extending constructive solid geometry to … csgdominant...

Introduction Basic CSG Dominant Sets Projections Examples Implementation Conclusion

Extending Constructive Solid Geometry toProjections and Parametric Objects

George Tzoumas

joint work with D. Michelucci and S. Foufou

CNRS UMR 5158, LE2I, University of Burgundy, France

TMCE 2014May 19–23, 2014, Budapest, Hungary

Extending CSG to Projections and Parametric Objects G. Tzoumas ∗ 1/27


Constructive Solid Geometry

Boolean constructions via set operators:I quantifier-free boolean formula

I atoms P < 0, P = 0, P ∈ PI operators ¬,∨,∧ for complement Ac , ∪, ∩I difference A− B = A ∩ Bc , De Morgan. . .

∩

circle parabola

CSG-tree representation



Motivation

Unified representation

I CSG primitives (boolean formulas)

I Projections (e.g. parametric solids)

I Minkowski Sums

I Extrusions or Sweeps

Basic functionality

I Test if set is empty. Force a set to be non-empty

I Compute topology (simplicial complex homotopy equiv.)



Previous work (list non-exhaustive)

Topological propertiesI semi-algebraic sets

I Cylindrical Algebraic Decomposition [Collins, 1975]

I semi-algebraic subsets homeomorphic to open boxes

I Interval Arithmetic [Delanoue et al, 2006]

I simplicial complex homotopy equivalent

I implicit surfacesI Interval Analysis & Morse Theory [Hart et al, 1997]



Primitives

Assume F (x , y) ≤ 0 (e.g. x2 + y2 − 1 ≤ 0)

I F (x , y) < 0 → points “inside”

I F (x , y) = 0 → points on the boundary

I F (x , y) > 0 → points “outside”

x : [. . .]y : [. . .]



Primitives“convert” inequality F (x , y) ≤ 0 to equality by adding a slack variable:

F (x , y)− s = 0, s ∈ (−∞, 0]

F (x , y) ≤ 0 −→ f (x , y ; s) = 0

I s < 0 → points “inside”

I s = 0 → points on the boundary

I s > 0 → points “outside”

I s : characteristic variable of set A

current newx : [. . .] s : [a, b]y : [. . .]...



Complement

GivenA : f (x; s) = 0

We obtain¬A : f (x;−s) = 0

Remark

complement still represented by points from the manifold



Disjunctive Normal Form

Given primitives A,B and expressions P,Qi

I A and ¬A are in DNF

I A ∪ B and A ∩ B are in DNF

I ¬(Q1 ∪ Q2) = ¬Q1 ∩ ¬Q2

I ¬(Q1 ∩ Q2) = ¬Q1 ∪ ¬Q2

I P ∩ (Q1∪Q2∪ . . .∪Qn) = (P ∩Q1)∪ (P ∩Q2)∪ . . .∪ (P ∩Qn)

projection distributes over union

π(Q1 ∪ Q2 ∪ . . . ∪ Qn) = π(Q1) ∪ π(Q2) ∪ . . . ∪ π(Qn)



Union

P = A1 ∪ . . . ∪ An

Consider each primitive separately:

A1 : f1(x; s1)

A2 : f2(x; s2)...

An : fn(x; sn)

Remark

x may not be uniquely associated with a primitive



Intersection

P = A1 ∩ . . . ∩ An

Problem

x should be uniquely associated with a primitive



Intersection as union of disjoint sets (Dominant Set)A dominates B: A|B

I x ∈ A ∩ B

I value of A ≥ value of B

A ∩ B = A|B ∪ B|A (max)

A

B

A|B

B|A

A|B1, . . . ,Bn := x ∈ Rd : 0 ≥ sA ≥ sBi



Properties of dominant sets

A ∩ B = A|B ∪ B|A(A|B)|C = A|B,C

A|(B|C ) ∪ A|(C |B) = A|B,CA ∩ B ∩ C = (A|B,C ) ∪ (B|C ,A) ∪ (C |A,B)

¬(A|B) = ¬A ∪ ¬B ∪ B|A

sA

sB

0 ≥ sA ≥ sB



ProjectionA : f (x , y , z ; s) = 0

πz(A) =

∃ z :f (x , y , z ; s) = 0s is minimal

Lagrangian: (f (x , y , z ; s) = x2 + y2 + z2 − 1− s)

L : s + λf (x , y , z ; s) (works, but ugly!)

(3×3) :

L′λ : f (x , y , z ; s) = 0L′s : 1− λ = 0 ⇒ λ = 1

L′z : λ∂f (x,y ,z;s)∂z = 0 ⇒ 2z = 0

⇒

s = x2 + y2 − 1λ = 1z = 0



Projection

Theorem

Let A : f (x; s) be a geometric primitive. When projecting down k dimen-sions (eliminating x1 . . . xk), the projection can be specified by:

πk(A) −→ ∂f

∂x1=

∂f

∂x2= . . . =

∂f

∂xk= 0



Projection (revisited)

(2× 2) :

{f (x , y , z ; s) = 0∂f (x,y ,z;s)

∂z = 0⇒{

s = x2 + y2 − 1z = 0



Join Set

Primitives of more than one equation

A ./ B := x ∈ Rd : sA = sB ∧ sA ≤ 0

fA(x; sA) = 0fB(x; sB) = 0sA − sB = 0



Join Set ProjectionRecall that we minimize s wrt projected variables xi . . .

I ./ → constraint (reduces dim.)

