Copyright @ 2003 by Jim X. Chen: [email protected] Introduction to Modeling Jim X. Chen George Mason University

Copyright @ 2003 by Jim X. Chen: [email protected]

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Introduction to Modeling

Jim X. ChenGeorge Mason University


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What is ModelingWhat is Modeling a process of constructing a virtual 3D

graphics object

Modeling tools: creating and constructing complex 3D models fast and easy.

Rendering is a process of creating images from graphics models.


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A graphics modelA graphics model geometrical descriptions (particles, vertices,

polygons) and associated attributes (colors, shadings, transparencies, materials)

can be saved in a file using a standard (3D model) file format.


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Models Organizational models: hierarchies representing institutional bureaucracies

Quantitative models: equations describing econometric, financial, socialogical, ... systems

Geometric models: collections of components with well-defined geometry and their interconnections

Deformable models: that change forms


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POLYGON MESHESPOLYGON MESHES

• list of vertices - polygon; list of edges - polygon• list of polygons -- objects• Plane equation from 3 vertices: Ax + By +Cz + D = 0

• Normal: (A,B,C) = k(P1P2 x P1P2)

• A, B, and C are proportional to the signed areas of the projections of the polygon onto the (y, z), (x, z), and (x, y) planes. If the polygon is parallel to the (x, y) plane, then A = B = 0.


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QUADRIC SURFACES

Quadric SurfacesThe implicit surface equation of the form

defines the family of quadric surfaces:

ax2+by2+cz2+2(dxy+eyz+fxz+gx,+hy+jz)+k=0

Sphere

x2+y2+z2=R2; or in parametric formx=rcoscos, y=rcossin, z=rsin

Ellipsoid

(x/a)2+(y/b)2+(z/c)2=1;or in parametric formx = a coscos, y = b cossin, z = c sin

Torus

[r-((x/a)2+(y/b)2)1/2]2+(z/c)2=1;orx = a(r+coscos, y = b(r+cossin, z = c sin


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Superquadrics

supperellipse: x = a coss, y = b sins

supperellipsoid

x=rcosscost, y=rcosssint, z=rsins

Blobby ObjectsSome objects do not maintain a fixed shape, but

change their surface characteristics in certain motions or when in proximaity to other objects.

One way to model: combinations of Gaussian density functions, or “bumps”:

where T is a threshold

and a and b are to adjust the amount of blobbiness

Tebk

zyXak

kkkk 222


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PARAMETRIC BICUBIC SURFACES

• General form of cubic curve: Q(u) = U ·M ·G where G, the geometry vector, is a constant • If we allow G to vary in 3D along some path:

Then, a functional description is often tesselated to produce a polygon-mesh approximation to the surface (trianglular polygon patches)

• For a fixed t1, Q(s, t1) is a curve because G(t1) is constant. If Gi(t) are cubics, the surface is said to be a parametric bicubic surface

)(

)(

)(

)(

)(),(

4

3

2

1

tG

tG

tG

tG

MStGMStsQ


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Hermite Surfaces

x

h

R

R

P

P

Muuuux

4

1

4

1

23 1)(

0001

0100

1233

1122

hM

,

)(

)(

)(

)(

),(

4

1

4

1

x

h

tR

tR

tP

tP

MStsx

)1,0(

)0,0(

)1,0(

)0,0(

)(1

xt

t

x

x

MTtP hx

Curve: and

Surface:

THxhx GMTtRtRtPtP )()()()( 4141Since:

we have:


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TThHxhh TMGMS

tR

tR

tP

tP

MStsx

)(

)(

)(

)(

),(

4

1

4

1

)1,1()0,1()1,1()0,1(

)1,0()0,0()1,0()0,0(

)1,1()0,1()1,1()0,1(

)1,0()0,0()1,0()0,0(

22

22

xts

xts

xs

xs

xts

xts

xs

xs

xt

xt

xx

xt

xt

xx

GHx

Where

Where x coordinates, coordinates of the tangent vectors and twists are specified


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Just as the Hermite cubic curves, the Hermite bicubic permits C 1 and G1 continuity from one patch to the next

1st, to have C0 continuity, the matching curves of the two patches must be identical, which means the control points for the two surfaces must be identical along the edge

To have C1 continuity, the control points along the edge and the tangent and twist vectors across the edge be equal.

To have G1 continuity, the tangent and twist vectors across the edge be in the same direction, but do not need to have the same magnitude. .


