SOLID MODELLING. PARAMETRICALLY WE CAN DEFINE SOLID AS- X=x(u,v,w) Y=y(u,v,w) Z=z(u,v,w) Tricubic solid- u,v,w Є [0,1]

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1 SOLID MODELLING PARAMETRICALLY WE CAN DEFINE SOLID AS- X=x(u,v,w) Y=y(u,v,w) Z=z(u,v,w) u w v Tricubic solid P(u,v,w) = Σ Σ Σ a ijk u i v j w k I=0 j=o k=0 u,v,w Є [0,1]

2 GRAPH BASED MODELS OR B-Rep MODELS B-REP STANDS FOR BOUNDARY REPRESENTATION. REPRESENT THE SOLID AS THE LIST OF THE OBJECT FACES. FACES AS THEIR SURFACE EQUATIONS. EDGES AS CURVE EQUATIONS AND END POINTS.

3 E 3 E 5 CONSIDER THE SIMPLEST SOLID TETRAHEDRON V 3 E 6 V 4 V 1 E 2 E 4 E 1 V 2 FOUR FACES- F 1 -V 1 V 3 V 2 F 2 -V 2 V 3 V 4 F 3 -V 1 V 4 V 2 F 4 -V 3 V 4 V 1

4 FACE F 1 STORES---Vertex list (V 1 V 3 V 2 ) Edge list (E 1 E 3 E 2 ) Adjacent face list (F 2 F 3 F 4 ) Similar definitions are for F 2 F 3 and F 4 EDGE E 1 STORES---Vertex list (V 1 V 2 ) Adjacent Edge list (E 2 E 3 E 4 E 5 ) connecting face list (F 1 F 3 ) Similar definitions are for other edges

5 VERTEX V 1 STORES--- connected edges (E 1 E 5 E 3 ) adjacent vertex list (V 2 V 3 V 4 ) adjacent faces (F 1 F 3 F 4 ) V 2 V 3 V 4 are described in the same fashion.

6 REPRESENTATION OF HOLES/POCKETS USING B-REP E V 3 4 V 3 V 8 E 7 V 7 E 8 E 4 E 6 V 5 E V 5 E 6 2 V 1 E 1 V 2 Here top face F1 is represented as- List of vertex list.[(v 1 V 2 V 3 V 4 ) (V 5 V 6 V 7 V 8 )] List of vertex list.[(e 1 E 2 E 3 E 4 ) (E 5 E 6 E 7 E 8 )]

7 The first vertex/edge list represents outermost boundary and all subsequent vertex/edge lists represent the holes or pockets inside. B-rep models has a lot of redundancy in its definition. E.g. edges are stored in surface definition as well in vertex definition. All the data should be consistent. Requires more storage shape. Speed of operation is an advantage.

8 To summarize--- FACES STORE- List of edges binding it. List of vertices. List of adjacent faces. Type of surface. Mathematical description of surface. EDGES- List of vertices. List of connecting edges. List of adjoining faces. Type of edges. Mathematical description of edges.

9 VERTICES- Coordinates List of edges connected to it. List of faces adjacent to it. List of vertices adjacent to it.

10 B-rep model is very complex and not user friendly.also B-rep is difficult to manipulate. Thus, B-rep is used at the back-end of computer and user interacts with it through some kind of user interface. USER CAD SYSTEM (B-REP) USER INTERFACE.

11 BOOLEAN MODELS OR CONSTRUCTIVE SOLID GEOMETRY(CSG) Regularized operators- 1. Union 2. Intersection. 3. Difference. C= (A U B) or (A I B) or (A B) C= A U B A, B Primitives stored in some other manner. C Stored as an expression AUB (binary tree).

12 Here properties of C are not known directly.(surfaces that constitute C are not known but can be evaluated from binary model.) Therefore, C=AUB is called UNEVALUATED or PROCEDURAL model. The leaf nodes like A, B etc. normally contain evaluated solid primitives.

13 Object C =AUB C= (T 1 A) U (T 2 B) -A is cube at origin -T 1 A is rectangular block -B is cylinder at origin. -T 2 B is cylinder at specified location and orientation. Evaluating a tree is an expensive process as new lines and surfaces are formed.

14 Evaluation is the process of finding out B-rep or Equivalent representation from CSG description and B-rep description for primitives. PRIMITIVES CSG DESCRIPTION UNEVALUATED/ PROCEDURAL MODEL EVALUATION FINAL MODEL (B-REP FORM) EVALUATED/ DESCRIPTIVE MODEL

15 STANDARD PRIMITIVES- 1. Unit cube. 2. Unit cylinder. 3. Unit sphere. 4. Unit cone. Advantages of CSG- User flexibility. Manipulation is easy. Less space required for storage. Easy to visualize. Disadvantage Computation or evaluation takes much time. Identifying features from CSG model is difficult, therefore it is not suitable for manufacturing.

16 FEATURE BASED MODELLING (FBM) -Basic operation are hole,slot, cut, chamfer etc. -As the basic definition itself contains manufacturing terms, feature recognition is easy. -E.g. a hole is defined on parameters as- - surface on which hole is made. - diameter of hole. - depth of hole. - position of hole.

17 Feature Hole in FBM

18 Feature chamfer in FBM d 45 0

19 QUAD TREE AND OCTREE REPRESENTATION QUAD TREE (2D) Suppose you want to the shaded area as surface- -we use the technique of space partitioning. I.e we divide the area in four equal parts and 4.

20 Part 3 is completely inside the shaded area---assign it a value 1. Part 1,3,4 are partially in..divide them again in 4 parts. If any part is completely outside shades are assign it value 0. This sub-dividing is carried out till the required resolution is achieved.

21 It is represented assurface

22 Octree model is same as quad-tree but extended to 3-D. A cube is divided into 8 parts. Representation is same as quad-tree. Advantages- -Easy to display. -Volume, area,mass calculation is easy. Disadvantages- -Mathematical description is lost. -For high resolution we get very long tree.