(s) Osculator Plane at M(s) Torsion 44
45 Geometry: Propeller Circular 2 d x d x = -R s in θ = -R c o s θ 2 x = R c o s d d θ θ θ 2 d y d y y = R s in θ, = R c o s θ, = -R s in θ 2 d θ d θ z = p θ d z 2 = p d z d θ = 0 2 d θ a in s i d s = d x + d y + d z = R + p e t d o n c s = R + p θ d x R 2 d x R = - s in θ = - c o s θ d s R + p d s R + p 2 d y R d τ d y R R τ = c o s θ e t = - s in θ a v e c ρ = d s 2 2 R + p d s d s R + p R + p 2 d z p d z = = d s R + p d s p s in θ 2 2 R p + -p d b p b = τ n c o s θ e t = T d o n c T = 2 2 R + p d s R + p R 2 2 R + p
46 Geometry: Propeller Circular Discretizing
47 Geometry: Frenet - Serret Reference, Equations, Curvature, Torsion The Frenet - Serret equations are a convenient framework for analyzing curvature. T(s) is the unit tangent to the curve as a function of path length s. N(s) is the unit normal to the curve B(s) is the unit binormal; the vector cross product of T(s) and N(s). Frenet Reference: Frenet Equations: For any parametric function f(t), the expression of the curvature and the torsion are the following: 47
48 Geometry: Curvature Gaussian, Average Gaussian Curvature Average Curvature Curvature Cmap 48
49 Geometry: Curvature Gaussian, Average Sphere Gaussian Curvature = 1 R 2 Average Curvature = 1 _ R 49
50 Geometry: Curvature Gaussian, Average Torus Gaussian Average 50
51 Geometry: Curvature Torus 51
52 Geometry: Surfaces of Revolution
53 Geometry: Particular Surfaces Mobius Strip: 'Endless Ribbon'
54 Geometry: Particular Surfaces The Klein Bottle 54
55 Geometry: Particular Surfaces The Klein Bottle 55
56 Geometry: Particular Surfaces The Klein Bottle: Curvature 56
57 Geometry: Particular Surfaces The Kuen Surfaces x=2*(cos(u)+u*sin(u))*sin(v)/(1+u*u*sin(v)*sin(v)) y=2*(sin(u)-u*cos(u))*sin(v)/(1+u*u*sin(v)*sin(v)) z=log(tan(v/2))+2*cos(v)/(1+u*u*sin(v)*sin(v)) 57
58 Geometry: Particular Surfaces The Dini Surfaces x=a*cos(u)*sin(v) y=a*sin(u)*sin(v) z=a*(cos(v)+log(tan((v/2))))+b*u a=1,b=0.2,u={ 0,4*pi},v={0.001,2} 58
59 Geometry: Particular Surfaces Asteroid x= pow (a*cos(u)*cos(v),3) y= pow (b*sin(u)*cos(v),3) z= pow (c*sin(v),3) 59
60 Geometry: Particular Surfaces The «Derviche» 60
61 Geometry: Curves Lagrange Interpolating Polynomial The Lagrange interpolating polynomial is the polynomial P(x) of degree <= (n-1) that passes through the n points (x 1,y 1 = f(x 1 )), x 2,y 2 = f(x 2 )),..., x n,y n = f(x n )), and is given by: 61
62 Geometry: Curves Lagrange Interpolating Polynomial Where: Written explicitly: 62
63 Geometry: Curves Cubic Spline Interpolating Polynomial A cubic spline is a spline constructed of piecewise third-order polynomials which pass through a set of m control points. The second derivative of each polynomial is commonly set to zero at the endpoints, since this provides a boundary condition that completes the system of m -2 equations. This produces a so-called "natural" cubic spline and leads to a simple tridiagonal system which can be solved easily to give the coefficients of the polynomials. However, this choice is not the only one possible, and other boundary conditions can be used instead. 63
64 Geometry: Curves Cubic Spline Interpolating Polynomial Consider 1-dimensional spline for a set of n+1 points (y 1, y 2,.., y n ), let the ith piece of the spline be represented by: Where t is a parameter and i = 0,, n-1 then Rearranging all these equations, leads to the following beautifully symmetric tridiagonal system: If the curve is instead closed, the system becomes 64
65 Geometry: Curves Cubic Spline/Lagrange
