Solid Modelling. Graphics Systems / Computer Graphics and Interfaces COLLEGE OF ENGINEERING UNIVERSITY OF PORTO

Transcription

1 Solid Modelling Graphics Systems / Computer Graphics and Interfaces 1

2 Solid Modelling In 2D, one set 2D line segments or curves does not necessarily form a closed area. In 3D, a collection of surfaces does not necessarily involve a closed volume. Solid modeling: In some applications it is important to: distinguish between the interior and exterior surface of a 3D object; and evaluate properties of objects that depend on this distinction. Ex: Simulation of mechanisms, volume, center of mass, application of finite elements to determine response to factors such as stress and temperature, etc. Applications: CAD / CAM and photo-realistic imaging. 2

3 Characteristics of a solid model 1. Should cover a domain representation broad enough to incorporate all kinds of objects we want to model. 2. The representation must be unambiguous and unique: A given representation must correspond to a single solid; and each object must have only one possible representation. The only representation allows us to compare two objects to determine equality. 3. Precision / Correctness: accurate modeling allows to represent the object without approximations. Systems that only accept representation by line segments only approximate curved surfaces. 4. Impossibility of creating invalid objects, i.e. that do not correspond to a solid. 5. Closed representation: The representation must remain valid after the application of any valid operations. 6. Compact representation to optimize the use of memory. 3

4 Characteristics of a solid model Example of invalid objects as a solid. - The representation of a) does not clearly identify the faces of the cube, only indicates edges. - We can consider that a sequence of 4 segments form a face? But solid b) would be (wrong) considered as a solid. In general, the representations used do not have all the features presented. 4

5 Boolean operations The combination of Boolean operations allows to define new objects, independently of the representation used. Operations are: union, difference / subtraction intersection. a) Objects A and B b) The U B c) A B d) A - B e) B - A 5

6 Operating two objects... Consider the two objects, CUBE and CYLINDER. 6

7 Example: CUBE - CYLINDER Subtraction 7

8 Union Example: CUBE U CYLINDER 8

9 Example: CUBE CYLINDER Intersection 9

10 Example - Ask Lego performed with Boolean operations. The solids are used cube and cylinder. This type of modeling is mainly used for regular objects as exemplified. 9

11 Types of Representation 1. Representation by Instantiating Primitives 2. Representation by Extrusion 3. Boundary Representation (Representação pela Fronteira) 4. Representation by Spatial Decomposition 5. Constructive Solid Geometry Representation (CSG) 10

12 Representation by Instantiating Primitives The modeling system has, pre-defined, a set of useful 3D solid for the desired modeling. The user can control the shape of the object defining the parameters that characterize it. It does not include the combination of objects as Boolean operations. Used for complex parts. 11

13 Representation by Extrusion The displacement of an object according to a trajectory defines another object: Translation (Extrusion) Rotation Ex: The translation a 2D rectangle along its perpendicular to the plane creates a parallelepiped. A simple extension is to modify the dimensions of the 2D object along the path. 12

14 Representation by Extrusion Using this method without path constraints can result in inefficient modeling of the object. Ex: If the object intersects itself complicates the calculation of volume. Can not generate a valid solid if the motion is in the plane containing the 2D shape. In general, software tools convert objects created by extrusion into other representations of the same objects. Boolean operations with objects created by extrusion. 13

15 Representation by Extrusion - Example 14

16 Representation by Extrusion - Example 1. Defining a way to make scanning for translation. 15

17 Representation by Extrusion - Example 2. Defining the shape of the section of the final object. 16

18 Representation by Extrusion - Example Object obtained by translation. 17

19 Representation by Extrusion - Example Object obtained by translation, rotation around the axis shifting and scaling along the route. 18

20 Boundary Representation (B-rep) The solids are described by its boundary surface. Uses the description by vertices, edges and faces. The most common representation is the boundary by a closed polygonal mesh. Will be considered only the solids with boundary 2-manifolds : Ie the neighbors of any point of the boundary surfasse are in a disk (that is to say that each edge is shared by two faces) (a) and (b) are 2-manifold (c) is not 2-manifold 19

21 Boundary Representation (B-rep) Polyhedron Solid delimited by a set of polygons whose edges belong to two polygons (solids 2- manifolds). Euler Formula A simple polyhedron without holes, obeys Euler's formula: V - E + F = 2 V - Vertex E - Edges (edges) F - Faces 20

22 By Boundary Representation (B-rep) The Euler's formula is necessary but it is not sufficient to ensure that an object is a simple polyhedron / valid solid. Additional conditions: 1. Each edge connects 2 vertices and is shared by 2 faces 2. At least three edges are at the same vertex 21

23 By Boundary Representation (B-rep) Generalization of Euler's Formula for polyhedra with holes: V - E + F - H = 2 (C - G) V - Vertex E - Edges (edges) F - Faces H - number of holes in the faces G - Number of holes crossing the object C - number of parts of the object 21

24 Exercise 22

25 Representation by Spatial Decomposition A solid is decomposed into: In a number of more primitive solids than the original The primitive solids are adjacent and do not intersect Types of Representation for Spatial Decomposition Cell Decomposition Enumeration of Space Occupation Octrees Binary trees Space Partition 23

26 Representation by Spatial Decomposition Cell Decomposition In Cell Decomposition: There is a set of primitive, parameterized cells Can be Curves Differs from Instance Primitives, by admitting the composition of more complex objects from other already established Gluing operation It is a union of cells that do not intersect a) Primitive cells to transform b) c) are the same final object created with different combinations 24

27 Representation by Spatial Decomposition Space Occupation Enumeration The Space Occupation Enumeration is a particular case of Cell Decomposition: Solid formed by identical cells of equal size, placed in a regular grid. The cells are designated Voxels (Volume elements) by analogy with pixels It controls only the presence or absence of each cell in the grid The most common form for cells is the cube The object is encoded by a single list of occupied cells 25

28 Representation for Spatial Decomposition Octrees The Octree is similar to quadtree The octree is 3D and the division of space is made by octants Number of nodes of an octree It is proportional to the object surface because of the need to subdivide occurs on the surface. Enumeration of octree 26

29 Representation for Spatial Decomposition Binary Space Partitioning Trees (BSP) At each step, the space is divided by a plane of arbitrary position and orientation Each internal node of the tree is associated with a plane and two pointers (one to inside and the other to outside). If a sub-space is homogeneous (fully indoors or outdoors), cease to be divided. 27

30 (CSG - Constructive Solid Geometry) The object is obtained by combining simple primitives using Boolean operators. The object is stored as a tree, where the interior nodes are operators and leaves are simple primitives Nodes represent Boolean operations and geometric transformations. 28

31 Exercise 29

32 References 3D Modeling & Surfacing Bill Fleming Morgan Kaufmann, Academic Press, 1999 Introduction to Computer Graphics James Foley, A. van Dam, S. Feiner, J. Hughes, R. Phillips Addison-Wesley Publishing Company,