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Geometry-I

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.

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Geometry-I

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  1. Geometry-I

  2. 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

  3. The Problem Before Us… The geometric representation of edges/co-edges/faces. • Edges/Coedges-part of a curve • Faces-part of a surface

  4. 2 Questions • How does one represent a curve/surface • How does one represent a part of either Implicit-surface as an equationf(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.

  5. An Example

  6. In general

  7. 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

  8. The Twisted Torus This occurs in a slanted blend • Parametric is difficult • Implicit is (practically) Impossible

  9. Easy Testing if point on curve/surface Deciding which side point of surface Hard Generating points on curve/surface Implicit-cost benefits

  10. Hard Testing if point on curve/surface Deciding which side point of surface Easy Generating points on curve/surface Parametric-cost benefits Exactly the Opposite!

  11. Our Decision-Parametric ! Reasons • Generating points on surfaces/curves isveryimportant • Interpolation/Approximation theory-creation of surfaces/curves from points is easy

  12. Our Decision-Parametric ! Reasons • Generating points on surfaces/curves isveryimportant • Interpolation/Approximation theory-creation of surfaces/curves from points is easy

  13. 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

  14. Parametric Representation-Edges Edge • End vertices v1 , v2 • Interval [a,b] • C: the curve function from parameter space [a,b] to model spaceR3 Edge – image of [a,b]

  15. Example 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] e2 e1

  16. Parametric Representation-Face Face • Domain D subset of R2 • S: surface function from parameter spaceR2 to model spaceR3 Face – image of D

  17. Example f1: part of cylinder X=1.2 +0.8 sin v Y=u Z=2.1 +0.8 cos v f2 f2:part of a plane X=u Y=v f1

  18. Domains Domains • P-curves in parameterspace • pi:[ai,bi] to parameter spaceR2 • Domain Loops (p1,-p2,p3,p4) • Normal Data

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

  20. P-Curves Parameter Space C2 v C1 C3 C5 C6 C4 • A total of 6 p-curves • All but c5 easy (lines) • c5 inverse image of a cylinder-cylinder intersection. • Only Approximately Computed! u

  21. 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!

  22. Recap

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