09 modelsolid en.ppt -...

COLLEGE OF ENGINEERING UNIVERSITY OF PORTO

COMPUTER GRAPHICS AND INTERFACES /GRAPHICS SYSTEMS JGB / AAS 2004

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Solid Modelling

Graphics Systems /

Computer Graphics and Interfaces



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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.



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Characteristics of a solid model1. 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.



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Characteristics of a solid modelExample 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.



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Boolean operationsThe combination of Boolean operations allows to define new objects, independently of the representation used. Operations are:

union,difference / subtractionintersection.

a) Objects A and B b) The U B c) A ∩ B d) A - B e) B - A

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



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SubtractionExample: CUBE - CYLINDER



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UnionExample: CUBE U CYLINDER



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IntersectionExample: CUBE ∩ CYLINDER





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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.



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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)



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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.



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Representation by ExtrusionThe 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.



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Representation by Extrusion

Boolean operations with objects created 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.



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Representation by Extrusion - Example



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1. Defining a way to make scanning for translation.



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2. Defining the shape of the section of the final object.



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Object obtained by translation.



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Object obtained by translation, rotation around the axis shifting and scaling along the route.



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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



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Boundary Representation (B-rep)

PolyhedronSolid delimited by a set of polygons whose edges belong to two polygons (solids 2-manifolds).

Euler FormulaA simple polyhedron without holes, obeys Euler's formula:

V - E + F = 2V - Vertex

E - Edges (edges)

F - Faces



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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



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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



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Exercise



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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



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Representation by Spatial DecompositionCell 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 operationIt 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



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Representation by Spatial DecompositionSpace 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



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Representation for Spatial DecompositionOctrees

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



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Representation for Spatial DecompositionBinary 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.



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(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.



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Exercise



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References

• 3D Modeling & Surfacing Bill FlemingMorgan Kaufmann, Academic Press, 1999

• Introduction to Computer GraphicsJames Foley, A. van Dam, S. Feiner, J. Hughes, R. PhillipsAddison-Wesley Publishing Company, 1996

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