Geometric and Solid Modeling

7/26/2019 Geometric and Solid Modeling

1/21

Seminar in

Design Computing

Geometric and Solid Modeling

Interactive Design in 3D

A representation that was general enough to

depict any shape

Editing operations that would allow shapes to

be modified limitlessly

Ian Braid

Bruce Baumgart

The Fathers of Solid Modeling (1973)

ACIS Parasolids

Methods for computing shapes rather than

drawing them


2/21

Seminar in

Design Computing



3/21

Seminar in

Design Computing


SOLID MODELINGData structure:

F

FF

E

E

E

Vertex adjacencies:

F

E

E

E

V

V

V

Face adjacencies:

F

FE VV

Edge adjacencies:

Geometry and topology

line :: edge

surface :: face

E + V = 212 + 8 = 2


4/21

Seminar in

Design Computing



Other issues:F

E

E

E

V

V

V

Face adjacencies:

F

E

E

E

V

V

V

Face - loop - edge adjacencies:

LF L

Faces bounded by

multiple loops Euler Equation:

F - E + V = 211 - 24 + 16 = ?? 11 - 24 + 16 - 1 = 2

F - E + V - L = 2


5/21

Seminar in

Design Computing



Other issues:

B

S1S2

Shapes bounded by

multiple shells

12 - 24 + 16 - 0 = ?

F - E + V - L = 2

24 + 16 - 0 = 2*2

- E + V - L = 2*S


6/21

Seminar in

Design Computing



Other issues:

Shapes with holes

through them

16 - 36 + 24 - 2 = 2*2 ??

F - E + V - L = 2*S F - E + V - L = 2*(S - H)

16 - 36 + 24 - 2 = 2*(2 - 1)

F - E + V - L = 2*(S - H)


7/21

Seminar in

Design Computing



Winged edge data structure

body

shell

face face face

loop loop

shell

edge edge edge

vert vert

Edge #

loop_ptr

p-edge_ptrs-edge_ptr

vert_ptr

F

FE VV

Edge adjacencies:


8/21

Seminar in

Design Computing



Winged edge data structure

body

shell

face face face

loop loop

shell

edge edge edge

vert vert

F

FE VV

Edge adjacencies:

Split edge data structure

Edge #

loop_ptr

p-edge_ptr

s-edge_ptr

vert_ptr

op_edge_ptr


9/21

Seminar in

Design Computing



How to construct - modify

shape topology

Euler operators

F1F2

F3F4


10/21

Seminar in

Design Computing



How to construct - modify

shape topology

Euler operators

Mbsflv( );

1-0+1=2

Mev( e1,e2, l);

0-1+1=0

Mef( e1,e2, l);

1-1+0=0

F

Keml( e1);

0+1-0-1=0

F


11/21

Seminar in

Design Computing


*body Mbsflv()

/* creates a minimal shape, consisting of a body and shell, with one

face bounded by a vertex */

vf

null

Kbsflv(s, body_too )

* body s;

flag body_too;

/* deletes a shell, consisting of all its faces, loops, edges and vertices.

The flag determines if it is the last shell, whether to delete the body also.

*/

nullvf

vv


12/21

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*edge Mev( *edge1,*edge2,*loop1 )

*edge edge1, edge2;

*loop loop1;

/* adds a new edge as successor of edge1. oppedge of new edgeis predecessor of edge2. all edges between edge1 and edge2 are

put on new vertex in the case where an edge is connected to a

vertex bounding a face, the loop of the bounding vertex is given */


13/21

Seminar in

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AB

C

*edge Mef( *edge1,*edge2, *loop1 )*edge edge1, edge2;*loop loop1;

/* makes an edge and faceloop; new face may be bound by single

edge; new edge is successor to edghe1 and its opposite its successorof edge2; newedge is on existing face;is edge1 = edge2, then newfaceis a single edgeloop */


14/21

Seminar in

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*edge Mekl(*edge1, edge2, loop1, loop2 )

*edge edge1, edge2;

*loop loop1,loop2;