66 Geometry: Curves Bézier Given a set of n + 1 control points P 0, P 1,.., P n, the corresponding Bèzier curve (or Bernstein- Bèzier curve) is given by: Where B i,n (t) is a Bernstein polynomial and. A "rational" Bézier curve is defined by: where p is the order, B i,p are the Bernstein polynomials, P i are control points, and the weight W i of P i is the last ordinate of the homogeneous Point P iw. These curves are closed under perspective transformations, and can represent conic sections exactly. 66
67 Geometry: Curves Bézier:Properties The Bézier curve always passes through the first and last control points. The curve is tangent to P1 P0 and Pn Pn-1 at the endpoints. The curve lies within the convex hull of the control points. 67
68 Geometry: Curves Bézier:Properties A desirable property is that the curve can be translated and rotated by performing these Operations on the control points. Undesirable properties of Bézier curves are their numerical instability for large numbers of control points, and the fact that moving a single control point changes the global shape of the curve. 68
69 Geometry: Curves Bézier: Bernstein Polynomials B in (t) = C ni (t-1) n-i t i, C ni = n! / i!(n-i)! B j i (t) = (1-t)B j-1 i (t) + t B j-1 i-1 (t) B 1 0 (t) = (1-t)B 0 0 (t) + t B -10 (t) B 1 1 (t) = (1-t)B 0 1 (t) + t B 00 (t) B 0 2 (t) = (1-t)B 0 1 (t) + t B -11 (t) 69
70 Geometry: Curves Bézier: Bernstein Polynomials B i j (t) = (1-t)B i j-1 (t) + t B i-1 j-1 (t) Unit Partition: Σ i=0,n B in (t) = 1 0<=B in (t)<= 1 B in (0) = 0 et B in (1) = 0 B 0n (0) = 1 B nn (1) = 1 70
71 Geometry: Curves B-Spline A B-Spline is a generalization of the Bézier curve. Let a vector known as the knot vector be defined T = {t 0, t 1,, t m }, where T is a no decreasing sequence with, and define control points P 0,..., P n. Define the degree as: p = m-n-1 The "knots t p+1,..., t m-p-1 are called internal knots. 71
72 Geometry: Curves B-Spline Define the basis functions as: Then the curve defined by: is a B-spline. Specific types include the non periodic B-spline (first p+1 knots equal 0 and last p+1 equal to 1; illustrated above) and uniform B-spline (internal knots are equally spaced). A B-spline with no internal knots is a Bézier curve. A curve is p - k times differentiable at a point where k duplicate knot values occur. 72
73 Geometry: Curves NURBS-Curve A non uniform rational B-spline curve defined by: where p is the order, N i,p are the B-Spline basis functions, P i are control points, and the weight W i of P i is the last ordinate of the homogeneous point P iw. These curves are closed under perspective transformations, and can represent conic sections exactly. 73
74 Geometry: Curves Conics P(t) = i = 3 i = 0 i = 3 wi Ni, 2( t) Pi( t) i =00 wi Ni, 2( t) Parabola w=1 (0,0,0,1,1,1) Hyperbole w=4 Ellipse w=1/4 74
75 Geometry: Curves Arcs NURBS of degree 2 Control points (Isosceles triangle) Knot vector (0,0,0,1,1,1) 75
76 Geometry: Curves Circles (0, 0, 0, 1/3, 1/3, 2/3, 2/3, 1, 1, 1) (0, 0, 0, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 1, 1, 1) 76
77 Geometry: Surfaces Bézier A given Bézier surface of order (n, m) is defined by a set of (n + 1)(m + 1) control points ki,j. evaluated over the unit square, where: is a Bernstein polynomial, and is the binomial coefficient. 77
78 Geometry: Surfaces Bézier S(u,v) = Σ i=0,n Σ j=0,m B in (u) B jm (v) P ij (n+1)(n+1) points P ij P i (v) = Σ j=0,m B jm (v)p ij 78
79 Geometry: Surfaces NURBS NURBS are nearly ubiquitous for computer-aided design (CAD), manufacturing (CAM), and engineering (CAE) and are part of numerous industry wide used standards, such as IGES, STEP, ACIS, Parasolid. 79
