Geometry-I. What’s done The two-tier representation Topology-the combinatorial structure Geometry –The actual parameters representing the geometric face.

Geometry-I

What’s done• The two-tier representation• Topology-the combinatorial

structure• Geometry –The actual

parameters representing the geometric face.

In this Talk (and henceforth!)

GEOMETRY

The Problem Before Us…

The geometric representation of edges/co-edges/faces.

• Edges/Coedges-part of a curve

• Faces-part of a surface

2 Questions

• How does one represent a curve/surface• How does one represent a part of eitherImplicit-surface as an equation f(x,y,z)=0 curve as two such equationsParametric-edge is (x(t),y(t),z(t)) surface is (x(t,u),y(t,u),z(t,u)) Thus surface /curve :parameters on which the

coordinates depend.

An Example

Implicit Parametric

Curve

(circle)

x^2 +y^2-1=0

z=0

x=(2t)/(1+t^2)

y=(1-t^2)/(1+t^2)

z=0

Surface

(cylinder)

x^2 +y^2 –1=0 x=(2t)/(1+t^2)

y=(1-t^2)/(1+t^2)

z=u

In general

Implicit Parametric

Curve F(x,y,z)=0

G(x,y,z)=0

X=x(t)

Y=y(t)

Z=z(t)

Surface F(x,y,z)=0 X=x(t,u)

Y=y(t,u)

Z=z(t,u)

Another Example-Torus

Torus-occurs as a blend

Parametric

x=(R+r sin u)cos t

y=(R+r sin u)sin t

z=r cos u

Implicit-tedious

x^2 +y^2=

(R +/- sqrt(r^2 –z^2))^2

The Twisted Torus

This occurs in a slanted blend

• Parametric is

difficult• Implicit is (practically)

Impossible

Implicit-cost benefits

Easy• Testing if point on

curve/surface

• Deciding which side point of surface

Hard

• Generating points on curve/surface

Parametric-cost benefits

Hard

• Testing if point on curve/surface

• Deciding which side point of surface

Easy

• Generating points on curve/surface

Exactly the Opposite!

Our Decision-Parametric !

Reasons• Generating points on

surfaces/curves is very important

• Interpolation/Approximation theory-creation of surfaces/curves from points is easy

Our Decision-Parametric !

Reasons• Generating points on

surfaces/curves is very important

• Interpolation/Approximation theory-creation of surfaces/curves from points is easy

So Then -Parametric

• Curves: One parameter

X=x(t) Y=y(t) Z=z(t)

Domain of definition: an interval

• Surfaces: Two parameters

X=x(u,v) Y=y(u,v) Z=z(u,v)

Domain of definition: an Area

Parametric Representation-Edges

Edge• End vertices v1 , v2

• Interval [a,b]

• C: the curve function from

parameter space [a,b] to model space R3

Edge – image of [a,b]

Example

e1

e2

e1: part of a line

X=1+t; Y=t, Z=1.2+t

t in [0,2.3]

e2: part of a circle

X=1.2 +0.8 cos t

Y=0.8+0.8 sin t

Z=1.2

T in [-2.3,2.3]

Parametric Representation-Face

Face• Domain D subset of R2

• S: surface function from

parameter space R2 to model space R3

Face – image of D

Example

f1

f2

f1: part of cylinder

X=1.2 +0.8 sin v

Y=u

Z=2.1 +0.8 cos v

f2: part of a plane

X=u

Y=v

Domains Domains• P-curves in parameter

space• pi:[ai,bi] to parameter space R2

• Domain Loops

(p1,-p2,p3,p4)

• Normal Data

Example Domain

f1

f2

Parameter Space

u

v

Part removed by the boss

Part of Cylinder

P-Curves

Parameter Space

u

vC1

C2

C3

C4

C5

C6A total of 6 p-curves

•All but c5 easy (lines)

• c5 inverse image of a cylinder-cylinder intersection.

•Only Approximately Computed!

Co-edges-The image of this p-curve is only an approximation to the correct intersection

-This results in 3 separate paramets of the same intersection curve

-all of these are required!

Recap

Entity Toplogical data Geometric Data

Face Loops,

^co-edges

^domain

Surface function S

Edge ^vertices

[a,b]

Curve function C

Co-edge [a,b] : P-curve domains

P-curve functions C