VEGAS Effective Geometric Algorithms for Surfaces and Visibility - - PowerPoint PPT Presentation

▶
vegas effective geometric algorithms for surfaces and
SMART_READER_LITE
LIVE PREVIEW

VEGAS Effective Geometric Algorithms for Surfaces and Visibility - - PowerPoint PPT Presentation

Nov 25, 2022 359 likes •739 views

VEGAS Effective Geometric Algorithms for Surfaces and Visibility Proposed in July 2004 Created in August 2005 Evaluation period: 2005-2006 p. 1 VEGAS project members Permanent: Sylvain Lazard CR INRIA Sylvain Petitjean CR CNRS Hazel

slide-1
SLIDE 1

VEGAS Effective Geometric Algorithms for Surfaces and Visibility

Proposed in July 2004 Created in August 2005 Evaluation period: 2005-2006

– p. 1

slide-2
SLIDE 2

VEGAS project members

Permanent: Sylvain Lazard CR INRIA Sylvain Petitjean CR CNRS Hazel Everett

  • Prof. Univ. Nancy 2

Xavier Goaoc CR INRIA (since May 2005) Laurent Dupont

  • Assist. Prof. Univ. Nancy 2 (since Sept. 2005)

Marc Pouget CR INRIA (since Sept. 2006)

  • PhD. Students:
  • Temp. Engineer:

(since Sept. 2006) Postdoc: Joint with GEOMETRICA (since Oct. 2006)

– p. 2

slide-3
SLIDE 3

Scientific Objectives

Contribute to the development of

effective geometric computing treating complex geometric objects

– p. 3

slide-4
SLIDE 4

Scientific Objectives: Focus

3D Visibility problems Geometric computing with curved objects

– p. 4

slide-5
SLIDE 5

Scientific Objectives: Example

How to generate high-quality rendered images of scenes modeled with low-degree algebraic surfaces?

– p. 5

slide-6
SLIDE 6

Scientific Objectives: Originality

We attack all the aspects from theory to practice needed for the development of

certified and effective geometric computing

dedicated to non-discretized non-linear problems

– p. 6

slide-7
SLIDE 7

Scientific Objectives: Non-linear Objects

Need to manipulate non-linear objects Classical approach: discretize into a mesh of triangles

  • universal, simple algorithms, hardware
  • numerical error, lots of triangles

Our approach: algorithms that take into account the exact geometry of objects

– p. 7

slide-8
SLIDE 8

Scientific Objectives: Non-linear Objects

Need to manipulate non-linear objects both concrete and abstract:

– p. 8

slide-9
SLIDE 9

Scientific Objectives: Robustness

Is R to the left, to the right, or on line PQ ?

P Q R

Compute the sign of

  • Px

Qx Rx Py Qy Ry 1 1 1

  • Finite-precision floating-point computation

Topological incoherences CRASH ! Robust algorithms:

  • treat all degenerate situations
  • implement geometric decisions exactly

– p. 9

slide-10
SLIDE 10

Scientific Achievements: Plan

3D visibility Geometric computing with curved surfaces

– p. 10

slide-11
SLIDE 11

3D Visibility: Panorama

3D Visibility problems

  • surface-to-surface visibility queries
  • shadows - limits of umbra and penumbra

Computations are discretized and extremely costly Objectives: Efficient algorithms

(No special use of graphics hardware)

Long-term goals: more efficient rendering of higher quality

– p. 11

slide-12
SLIDE 12

3D Visibility: Research directions

  • Theory of lines in space
  • Algorithmic and implementation
  • f 3D vis. data structures
  • 3D visibility and applications

– p. 12

slide-13
SLIDE 13

3D Visibility: Theory of lines in space

4 lines in 3D admit at most 2 (cc) transversals 4 segments in 3D admit at most 4 (cc) transversals

[BELSW 2005]

– p. 13

slide-14
SLIDE 14

3D Visibility: Theory of lines in space

Maximum number of lines tangent to 4 triangles?

– p. 14

slide-15
SLIDE 15

3D Visibility: Theory of lines in space

Maximum number of lines tangent to 4 triangles? Lower bound: 62 Upper bound: 162 (naive 4·34 = 324) Example with “fat” triangles: 40

[BDLS 2005]

– p. 14

slide-16
SLIDE 16

3D Visibility: Theory of lines in space

4 spheres admit infinitely many tangents iff aligned centers and at least one common tangent

[BGLP 2006]

– p. 15

slide-17
SLIDE 17

3D Visibility: Theory of lines in space

Given a polyhedron Pn with n faces approximating a surface in a reasonable way On average over all viewpoints the silhouette of Pn is of size O(√n)

[G 2006]

– p. 16

slide-18
SLIDE 18

3D Visibility: Theory of lines in space

Given n disjoint unit balls in Rd Number of ordering in which a line pierces all the balls?

A B C

A B C l1 l2 l3

l1 : ABC l2 : ACB l3 : BAC

The set of balls admits at most 2 distinct geometric permutations if n > 8 and at most 3 if n 8

Tight bounds except for n = 4,...,8

[CGN 2005]

– p. 17

slide-19
SLIDE 19

3D Visibility: Theory of lines in space

Given n disjoint unit balls in Rd Helly-type theorem for transversals to balls: If every subset of 4d −1 balls admit a line transversal then the set of balls admits a line transversal

[CGHP 2006]

– p. 18

slide-20
SLIDE 20

3D Vis.: Complexity, Algorith. & Implem.

