Introduction to solid modeling

Question 1

Objective of solid modeling

Answer 1

To find a computational representation for physical objects, starting from their Mathematical model

Question 2

Properties of a representation scheme

Answer 2

• Formal properties
  1. domain
  2. validity
  3. completeness
  4. uniqueness
• Desirable properties
  1. conciseness
  2. ease of creation
  3. efficiency in application

Question 3

Domain

Answer 3

subset of the model space M containing models which have a representation r

Question 4

Validity of a representation scheme

Answer 4

A representation scheme is syntactic correct if all models of the domain can be represented;
A representation scheme is semantic correct if each representation has a corresponding model
A representation scheme is valid if and only if it is syntactically and semantically correct

Question 5

Completness of a representation scheme

Answer 5

A representation scheme is ambiguous if two models in the domain have the same representation
A representation scheme is complete if it is valid and unambiguous

Question 6

Uniqueness of a representation scheme

Answer 6

A representation scheme is unique if for any model in the domain, a representation exists, and it is unique

Question 7

Most used representation schemes

Answer 7

1. Wire frame
2. Parametrized primitive instancing
3. spacial occupancy enumeration
4. cell decomposition
5. sweeping
6. constructive solid geometry
7. boundary representation

Question 8

wire frame

Answer 8

• simple and historical representation
• it represents an object using edges and vertices
• problems:
• nonsense objects - > not valid
• ambiguity - > not complete
• verbose

Question 9

parametrized primitive instancing

Answer 9

• basic representations defined as a function of some parameters (sphere, cube, cylinder…) defines a family of object
• changing the parameters it is possible to obtain many different solids
• it is unambiguoous and may be unique
• very compact

Question 10

spacial occupancy enumeration

Answer 10

• volume divided into small cubes (voxels) of fixed size
• solid is represented by full or empty cubes
• each cell may be represented by its centre
• unambiguous but not unique

Question 11

Cell decomposition (octree)

Answer 11

• recursive division of the 3D space in a tree data structure
• internal node is partially full
• each node is divided into 8 cubes, which may be full, empty or nodes
• it’s a more efficient spatial enumeration

Question 12

sweeping

Answer 12

• generatrix: a moving point-set (line, surface or volume)
• directrix: a trajectory
• the motion of the generatrix along the directrix represents the solid
• a non solid result can be obtained with a wrong choice of generatrix or directrix
• dangling edges and faces must be avoided

Question 13

constructive solid geometry

Answer 13

• rigid solid are represented as binary trees with Boolean contructions of primitives
• regularized set of operations must be used
• terminal nodes are primitive solids
• non-terminal nodes are rigid transformations or regularized operations

Question 14

regularized boolean operations

Answer 14

to maintain boundary determinism, the regularized boolean operations must be used
these are the closure of the classical operations

Question 15

boundary representation

Answer 15

• solid represented as a collection of connected surface elements, the boundary between solid and non-solid
• composed of topology and geometry
• geometry: surfaces, curves, points
• topology: faces, edges, vertices
• valid, unambiguous
• not unique