/* makes a new edge between 2 loops on the same face, kills one loop; new edge

is successor of edge1 and oppedge is successor of edge2 */


15/21

Seminar in

Design Computing


SOLID MODELINGData structure: Euler operators

Operator F - E + V - L = 2 * (S - H)

mbsfv 1 - 0 + 1 - 0 = 2 * (1 - 0)kbsfv n - m + p - q = 2 * (1 - 0)

mev 0 - 1 + 1 - 0 = 2 * (0 - 0)

kev 0 + 1 - 1 - 0 = 2 * (0 - 0)

mef 1 - 1 + 0 - 0 = 2 * (0 - 0)kef -1 + 1 + 0 - 0 = 2 * (0 - 0)

Mekl 0 - 1 + 0 + 1 = 2 * (0 - 0)

Keml 0 + 1 + 0 - 1 = 2 * (0 - 0)

FglueMhole -2 - n + n - 0 = 2 * (0 - 1) glue end faces to make torus

Mlamina 2 - n + n - 0 = 2 * (1 - 0) cuts thru shape & makes lamina

KfaceMhole -2 + m - m - 0 = 2 * (0 - 1) merges two pockets into a hole

Fglue -2 + m - m - 0 = 2 * (-1 - 0) glues two coincident faces

KLamina -2 + m - m - 0 = 2 * (-1 - 0) deletes lamina

Completeness?


16/21

Seminar in

Design Computing


BOOLEAN OPERATIONS (SIMPLE)OVERVIEW

1. partition faces sothey are uniquelyinside or outsidethe other shape.

2. kill all facesinside (outside theother shape.

3. Glue facestogether



17/21

Seminar in

Design Computing

Geometric and Solid ModelingGLUING TWO FACES WITHOUT INNER LOOPS:

This operation deletes N edges and N vertices and 2 faces and either (deletes

one shell) or (increases the genus (number of holes) by 1).

PARAMETERS PASSED: half-edge on each face that is to be merged

A. CHECK FOR LEGAL GLUE:1.same number of edges in g lued fa ces2.check each glued face has only one loop

B. SWAP HALF EDGES ON GLUED FACES:1.switch opposite references on opposite half edgesof glued faces

C. MERGE VERTICES TO A SINGLE SET:1.move all half edge references to one set of vertices2. traverse around all vertices to be removed, retargettingedge references to kept vertices

D. DELETE EXTRA VERTICES, EDGES, LOOPSAND FACES:1.delete N vertices on one of glued faces2. delete 2*N half edges on both glued faces

3. delete 2 loops and faces that are glued

FACE FACE FACE


18/21

Seminar in

Design Computing


E. IF GLUING TWO SHELLS:1. merge in faces to single list2. delete shell not used in new shapeIF GLUING TWO SHAPES3.merge in vertex lists4.delete shape not used

F. DO NEEDED BOOKKEEPING:1.re-index all faces, edges and vertices2. update euler count and change genus, if necessary

SHELL

FACE FACE FACE

FACE FACEFACE

Add 2 faces, with n edges

and n vertices

MakeLamina

A B

C

Still connected, but withlamina in middle


19/21

Seminar in

Design Computing


glue end faces together, removing 2

faces; deletes a set of edges and

vertices bounding lamina, adds hole

FaceGlueMakeHole

glue end faces together, removing 2

faces; deletes a set of edges and

vertices bounding lamina, adds hole

KillFaceMakehole


20/21

Seminar in

Design Computing


What is a legal solid shape? All operations must

respect a set of basic well-formedness conditions

Continuous

No infinite dimensions

No self-intersections

Mobius Law regarding manifolds

S S

An algebra


21/21

Seminar in

Design Computing


Assignment A:

The Expanded Euler equation, with the parameters F,E,V,L,S,H, defines a

shape topology appropriate for a family of shapes.Is there a unique shape topology for a particular set of parameters? If we

stored shape topologies by their parameters, would there be a unique

topology for a given set of Euler parameters?

Please (informally) provide a convincing argument one way or the other.

Assignment B:

Define the sequence of Euler operators needed to define an

extrusion, on M sides in the base.

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Geometric and Solid Modeling