manifold (good)

8/4/2019 Manifold (Good)

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Solid Modeling

Reference:

Computer Graphics. Principles and Practice

Foley - van dam

Feiner

Hugheschapter 12


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Solid representation - DesiderataRequicha 80

Domain:A representation should be able to represent auseful set of geometric objects.

Unambiguity:When you see a representation of a solid, youwill know what is being represented withoutany doubt. An unambiguous representation isusually referred to as a completeone.

Uniqueness:from the representations it is immediatelypossibile to determine if two solids are identical


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DesiderataRequicha 80

Accuracy

Little or no approximation

ValidnessA representation should not create any phisically

unrealizable or impossible solids. Closure

Solids will be transformed and used with otheroperations such as union and intersection. "Closure"

means that transforming a valid solid always yields avalid solid.

Compactness and Efficiencyto save memory and computation time


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2-ManifoldTorus

Intuitively, we can deform the surface locally into

a plane without tearing it or identifying separatepoints with each other.


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A non manifold object

Surfaces thatintersect or touchthemselves are non

manifolds.


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Boolean Operations on Solids

Boolean operation on valid solids mayproduce non valid solids.

Example: Intersecting two cubes that touchalong a face produces a face, i.e. a nonvalid solid


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Regularized Boolean operations

Goal: eliminate dangling" lower-dimensionalstructures that may be produced byBoolean operations

Notation:

A*

B, A

* B, A -* B


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Regularized IntersectionExample


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Regularized Boolean Intersection

Compute A * B

Three steps:

1. Compute A

B2. Take the interior of A B

3. Form the closure of this interior by adding allboundary points adjacent to some interior

neighborhood.

A * B = closure(interior A B )


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Some definitions

The interior consists of all those points p of A Bsuch that an open ball of radius , centered at p, consists only of points

of A B, for a sufficiently small radius .

A point q that is a boundary point of A B is adjacent to the interior ifwe can find a curve segment (q; r) of sufficiently small length,between q and another point r of A B, such that all points of thissegment are interior points of A B, except q.


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Regularized Boolean operationson manifolds

The result may be a non manifold solid.

As an example, the figure below may be obtained by the regularized

union of two L shaped solids.


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Constructive Solid Geometry(CSG)

A solid is represented as a set theoretic

Boolean expression of primitive solidobjects, of a simpler structure.

Both the surface and the interior of anobject are defined, albeit implicitly.


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Primitives

Blocks, cylinders, spheres, cones, andtori

A primitive object is instantiated byspecifying some parameters Side lengths of blocks, the diameter and

length of cylinders, placing the primitives with respect to a

common world coordinate frame


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Combining the primitives

A solid can be considered as the result ofapplying regularized Boolean operators toa set of instantiated and transformed CSGprimitives

CSG representations are not unique


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Example


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Example(from Shene, Michigan)

(trans(Block1) + trans(Block2)) - trans(Cylinder)


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B-reps

Boundary representationa solid is modeled as a set of surfaces forming its boundary.

Topology and Geometry

1. Topology

how the surfaces are connectedtogether.(incidences and adjacencies ) and orientation of vertices,edges, and faces.

2. Geometry where the surfaces actually are in space.

for example, the equations of the surfaces of which thefaces are a subset.

3. Topology and Geometry are strongly linked.


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B-reps (continued)

Surfaces must be orientable.

Boundaries may be linear (planar)polygons or curved patches.

Must be able to handle holes in faces.

Keep track of interior loops in faces.


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Euler-Poincare formula:

For a polyhedron

V - E + F - 2 = 0 V = Vertices

E = Edges F = Faces

Example: A tetrahedron has four vertices, fourfaces, and six edges

4-6+4=2.


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Proof by induction

Base: the formula is true for tetrahedaInduction step: Suppose it is true for two solids, Sol1 and Sol2, Suppose the solids each have a face which is the mirror image of

the corresponding face on the other. Assume the faces each have n

edges.

Join the solids together at those faces and obtain object Sol3The, the two faces disappear and:

F3 = F1 + F2 - 2

V3 = V1 + V2 - nE3 = E1 + E2 - n

Thus:

V3 + F3 - E3 = (F1 + V1 - E1) + (F2 + V2 - E2 ) - 2 + nn = 2


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Euler Poincare Formula

Necessary but not sufficient condition for a validrepresentation.