80 Geometry: Surfaces NURBS: Properties NURBS curves and surfaces are useful for a number of reasons: They are invariant under affine as well as perspective transformations. They offer one common mathematical form for both standard analytical shapes (e.g., conics) and free-form shapes. They provide the flexibility to design a large variety of shapes. They reduce the memory consumption when storing shapes (compared to simpler methods). They can be evaluated reasonably quickly by numerically stable and accurate algorithms. They are generalizations of non-rational B-Splines and non-rational and rational Bézier curves and surfaces. 80
3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
More information3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
More informationIntroduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry
Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry Representation Ab initio design Rendering Solid modelers Kinematic
More informationCOMPUTER AIDED ENGINEERING DESIGN (BFF2612)
COMPUTER AIDED ENGINEERING DESIGN (BFF2612) BASIC MATHEMATICAL CONCEPTS IN CAED by Dr. Mohd Nizar Mhd Razali Faculty of Manufacturing Engineering mnizar@ump.edu.my COORDINATE SYSTEM y+ y+ z+ z+ x+ RIGHT
More informationLECTURE #6. Geometric Modelling for Engineering Applications. Geometric modeling for engineering applications
LECTURE #6 Geometric modeling for engineering applications Geometric Modelling for Engineering Applications Introduction to modeling Geometric modeling Curve representation Hermite curve Bezier curve B-spline
More informationGL9: Engineering Communications. GL9: CAD techniques. Curves Surfaces Solids Techniques
436-105 Engineering Communications GL9:1 GL9: CAD techniques Curves Surfaces Solids Techniques Parametric curves GL9:2 x = a 1 + b 1 u + c 1 u 2 + d 1 u 3 + y = a 2 + b 2 u + c 2 u 2 + d 2 u 3 + z = a
More informationCS3621 Midterm Solution (Fall 2005) 150 points
CS362 Midterm Solution Fall 25. Geometric Transformation CS362 Midterm Solution (Fall 25) 5 points (a) [5 points] Find the 2D transformation matrix for the reflection about the y-axis transformation (i.e.,
More informationDesign considerations
Curves Design considerations local control of shape design each segment independently smoothness and continuity ability to evaluate derivatives stability small change in input leads to small change in
More informationGeometric Modeling Systems
Geometric Modeling Systems Wireframe Modeling use lines/curves and points for 2D or 3D largely replaced by surface and solid models Surface Modeling wireframe information plus surface definitions supports
More informationRational Bezier Curves
Rational Bezier Curves Use of homogeneous coordinates Rational spline curve: define a curve in one higher dimension space, project it down on the homogenizing variable Mathematical formulation: n P(u)
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationBezier Curves. An Introduction. Detlef Reimers
Bezier Curves An Introduction Detlef Reimers detlefreimers@gmx.de http://detlefreimers.de September 1, 2011 Chapter 1 Bezier Curve Basics 1.1 Linear Interpolation This section will give you a basic introduction
More informationGeometric Modeling of Curves
Curves Locus of a point moving with one degree of freedom Locus of a one-dimensional parameter family of point Mathematically defined using: Explicit equations Implicit equations Parametric equations (Hermite,
More informationBézier Splines. B-Splines. B-Splines. CS 475 / CS 675 Computer Graphics. Lecture 14 : Modelling Curves 3 B-Splines. n i t i 1 t n i. J n,i.
Bézier Splines CS 475 / CS 675 Computer Graphics Lecture 14 : Modelling Curves 3 n P t = B i J n,i t with 0 t 1 J n, i t = i=0 n i t i 1 t n i No local control. Degree restricted by the control polygon.