I even more constraints → Jacobian

Lemma

Let A : f0(x; s) and Bi : fi (x; s), i = 1 . . . n geometric primitives. Then

πk(A ./ B1 ./ · · · ./ Bn)→{

∅, k ≤ nJi0i1...in(f0, f1, . . . , fn) = 0, k > n

,

where 1 ≤ i0 < i1 < . . . < in ≤ k



Dominant Set Projection

A

B

A|B

B|A

Theorem

πk(A|B) = πk(A)|B ∪ πk(A ./ B)

Constrained optimization problem. Critical value of s:

I inside primitive

I boundary conditions (joins)



Dominant Set Projection (generalized)

Theorem

πk(A|[Bm]1:n) =

πk(A)|[Bm]1:n⋃ni=1 πk(A ./ Bi )|[Bm]1:nm 6=i⋃ni ,j=1i<j

πk(A ./ Bi ./ Bj)|[Bm]1:nm 6=i ,m 6=j⋃ · · ·⋃πk(A ./ B1 ./ · · · ./ Bn)



Projection of Intersection

Basic transformation

π(A ∩ B) = π(A)|B ∪ π(B)|A ∪ π(A ./ B)

I contributing points from each primitive (critical points of s)

I contributing points from joint (boundary condition)



Intersection of projection of intersections

π(E1 ∩ E2) ∩ π(E1 ∩ E3)

= [π(E1|E2) ∪ π(E2|E1)] ∩ [π(E1|E3) ∪ π(E3|E1)] A ∩ B = A|B ∪ B|A= [π(E1|E2) ∩ π(E1|E3)] ∪ [π(E1|E2) ∩ π(E3|E1)]∪ (A ∪ B) ∩ (C ∪ D)→

[π(E2|E1) ∩ π(E1|E3)] ∪ [π(E2|E1) ∩ π(E3|E1)] DNF= [π(E1)|E2 ∪ E1 ./ E2] ∩ [π(E1)|E3 ∪ E1 ./ E3] ∪ · · · π(A|B) == [π(E1)|E2 ∩ π(E1)|E3] ∪ · · · π(A)|B ∪ 6 π(A ./ B)= [π(E1)|E2, π(E1)|E3)] ∪ [π(E1)|E3, π(E1)|E2] ∪ · · ·=

⋃18i=1 Si



Intersection of projection of intersections

π(E1 ∩ E2) ∩ π(E1 ∩ E3)

set contributing set formula

S1 π(E1)|E2, π(E1)|E3

S2 π(E1)|E3, π(E1)|E2

S3 π(E1) π(E1)|E2,E1 ./ E3

S4 π(E1)|E3,E1 ./ E2

S5 π(E1)|E2, π(E3)|E1

S6 π(E1)|E3, π(E2)|E1

S7 π(E2)|E1, π(E1)|E3

S8 π(E2) π(E2)|E1,E1 ./ E3

S9 π(E2)|E1, π(E3)|E1

S10 π(E3)|E1, π(E1)|E2

S11 π(E3) π(E3)|E1,E1 ./ E2

S12 π(E3)|E1, π(E2)|E1

S13 E1 ./ E2|(π(E1)|E3)S14 E1 ./ E2 E1 ./ E2|E1 ./ E3

S15 E1 ./ E2|(π(E3)|E1)S16 E1 ./ E3|(π(E1)|E2)S17 E1 ./ E3 E1 ./ E3|E1 ./ E2

S18 E1 ./ E3|(π(E2)|E1)



Parametric Disk

Via projections: project (eliminate) parameters

X : x − r cos θ = 0Y : y − r sin θ = 0R : r2 − 1 = s

,θ ∈ [−π, π)r ∈ [0, 1]

6 πr ,θ(R ./ X ./ Y )



Parametric Annulus

F : (x − cos t)2 + (y − sin t)2 − 1

4− s = 0 πt(F )→ ∂F

∂t= 0



Python/SAGE

x,y,z = SR.var(’x,y,z’)

A=PrimitiveSet((x− 34)2 +(y− 3

4)2 +(z− 3

4)2 < 2

3,

{x:RIF(-2,2), y:RIF(-2,2),z:RIF(-2,2)})B=PrimitiveSet((x− 1

4)2 +(y − 1

4)2 +(z− 1

4)2 < 1,

{x:RIF(-2,2), y:RIF(-2,2),z:RIF(-2,2)})G=ProjectionSet(IntersectionSet(A,B),set([z]))

Quimper [Chabert-Jaulin’09]

1.

116

(4z − 3)2 + 116

(4y − 3)2+

+ 116

(4x − 3)2 − s0 − 23

= 0

116

(4z − 1)2 + 116

(4y − 1)2+

+ 116

(4x − 1)2 − s1 − 1 = 0

2z − 32= 0, s0 − s1 ≥ 0

2.

116

(4z − 3)2 + 116

(4y − 3)2+

+ 116

(4x − 3)2 − s0 − 23

= 0

116

(4z − 1)2 + 116

(4y − 1)2+

+ 116

(4x − 1)2 − s1 − 1 = 0

s0 − s1 = 0

3.

116

(4z − 3)2 + 116

(4y − 3)2+

+ 116

(4x − 3)2 − s0 − 23

= 0

116

(4z − 1)2 + 116

(4y − 1)2+

+ 116

(4x − 1)2 − s1 − 1 = 0

2z − 12= 0, s1 − s0 ≥ 0

4. Identical to 2.

π(A ∩ B) = π(A)|B ∪ π(A ./ B) ∪ π(B)|A ∪ π(B ./ A)



Simplicial complex homotopy equivalent[Delanoue et al, 2006] extension for more types of sets

I projections, Minkowski sums

I star test

interior point s is a star

I if any segment from s lies inside

I if tangent planes at the boundary leave s on the same side(do not pass through!)



Work in progress/Future work

I Star test(boundary, gradient?)

I other constructions(medial axis)

I Minkowski Sums(characteristic variable?)

Thank you!


extending constructive solid geometry to … csgdominant...

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