Example: 8 vertices, 12 edges, 6 faces


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Extension of solids

A solid can have holes

A face may have a loop or ring of vertices`floating, i.e. unconnected by edges to the

other vertices of the face


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Extension of Euler-Poincareformula to 2-manifolds

V-E+F-H=2(C-G) V = Vertices E = Edges F = Faces H = Holes in faces C = Components (or shells) G = Genus (holes through solid)

Tweaking (deformations, twistings, and stretchings butnot tearing, or cutting ) solids modifies the solid withoutchanging the topology or the above numbers.


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A solid with holes and loopsExample

V E + F H = 2 (C G)

24 36 + 15 3 = 2( 1 1)


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ExampleA surface with genus 2


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Validity of a B-rep

Ensuring validity of a B-rep can be very difficult

Topological validity Orientability adjacent faces must have correct

orientationsGeometric validity:Determining position/orientation of

vertices/planes, etc.

Inaccuracies in geometry and computation canlead to topological inconsistencies.


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Robustness Problems

Numerical errors: In representation or arising in computation. Cause conflict between geometric and topological

information, or conflict within geometric information.

Example 1: Plane equations say that an edge should liewithin a solid, but topological information says it liesoutside.

Example 2: Intersection of 3 plane equations for facesyields coordinates that are different than those of thevertex shared by those faces.


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Winged-Edge RepresentationBaumgart 1974

Polyhedral objects represented by

vertices, edges and faces

Each vertex has pointer to one of its edges

Each face has a pointer to one of its edges

Each edge has 8 pointers to ..


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Each edge has 8 pointers to:its two faces:

Pface and Nface

its two vertices

each of its four neighboringedges (wings)

Traversal of V1 and V2 in

opposte directions whentraversing Pface and Nfacein counterclockwise order

The two edges at N are theprevious counterclockwiseon the Pface and theprevious clockwise on theNface

V1=N

V2=P

PfaceNface

PccwePcwe

NccweNcwe


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Edge nodes -> topology

vertex nodes -> geometry

face nodes -> photometry


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Key properties

constant element size

topological consistency

efficient search for lists of edges of the faces

efficient perimeter calculation


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CSG and B-repsummary

They have different inherent strengths andweaknesses.

CSG object always valid its surface is closed and orientable and

encloses a volume, provided the primitivesare valid.

A B-rep object is easily rendered on agraphic display system.


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Sweeping Representations

Sweeping a contour along a space curve

Translational

Rotational

Generalized


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Examples


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Spatial representations


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Cell decomposition


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QuadtreeSameh 84

Recursive binary subdivision of the plane intoquadrants.

A quadrant may be full, partially full or emptydepending on the portion of the quadrantoccupied by the object.

Subdivision continues until all quadrants areeither full or empty

Hi hi l d t t t


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Hierarchical data structureQuadtree example

Partially full (or gray) nodeFull (or black) node

Empty (or white) node


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Quadrant numbering or labeling

2NE

3NW

0SE

1SW


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Boolean OperationsUnion

Given two trees S and T

Traverse the two trees from top-down

Consider a matching pair of nodes

(at the same position in the two trees)

Three cases:

1. If either node is blackthen a corresponding blacknode is addedto the new tree

2. If one of the nodes is white then a node is added with the color ofthe other node of the pair

3. If both nodes are gray, then a gray node is added and thealgorithm is applied recursively to the children nodes


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Is that all?

For case 3, if all children of the addednode turn out to be black then they aredeleted and the node itself become black


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Intersection operation

Similar to union with the role of black andwhite are interchanged.

Fi di th i hb f d


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Finding the neighbor of a nodein a given direction

Find the neighbor of the red node in the N direction


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Algorithm

Traverse the tree upward until the lowestcommon ancestor of the node and its neighboris found

This is done by moving up towards the root until anode is reached from a south S ( 0 or 1) link

Then move downward in a mirror fashion until aleaf of the tree

Li i


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Linear representationof a quadtree

Post-order list of the nodes

A node is represented a string of digits,each digit corresponding to a level in thetree


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Example

00X 02X 03X 200 202 203 22X 23X

Xs are added to nodes not at lowest level of the tree

Only full nodes are represented

0 1 2 3


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Octree

Similar idea to quadtree in 3D space

A node has 26 neighbors

6 along a face

12 along an edge

8 along a vertex

BSP


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BSPBinary Space Partitioning Trees

Space-partitioning based on planes (or, inhigher dimensions,hyperplanes)

A plane divides the space points into two

regions corresponding to the two halfspaces defined by the plane.

A recursive subdivision using planes until

the partitioning satisfies somerequirements produces a BSP tree

E l


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ExamplePolygon


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Protein BSP

manifold (good)

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