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationCurve 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 informationCentral 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 informationGEOMETRIC TOOLS FOR COMPUTER GRAPHICS
GEOMETRIC TOOLS FOR COMPUTER GRAPHICS PHILIP J. SCHNEIDER DAVID H. EBERLY MORGAN KAUFMANN PUBLISHERS A N I M P R I N T O F E L S E V I E R S C I E N C E A M S T E R D A M B O S T O N L O N D O N N E W
More informationCS 475 / CS Computer Graphics. Modelling Curves 3 - B-Splines
CS 475 / CS 675 - Computer Graphics Modelling Curves 3 - Bézier Splines n P t = i=0 No local control. B i J n,i t with 0 t 1 J n,i t = n i t i 1 t n i Degree restricted by the control polygon. http://www.cs.mtu.edu/~shene/courses/cs3621/notes/spline/bezier/bezier-move-ct-pt.html
More informationSuggested List of Mathematical Language. Geometry
Suggested List of Mathematical Language Geometry Problem Solving A additive property of equality algorithm apply constraints construct discover explore generalization inductive reasoning parameters reason
More information2D Spline Curves. CS 4620 Lecture 13
2D Spline Curves CS 4620 Lecture 13 2008 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes [Boeing] that is, without discontinuities So far we can make things with corners
More information2D Spline Curves. CS 4620 Lecture 18
2D Spline Curves CS 4620 Lecture 18 2014 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes that is, without discontinuities So far we can make things with corners (lines,
More informationConstruction and smoothing of triangular Coons patches with geodesic boundary curves
Construction and smoothing of triangular Coons patches with geodesic boundary curves R. T. Farouki, (b) N. Szafran, (a) L. Biard (a) (a) Laboratoire Jean Kuntzmann, Université Joseph Fourier Grenoble,
More informationParametric curves. Brian Curless CSE 457 Spring 2016
Parametric curves Brian Curless CSE 457 Spring 2016 1 Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics
More informationCurve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur
Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur Email: jrkumar@iitk.ac.in Curve representation 1. Wireframe models There are three types
More informationLecture IV Bézier Curves
Lecture IV Bézier Curves Why Curves? Why Curves? Why Curves? Why Curves? Why Curves? Linear (flat) Curved Easier More pieces Looks ugly Complicated Fewer pieces Looks smooth What is a curve? Intuitively:
More informationNeed for Parametric Equations
Curves and Surfaces Curves and Surfaces Need for Parametric Equations Affine Combinations Bernstein Polynomials Bezier Curves and Surfaces Continuity when joining curves B Spline Curves and Surfaces Need
More informationComputergrafik. Matthias Zwicker Universität Bern Herbst 2016
Computergrafik Matthias Zwicker Universität Bern Herbst 2016 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling 2 Piecewise Bézier curves Each
More informationGeometric Modeling Topics
Geometric Modeling Topics George Allen, george.allen@siemens.com Outline General background Convergent modeling Multi-material objects Giga-face lattices Page 2 Boundary Representation (b-rep) Topology
More informationShape Representation Basic problem We make pictures of things How do we describe those things? Many of those things are shapes Other things include
Shape Representation Basic problem We make pictures of things How do we describe those things? Many of those things are shapes Other things include motion, behavior Graphics is a form of simulation and
More informationVALLIAMMAI 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 informationCurves and Curved Surfaces. Adapted by FFL from CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006
Curves and Curved Surfaces Adapted by FFL from CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006 Outline for today Summary of Bézier curves Piecewise-cubic curves, B-splines Surface
More informationSpline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong
More informationGeometric Modeling. Introduction
Geometric Modeling Introduction Geometric modeling is as important to CAD as governing equilibrium equations to classical engineering fields as mechanics and thermal fluids. intelligent decision on the
More informationReview 1. Richard Koch. April 23, 2005
Review Richard Koch April 3, 5 Curves From the chapter on curves, you should know. the formula for arc length in section.;. the definition of T (s), κ(s), N(s), B(s) in section.4. 3. the fact that κ =
More informationCurriculum Vitae of the Authors
Curriculum Vitae of the Authors Mario Hirz has been awarded an M.S. degree in mechanical engineering and economics, a Ph.D. in mechanical engineering, and a venia docendi in the area of virtual product
More informationWe 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 informationSurfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
More informationGeometric 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 informationGeometric and Solid Modeling. Problems
Geometric and Solid Modeling Problems Define a Solid Define Representation Schemes Devise Data Structures Construct Solids Page 1 Mathematical Models Points Curves Surfaces Solids A shape is a set of Points
More information9. Three Dimensional Object Representations
9. Three Dimensional Object Representations Methods: Polygon and Quadric surfaces: For simple Euclidean objects Spline surfaces and construction: For curved surfaces Procedural methods: Eg. Fractals, Particle
More informationUntil now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple
Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of
More informationCS Object Representation. Aditi Majumder, CS 112 Slide 1
CS 112 - Object Representation Aditi Majumder, CS 112 Slide 1 What is Graphics? Modeling Computer representation of the 3D world Analysis For efficient rendering For catering the model to different applications..
