Lecture 4b. Surface. Lecture 3 1

Transcription

1 Lecture 4b Surface Lecture 3 1

2 Surface More complete and less ambiguous representation than its wireframe representation Can be considered as extension to wireframe representation In finite element, surface is used in meshing Widely used in animation, virtual reality, automotive, ship etc Lecture 3 2

3 Types of Surface path 2D Profile (a) (b) Swept surface Bezier Curves Freeform surfaces (a) (b) (c) Surface of revolution (a) Lecture 3 3 (b) Freeform surface

4 Surface Surface is the extension of parametric curves Parametric Curve P(t) = [x(t) y(t), z(t)] Surface Representation P(s,t) = [x(s,t), y(s,t), z(s,t)] Lecture 3 4

5 Fact on Surface Freeform surfaces are created by parametric curves. The smoothness of the curves depends on the curves they are created. NURBS is created by non-uniform rational B Spline curve Lecture 3 5

6 Lecture 4c Geometric Modelling Lecture 3 6

7 Criteria for Solid Modeling Boundary 3D homogenous objects: no line or surface attach to one of the face Finite: must have size and shape Lecture 3 7

8 Type of solid modeling Primitive objects Extrude and revolved bodies Cellular decomposition Simple regular grid, octree adaptive grid, general adaptive grid Constructive Solid geometry Boundary Representation Cloud of points Lecture 3 8

9 Primitive Object Sphere Cone Cylinder Torus Box Extruded revolved 2D Profile bodies Axis Extrusion Revolved Lecture 3 9

10 Cellular decomposition Disadvantages Lecture 3 10

11 Octree adaptive grid Lecture 3 11

12 Constructive Solid Geometry A B C D E F G A U B C U D F U G H (C U D) - (A U B) J ((C U D) - (A U B)) - (F U G) - H - J Lecture 3 12

13 Half space in CSG Top Planar surface Left side Planar surface Cylindrical surface Rear Right Side Bottom 6 Half space plane face Front (a) (b) Lecture 3 13

14 Boundary Representation Topology Geometry Solid Vertex Edge Loop Face Loop Surface Edge Curve Face Solid Vertex Node Lecture 3 14

15 B-Reps: Geometry Point : has no dimensions, only position (x, y, z) Curve: is one-dimensional (parametric equation) Surface: two dimensional (2D) Solid : three-dimensional (3D) Therefore, in order to get the whole geometrical properties of the solid, all the topology and geometry must be visited. Topology Solid Face Loop Geometry Surface Edge Curve Vertex Node Lecture 3 15

16 B-Reps: Surface Normal Vector : N Surface vector : V Ref vertex: P P N Plane V N P V Cylinder Types of surface depends on the modeller Some modeller or translator combines cylindrical and conical surface as one surface P N V P N V Surface is described by normal vector, surface vector and ref vertex Cone Surface Sphere Based on the description, surface has infinite area Lecture 3 16

17 Surface: Plane N A C B Implicit Equations Normal Vector N = < a, b, c > Plane eq a (x x0)+b(y y0)+c(z z0) = 0 Parametric Equations Based on 3 points on the plane R( s, t) OA sab t AC Therefore, equation shows that plane will have infinite area Question: How do plane surface maintains its boundary? Lecture 3 17

18 Plane: Example N R( s, t) OA sab t AC A C B A = ( 0, 1, -1), B = (2, -1,0), C=(0,0,1) OA 0,1, 1 AB 2, 2,1 AC 0, 1, 2 R( s, t) 0,1, 1 s 2, 2,1 t 0, 1, 2 Lecture 3 18

19 Surface: Cylindrical and N P V Cone N P V Cylinder Implicit Equations x 2 + y 2 = R 2 and z= z Parametric Equation X = R cos(2*pi*t) Y = R sin(2*pi*t) Z= z(s) Since z relies on other free parameter, cylindrical surface can have infinite length Moreover, if radius is governed by equation that causes the other radius will be larger, smaller or equal to 0 cylindrical and cone can be grouped together. Lecture 3 19

20 Loop Loop is topology without geometry The purpose of loop is to maintain the void and solid of the model, as such point can be detected whether to be on, inside or outside the solid Two type of loops Outer loop: ccw direction Inner loop: cw direction Lecture 3 20

21 Loop: Invalid closed Loop Crossed Edges Open Profile Invalid Nested Profile Lecture 3 21

22 B-Reps: Geometry Curve: as discussed in parametric curve Parametric Curve Node: coordinate The geometry of curve is coordinate (x,y, z) Lecture 3 22

23 How to access topology and geometry data swface = swbody.getfirstface() for (i = 0; i < no_face ; i++) no_loop =swface.get_loop_no swloop = swface.getfirstloop() for (j = 0; j < no_loop ; j++) no_egde =swloop.get_face_no() swedge = swloop.getfirstedge() for (k = 0; k < no_edge; k++) swcurve = swedge.getcurveparam() no_vertex = swedge.getnovertex() swvertex = swedge.getfirstvertex() for ( l = 0; l < no_vertex ; l++) point = swvertex.getvertexparam(). swvertex = swedge.getnextvertex() end for swedge = swloop.getnextedge() end for swloop = swloop.getnextloop() end for surface = swface.getfaceparam() swface = swbody.getnextface() End for Lecture 3 23

24 Case: to retrieve the boundary of a face BODY FACE On each face; retrieve the geometrical properties of the surface Reference point Normal Vector surface vector Can be used to determine the orientation of the face LOOP EDGE VERTEX On each vertex; retrieve the point. Compile all the vertices from all the edges the boundary of the face can be determined Lecture 3 24

25 B-Reps: Graph Representation E E edge edge V vertex face face F E F vertex V edge edge E E Lecture 3 25

26 Euler Poincare Law 2(B-G)=V-E+F-I B: Body G: genus V: vertex E: edges F: Face I: Inner loop V = 8, E = 12, F = 6, I = 0 G = 0 2(B -G) = V E + F I B = 1 (one object only) V = 16, E = 24, F = 10, I = 2 G = 1 B = 1 (one object only) V=2, E=3, F=3, I = 0 G = 0 B = 1 (one object only) V = 12, E = 16, F = 6, I = 1 G = 0 B = 0.5 (object cannot be built) Lecture 3 26

27 CSG and B-Reps: understanding Using the Euler- Poincare Law, show that both bodies is 1 object? Give the description of these models according to CSG and B-Reps Modeler? *The description should be as detail as to half space for CSG and topology and geometry information for B-Reps Lecture 3 27

28 Part 1 E = 42 V = 28 F = 16 I = 0 G = 0 Part 2 E = 18 V = 12 F = 8 I = 4 G = 2 Lecture 3 28

29 Solid Kernel ACIS AutoCAD, Inventor ParaSolid Solid Edge, Unigraphics Lecture 3 29

30 Cloud of points Reverse engineering using 3D scanner or Contact Measuring Machines (CMM) digitizer List points Surface Solid Sample software: Geomagic, RapidForm, HandyScanning Module for CATIA "Piggy bank" one scan, 150,000 points, 1.1 Mb four scans, 600,000 points, 3.7 Mb Lecture 3 30

31 Issue on cloud point Efficient mathematical model such as smoothing, capacity to handle huge data, error detection, issue of controlling curves or subparts Application: design of cloth, 3D anthropometric data Lecture 3 31

32 What is the difference? Wireframe model Solid model Lecture 3 32