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 either Implicit-surface as an equation f(x,y,z)=0 curve as two such equations Parametric-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 ImplicitParametric 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=0x=(2t)/(1+t^2) y=(1-t^2)/(1+t^2) z=u

In general ImplicitParametric CurveF(x,y,z)=0 G(x,y,z)=0 X=x(t) Y=y(t) Z=z(t) SurfaceF(x,y,z)=0X=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/Approxi mation theory- creation of surfaces/curves from points is easy

Our Decision-Parametric ! Reasons Generating points on surfaces/curves is very important Interpolation/Approxi mation 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 v 1, v 2 Interval [a,b] C: the curve function from parameter space [a,b] to model space R 3 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= cos t Y= sin t Z=1.2 T in [-2.3,2.3]

Parametric Representation-Face Face Domain D subset of R 2 S: surface function from parameter space R 2 to model space R 3 Face – image of D

Example f1 f2 f1: part of cylinder X= sin v Y=u Z= cos v f2: part of a plane X=u Y=v

Domains P-curves in parameter space p i :[a i,b i ] to parameter space R 2 Domain Loops (p 1,-p 2,p 3,p 4 ) Normal Data

Example Domain f1 f2 Parameter Space u v Part removed by the boss Part of Cylinder

P-Curves Parameter Space u v C1 C2 C3 C4 C5 C6 A 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 EntityToplogical dataGeometric Data FaceLoops, ^co-edges ^domain Surface function S Edge^vertices [a,b] Curve function C Co-edge[a,b] : P-curve domains P-curve functions C