More informationCurves 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 informationComputergrafik. Matthias Zwicker. Herbst 2010
Computergrafik Matthias Zwicker Universität Bern Herbst 2010 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling Piecewise Bézier curves Each segment
More informationSpline Notes. Marc Olano University of Maryland, Baltimore County. February 20, 2004
Spline Notes Marc Olano University of Maryland, Baltimore County February, 4 Introduction I. Modeled after drafting tool A. Thin strip of wood or metal B. Control smooth curved path by running between
More informationCurves and Surfaces. CS475 / 675, Fall Siddhartha Chaudhuri
Curves and Surfaces CS475 / 675, Fall 26 Siddhartha Chaudhuri Klein bottle: surface, no edges (Möbius strip: Inductiveload@Wikipedia) Möbius strip: surface, edge Curves and Surfaces Curve: D set Surface:
More informationCurves. Computer Graphics CSE 167 Lecture 11
Curves Computer Graphics CSE 167 Lecture 11 CSE 167: Computer graphics Polynomial Curves Polynomial functions Bézier Curves Drawing Bézier curves Piecewise Bézier curves Based on slides courtesy of Jurgen
More informationChapter 4-3D Modeling
Chapter 4-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 1 The 3D rendering
More information(Discrete) Differential Geometry
(Discrete) Differential Geometry Motivation Understand the structure of the surface Properties: smoothness, curviness, important directions How to modify the surface to change these properties What properties
More informationTO DUY ANH SHIP CALCULATION
TO DUY ANH SHIP CALCULATION Ship Calculattion (1)-Space Cuvers 3D-curves play an important role in the engineering, design and manufature in Shipbuilding. Prior of the development of mathematical and computer
More information08 - Designing Approximating Curves
08 - Designing Approximating Curves Acknowledgement: Olga Sorkine-Hornung, Alexander Sorkine-Hornung, Ilya Baran Last time Interpolating curves Monomials Lagrange Hermite Different control types Polynomials
More informationCSG obj. oper3. obj1 obj2 obj3. obj5. obj4
Solid Modeling Solid: Boundary + Interior Volume occupied by geometry Solid representation schemes Constructive Solid Geometry (CSG) Boundary representations (B-reps) Space-partition representations Operations
More informationIntroduction to Geometry. Computer Graphics CMU /15-662
Introduction to Geometry Computer Graphics CMU 15-462/15-662 Assignment 2: 3D Modeling You will be able to create your own models (This mesh was created in Scotty3D in about 5 minutes... you can do much
More informationCOMPUTER 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 informationCurves and Surfaces for Computer-Aided Geometric Design
Curves and Surfaces for Computer-Aided Geometric Design A Practical Guide Fourth Edition Gerald Farin Department of Computer Science Arizona State University Tempe, Arizona /ACADEMIC PRESS I San Diego
More informationCurves D.A. Forsyth, with slides from John Hart
Curves D.A. Forsyth, with slides from John Hart Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction
More informationEstimating normal vectors and curvatures by centroid weights
Computer Aided Geometric Design 21 (2004) 447 458 www.elsevier.com/locate/cagd Estimating normal vectors and curvatures by centroid weights Sheng-Gwo Chen, Jyh-Yang Wu Department of Mathematics, National
More informationCS130 : Computer Graphics Curves. Tamar Shinar Computer Science & Engineering UC Riverside
CS130 : Computer Graphics Curves Tamar Shinar Computer Science & Engineering UC Riverside Design considerations local control of shape design each segment independently smoothness and continuity ability
More informationComputer Graphics Curves and Surfaces. Matthias Teschner
Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves
More information2Surfaces. Design with Bézier Surfaces
You don t see something until you have the right metaphor to let you perceive it. James Gleick Surfaces Design with Bézier Surfaces S : r(u, v) = Bézier surfaces represent an elegant way to build a surface,
More information(Refer Slide Time: 00:02:24 min)
CAD / CAM Prof. Dr. P. V. Madhusudhan Rao Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture No. # 9 Parametric Surfaces II So these days, we are discussing the subject
More informationFrom curves to surfaces. Parametric surfaces and solid modeling. Extrusions. Surfaces of revolution. So far have discussed spline curves in 2D