Sets of free line segments tangent to objects Take into account the structure of the objects For n triangles organized into k convex polytopes

  • Size Θ(n2k2) in the worst case
  • Algorithm Θ(n2k2logn) in the worst case

[BDD+ 2006]

– p. 19

slide-21
SLIDE 21

3D Visibility: Conclusion

  • Many fundamental results on the properties of free lines

and line segments in space

  • Practical algorithm for computing 3D visibility global

data structures

  • Ongoing implementation

– p. 20

slide-22
SLIDE 22

Scientific Achievements: Plan

3D visibility Geometric computing with curved surfaces

– p. 21

slide-23
SLIDE 23

Geometric comput. with curved surfaces

Low-degree surfaces are everywhere 95 % of surfaces of mechanical objects are made up of quadrics and torii

– p. 22

slide-24
SLIDE 24

Intersection of quadrics

Compute an exact parametric form

  • f the intersection of two

arbitrary implicit quadric surfaces

Input:

Q1 : 4x2 +z2 −1 = 0 Q2 : x2 +4y2 −z2 −1 = 0

Output: Smooth quartic

     2u3 −6u 7u2 +3 10u2 −6 2u3 +18u     ±      −2 u 2u 2     ·

  • ∆(u)

∆(u) = −3u4 +26u2 −3

– p. 23

slide-25
SLIDE 25

Intersection of quadrics

Compute an exact parametric form

  • f the intersection of two

arbitrary implicit quadric surfaces

Input:

Q1 : z2 +xy = 0 Q2 : x+yz = 0

Output:

Line Cubic

     u 1           −u3 4u −2u2 −8     

– p. 23

slide-26
SLIDE 26

Intersection of quadrics

Algorithm

  • Classification of the type of intersection in P3(R)

(smooth quartic, cubic & line, 2 conics, etc.)

  • Parameterization of each component
  • Rational parameterization if one exists
  • Optimal or almost optimal parameterization in the

degree of the extension of Z of the coefficients

[DDLP03, D04, DLLP 05a, DLLP 05b, DLLP 05c]

– p. 24

slide-27
SLIDE 27

Intersection of quadrics

Implementation

[LPP 2006]

  • Efficient C++ implementation (∼ 20 000 lines)
  • On-line web server
  • Code distributed: QI (INRIA License)

– p. 25

slide-28
SLIDE 28

Intersection of quadrics

Implementation

[LPP 2006]

  • Efficient C++ implementation (∼ 20 000 lines)
  • On-line web server
  • Code distributed: QI (INRIA License)

Applications

  • Interactions between potential energy surfaces
  • Dept. of Chemistry, Imperial College, London
  • Image of conics seen by a catadioptric camera with a

paraboloidal mirror

IRIT (CNRS - Univ) Toulouse

  • Spacecraft thermal radiation analysis

– p. 25

slide-29
SLIDE 29

Intersection of quadrics

Experiments

[LPP 2006]

  • Random quadrics, coeffs up to 10 digits: < 50ms
  • Chess set:
  • 6 pieces (108 quadrics), 971 intersections
  • 3.4 ms on average

– p. 26

slide-30
SLIDE 30
  • Comput. with curved surf.: Conclusions

Major leap forward on certified computations with quadrics

– p. 27

slide-31
SLIDE 31

Objectives of the project proposal

  • 3D visibility
  • Theory of lines in space
  • Algorithmics and implementation
  • Visibility and applications
  • Geometric computing with curved objects

– p. 28

slide-32
SLIDE 32

Objectives for the next four years

  • Theory of lines in space
  • 3D visibility
  • Theory of lines in space
  • Algorithmics and implementation
  • Visibility and applications (longer term)
  • Geometric computing with curved objects

– p. 28

slide-33
SLIDE 33

Objectives for the next four years

Theory of lines in space

  • Discrete properties of sets of lines

Combinatorial geometry: Helly-type theorems, geometric permutations

  • Combinatorial complexity

Complexity of sets of lines in space

  • Efficient and effective computing on sets of lines

Predicates for line transversals

– p. 29

slide-34
SLIDE 34

Objectives for the next four years

3D visibility

  • Theory & Algorithms

Visibility skeleton for non-convex polyhedra Shadows

  • Implementation

Visqueux: Visibility skeleton

– p. 30

slide-35
SLIDE 35

Objectives for the next four years

Geometric computing with curved objects

  • Certified computation with low-degree surfaces

Boundary evaluation, Medial axis of polyhedra Geometric computing with algebraic tools Improving QI

  • Math. investigation of geometric features on surfaces

Differential geometry for smooth and discrete objects “Visual event” curves on smooth surfaces

– p. 31

slide-36
SLIDE 36

Collaborations

GEOMETRICA, SALSA McGill (Montréal), Poly. Univ (New-York), KAIST (Korea) and many more punctual collaborations Also close contacts with CACAO, ALICE, ARTIS

– p. 32

slide-37
SLIDE 37

Conclusions

Robust and efficient geometric algorithms dealing with curved objects may be obtained by using the right mathematical tools making thorough treatment of degenerate cases and a careful implementation of the primitive operations

– p. 33