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Describe the following 3D representation methods:- a) B-REP. b.) Sweep representation. c.) CSG.

Mumbai university > Comp > SEM 4 > Computer Graphics

Marks: 10M

Year: May 2014

1 Answer
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B-Rep:

  1. B-Rep stands for Boundary Representation.

  2. It is an extension to the wireframe model.

  3. B-Rep describes the solid in terms of its surface boundaries: Vertices, edges and faces as shown in figure 35

enter image description here

  1. It is a method for representing shapes using the limits.

  2. A solid is represented as a collection of connected surface elements, the boundary between solid and non-solid.

  3. There are 2 types of information in a B – rep topological and geometric.

  4. Topological information provides the relationships among vertices, edges and faces similar to that used in a wireframe model.

  5. In addition to connectivity, topological information also includes orientation of edges and faces.

  6. Geometric information is usually equations of the edges and faces.

  7. The B-rep of 2 manifolds that have faces with holes satisfies the generalized Euler’s formula:

$$V – E + F – H = 2 (C - G)$$

Where, V = Number of vertices.

E = Number of edges.

F = Number of faces.

H = Number of holes in the faces.

C is the number of separate components (parts).

G is the genus (for a torus G = 1)

enter image description here

Sweep Representations:

Sweep representations are used to construct 3D object from 2D shape that have some kind of symmetry.

For example, a prism can be generated using a translational sweep and rotational sweeps can be used to create curved surfaces like an ellipsoid or a torus.

In both cases, you start with a cross-section and generate more vertices by applying the appropriate transformation.

More complex objects can be formed by using more complex transformations. Also, we can define a path for a sweep that allows you to create more interesting shapes.

CSG:

  1. CSG stands for Constructive Solid Geometry.

  2. It is based on set of 3D solid primitives and regularized set theoretic operations.

  3. Traditional primitives are: Block, cones, sphere, cylinder and torus.

  4. Operations: union, intersection, difference + translation and rotation.

  5. A complex solid is represented using with a binary tree usually called as CSG tree.

  6. CSG tree is shown below in figure 37.

enter image description here

  1. Ray casting is a method used for determining boundaries of the resulting object if you start with a boundary representation.

  2. Octree representations are designed to make this process easier.

    Example:

enter image description here

Figure 38 (b) is the intersection of original solid.

Figure 38 (c) is the difference between 2 original solid.

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