From curves to surfaces Parametric surfaces and solid modeling CS 465 Lecture 12 2007 Doug James & Steve Marschner 1 So far have discussed spline curves in 2D it turns out that this already provides of
More informationWeek 7 Convex Hulls in 3D
1 Week 7 Convex Hulls in 3D 2 Polyhedra A polyhedron is the natural generalization of a 2D polygon to 3D 3 Closed Polyhedral Surface A closed polyhedral surface is a finite set of interior disjoint polygons
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationMotivation. Parametric Curves (later Surfaces) Outline. Tangents, Normals, Binormals. Arclength. Advanced Computer Graphics (Fall 2010)
Advanced Computer Graphics (Fall 2010) CS 283, Lecture 19: Basic Geometric Concepts and Rotations Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283/fa10 Motivation Moving from rendering to simulation,
More informationFlank 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 information3.2 THREE DIMENSIONAL OBJECT REPRESENTATIONS
3.1 THREE DIMENSIONAL CONCEPTS We can rotate an object about an axis with any spatial orientation in threedimensional space. Two-dimensional rotations, on the other hand, are always around an axis that
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationGeometry. Chapter 5. Types of Curves and Surfaces
Chapter 5. Geometry Geometry refers to the physical items represented by the model (such as points, curves, and surfaces), independent of their spatial or topological relationships. The ACIS free form
More informationMathematical Tools in Computer Graphics with C# Implementations Table of Contents
Mathematical Tools in Computer Graphics with C# Implementations by Hardy Alexandre, Willi-Hans Steeb, World Scientific Publishing Company, Incorporated, 2008 Table of Contents List of Figures Notation
More informationImplicit Surfaces & Solid Representations COS 426
Implicit Surfaces & Solid Representations COS 426 3D Object Representations Desirable properties of an object representation Easy to acquire Accurate Concise Intuitive editing Efficient editing Efficient
More informationParametric curves. Reading. Curves before computers. Mathematical curve representation. CSE 457 Winter Required:
Reading Required: Angel 10.1-10.3, 10.5.2, 10.6-10.7, 10.9 Parametric curves CSE 457 Winter 2014 Optional Bartels, Beatty, and Barsky. An Introduction to Splines for use in Computer Graphics and Geometric
More informationCS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside
CS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside Blending Functions Blending functions are more convenient basis than monomial basis canonical form (monomial
More informationLecture 4b. Surface. Lecture 3 1
Lecture 4b Surface Lecture 3 1 Surface More complete and less ambiguous representation than its wireframe representation Can be considered as extension to wireframe representation In finite element, surface
More informationBézier and B-spline volumes Project of Dissertation
Department of Algebra, Geometry and Didactics of Mathematics Faculty of Mathemathics, Physics and Informatics Comenius University, Bratislava Bézier and B-spline volumes Project of Dissertation Martin
More information1. INTRODUCTION ABSTRACT
Copyright 2008, Society of Photo-Optical Instrumentation Engineers (SPIE). This paper was published in the proceedings of the August 2008 SPIE Annual Meeting and is made available as an electronic preprint
More informationMeasuring 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 informationGrade 9 Math Terminology
Unit 1 Basic Skills Review BEDMAS a way of remembering order of operations: Brackets, Exponents, Division, Multiplication, Addition, Subtraction Collect like terms gather all like terms and simplify as
More informationAnswer Key: Three-Dimensional Cross Sections
Geometry A Unit Answer Key: Three-Dimensional Cross Sections Name Date Objectives In this lesson, you will: visualize three-dimensional objects from different perspectives be able to create a projection
More informationA New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces
A New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces Mridula Dube 1, Urvashi Mishra 2 1 Department of Mathematics and Computer Science, R.D. University, Jabalpur, Madhya Pradesh, India 2
More informationThe goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a
The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with
More informationECE 600, Dr. Farag, Summer 09
ECE 6 Summer29 Course Supplements. Lecture 4 Curves and Surfaces Aly A. Farag University of Louisville Acknowledgements: Help with these slides were provided by Shireen Elhabian A smile is a curve that