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Bezier Surfaces

The Bezier bicubic formulation can be derived in exactly the same way

as above. The results are:

B-Spline SurfacesThe B-Spline bicubic formulation can be derived in exactly the same

way also. The results are:

Normals to SurfacesThe cross product between the s and t tangent vectors of the surface Q(s, t) results in the normal at given s and t:

TTbBxb TMGMStsx ),(

TTBsBSxBs TMGMStsx ),(

),(),( tsQt

tsQs


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Solid Modeling MethodsSolid Modeling Methods(Modeling Solids)(Modeling Solids)


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Creating solid models.

A solid model is defined by volumes.

Hierarchy of entities from low to high: keypoints lines areas volumes.

You cannot delete an entity if a higher-order entity is attached to it.

Volumes

Areas

Lines &Keypoints

Keypoints

Lines

Areas

Volumes


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File formats Representing Solids (solid models)

The domain of representation should be large to allow a useful set of physical objects (solids)

The representation should be unambiguous

Modeling Tools have their own file formats

Volumes

Areas

Lines &Keypoints

Keypoints

Lines

Areas

Volumes


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Simple 3D Half-Spaces Sphere Cylinder Cone Torus Box Plane

it splits space into two infinite half-spaces you can use an infinite cylinder and two planes to make a

capped cylinder You can also get a box from 6 planes…


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Modeling Approaches

Two approaches to creating a solid model: Top-down Bottom-up

Top-down modeling starts with a definition of volumes (or areas), which are then combined in some fashion to create the final shape.

add


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Approaches Bottom-up modeling starts with keypoints, from which

you “build up” lines, areas, etc.

You may combine both methods.


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Top-Down Modeling Top-down modeling starts with a

definition of volumes (or areas), which are then combined in some fashion to create the final shape. The volumes or areas that you initially

define are called primitives. Primitives are located and oriented with the

help of the working plane. The combinations used to produce the final

shape are called Boolean operations.


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Primitives 2-D primitives include rectangles,

circles, triangles, and other polygons.


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Primitives 3-D primitives: blocks, cylinders,

prisms, spheres, and cones.


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Top-Down Modeling

...Primitives When you create a 2-D primitive, a modeling tool

usually defines an area, along with its underlying lines and keypoints.

When you create a 3-D primitive, a modeling tool usually defines a volume, along with its underlying areas, lines and keypoints.


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Top-Down Modeling

...Primitives You can create primitives by specifying

their dimensions or by picking locations in the graphics window.


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Top-Down Modeling

Boolean Operations Boolean operations: combinations of geometric

entities: add, subtract, intersect, divide, glue, and overlap, etc.

The “input” to Boolean operations: geometric entities, simple primitives or complicated volumes imported from a CAD system.

add

Input entities Boolean operation Output entity(ies)


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Boolean CSG Operations Union

Addition, A B

Intersection A B

Difference Subtraction, A – B, A not B Difference is not commutative


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A more complicated example Difference of:

Intersection of Sphere and Cube Union of 3 Cylinders

- =


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Bottom-Up Modeling

Most modeling tools use top-down approach Low level programming systems usually

adopts with bottom-up modeling


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Primitive Instancing In a hierachical model, there are parts that are exactly

the same.

For example, all four wheels of a car can be the same model.

Instead of saving four copies of the model, we save just one primitive model and three instances

If we modify the primitive, we know that the primitive and the instances are identically changed.


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Sweep Representations a 2D area swept along a linear path

normal to the plane of the area to create a volume

2D/3D along a trajectory through space defines a new object -- sweep


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More representations Boundary Representations

describe object by its surface boundaries: shared edges have pointers to 2 polygons, etc.

Spatial-partitioning representations

Spatial-Occupancy Enumeration

identical solids called voxels (volume elements) arranged in a fixed regular grid.

Octrees: divide-&-conquer power of subdivision.

Binary Space-partitioning Trees: recursively divide space into pairs of subspaces, each separated by a plane. Originally used in determining visible surfaces in graphics.


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Physics-based Modeling Modeling triangular polyhedron: all

vertices and the surrounding triangular shapes are congruent


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Modeling Tools http://www.cs.gmu.edu/~jchen/graphi

cs Examples in 3D Studio MAX 3, the

book

Documents

Copyright @ 2003 by Jim X. Chen: [email protected] Introduction to Modeling Jim X. Chen George Mason University