More informationComputational Physics PHYS 420
Computational Physics PHYS 420 Dr Richard H. Cyburt Assistant Professor of Physics My office: 402c in the Science Building My phone: (304) 384-6006 My email: rcyburt@concord.edu My webpage: www.concord.edu/rcyburt
More informationPhysically-Based Modeling and Animation. University of Missouri at Columbia
Overview of Geometric Modeling Overview 3D Shape Primitives: Points Vertices. Curves Lines, polylines, curves. Surfaces Triangle meshes, splines, subdivision surfaces, implicit surfaces, particles. Solids
More informationModeling 3D Objects: Part 2
Modeling 3D Objects: Part 2 Patches, NURBS, Solids Modeling, Spatial Subdivisioning, and Implicit Functions 3D Computer Graphics by Alan Watt Third Edition, Pearson Education Limited, 2000 General Modeling
More informationCurves and Surfaces 1
Curves and Surfaces 1 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized Modeling Techniques 2 The Teapot 3 Representing
More informationThe Free-form Surface Modelling System
1. Introduction The Free-form Surface Modelling System Smooth curves and surfaces must be generated in many computer graphics applications. Many real-world objects are inherently smooth (fig.1), and much
More informationComputer Graphics Splines and Curves
Computer Graphics 2015 9. Splines and Curves Hongxin Zhang State Key Lab of CAD&CG, Zhejiang University 2015-11-23 About homework 3 - an alternative solution with WebGL - links: - WebGL lessons http://learningwebgl.com/blog/?page_id=1217
More informationCSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016
CSE 167: Introduction to Computer Graphics Lecture #11: Bezier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016 Announcements Project 3 due tomorrow Midterm 2 next
More informationPS Geometric Modeling Homework Assignment Sheet I (Due 20-Oct-2017)
Homework Assignment Sheet I (Due 20-Oct-2017) Assignment 1 Let n N and A be a finite set of cardinality n = A. By definition, a permutation of A is a bijective function from A to A. Prove that there exist
More informationalgebraic representation algorithm alternate interior angles altitude analytical geometry x x x analytical proof x x angle
Words PS R Comm CR Geo R Proof Trans Coor Catogoriers Key AA triangle similarity Constructed Response AAA triangle similarity Problem Solving AAS triangle congruence Resoning abscissa Communication absolute
More informationThe radius for a regular polygon is the same as the radius of the circumscribed circle.
Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.
More informationCSE 167: Introduction to Computer Graphics Lecture #13: Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2017
CSE 167: Introduction to Computer Graphics Lecture #13: Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2017 Announcements Project 4 due Monday Nov 27 at 2pm Next Tuesday:
More informationShape Modeling and Geometry Processing
252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry
More informationCOMP3421. Global Lighting Part 2: Radiosity
COMP3421 Global Lighting Part 2: Radiosity Recap: Global Lighting The lighting equation we looked at earlier only handled direct lighting from sources: We added an ambient fudge term to account for all
More informationCurves and Surfaces. Shireen Elhabian and Aly A. Farag University of Louisville
Curves and Surfaces Shireen Elhabian and Aly A. Farag University of Louisville February 21 A smile is a curve that sets everything straight Phyllis Diller (American comedienne and actress, born 1917) Outline
More informationLog1 Contest Round 2 Theta Circles, Parabolas and Polygons. 4 points each
Name: Units do not have to be included. 016 017 Log1 Contest Round Theta Circles, Parabolas and Polygons 4 points each 1 Find the value of x given that 8 x 30 Find the area of a triangle given that it
More informationReading. Parametric surfaces. Surfaces of revolution. Mathematical surface representations. Required:
Reading Required: Angel readings for Parametric Curves lecture, with emphasis on 11.1.2, 11.1.3, 11.1.5, 11.6.2, 11.7.3, 11.9.4. Parametric surfaces Optional Bartels, Beatty, and Barsky. An Introduction
More informationOne simple example is that of a cube. Each face is a square (=regular quadrilateral) and each vertex is connected to exactly three squares.
Berkeley Math Circle Intermediate I, 1/23, 1/20, 2/6 Presenter: Elysée Wilson-Egolf Topic: Polygons, Polyhedra, Polytope Series Part 1 Polygon Angle Formula Let s start simple. How do we find the sum of
More information