Topologically Guaranteed
Bivariate Solutions of
Under-constrained
Multivariate Polynomial
Systems
Yonathan Mizrahi
Prof. Gershon Elber
Department of Mathematics
Technion
Department of Computer Science
Technion
1
Outline
 Topology
– basic concepts:
General intro
Homeomorphisms
Manifolds
Classification theorems
solvers – a quick review?
 Bivariate solutions with topological
guarantee
 Conclusion
 Subdivision
Center for Graphics and Geometric Computing, Technion
2
Topology – basic concepts
3
Topology
The study of properties of space that are
invariant under continuous mappings, for
example:
 Closed/Open sets
 Compactness
 Connectedness
 Interior/boundary points
Center for Graphics and Geometric Computing, Technion
4
Topology
A topological space is a set 𝑋 equipped with a
family of subsets 𝜏 ⊂ 2 𝑋 called “open”, that
must satisfy a list of requirements
Topological spaces are considered equivalent
if there’s a continuous bijection with a
continuous inverse between them. Such a
function is called a homeomorphism.
Center for Graphics and Geometric Computing, Technion
5
Equivalent spaces - examples
open intervals in ℝ
 An open interval and the entire line
 Two open discs in the plane
 A sphere and the boundary of a cube
 Two
Center for Graphics and Geometric Computing, Technion
6
Manifolds
Topological spaces* that are locally
homeomorphic to ℝ𝑛 are called 𝑛-manifolds
 A curve with no self intersections is a
manifold of dimension one
 A two dimensional manifold is also called a
regular surface
 A sub-manifold is a manifold w.r.t to the
induced topology (defined by intersection
with open sets of the embedding space)
Center for Graphics and Geometric Computing, Technion
7
Classification theorems
Connected compact manifolds:
 Dimension zero: a point
 Dimension one: a closed interval or a closed
loop
 Dimension two: determined by the genus
and the number of boundary loops
(orientable case)
Center for Graphics and Geometric Computing, Technion
8
Subdivision solvers
A quick review?
9
Algebraic constraints
Problems (often…) reduce to:
𝐹 𝑥 =0
The Stewart Platform
(Kinematics)
http://www.physikinstrumente.com
Bisectors
(CAGD)
Center for Graphics and Geometric Computing, Technion
10
The general problem
Let 𝐹: 𝐷 ⊂ ℝ𝑛 → ℝ𝑘 𝑘 ≤ 𝑛 , where 𝐷
is the hyper-box:
𝐷 = 𝑎1 , 𝑏1 × ⋯ × 𝑎𝑛 , 𝑏𝑛 .
Consider the problem of finding:
𝑘
𝑍 = 𝑥 ∈ 𝐷: 𝐹 𝑥 = 0 ∈ ℝ
Center for Graphics and Geometric Computing, Technion
11
Basic assumptions
1. The underlying function 𝐹 is piecewise 𝐶 1 smooth
2. The gradients of the scalar
components of 𝐹 = 𝑓1 , … , 𝑓𝑘 are
linearly independent at solution
points (i.e. a maximal rank
differential)
Center for Graphics and Geometric Computing, Technion
12
Basic solution set properties
Under the regularity assumption, the
solution set:
𝑍 = 𝑥 ∈ 𝐷: 𝐹 𝑥 = 0 ∈ ℝ𝑘
Is an 𝑛 − 𝑘 dimensional sub-manifold of
𝑛
ℝ
(Inequalities: further trim, post-process)
Center for Graphics and Geometric Computing, Technion
13
Function representation
Useful properties of the Bézier (or Bspline) representation:
Convex hull property
Convergence of coefficients via
domain subdivision
𝐹 𝑥1 , … , 𝑥𝑛 = 𝑓1 , … , 𝑓𝑘
=
…
𝑖1
𝑃𝑖1…𝑖𝑛 𝐵𝑖1 ,𝑚1 𝑥1 … 𝐵𝑖𝑛 ,𝑚𝑛 𝑥𝑛 .
𝑖𝑛
Center for Graphics and Geometric Computing, Technion
14
Recursive subdivision solvers
The key idea!
(extended next to
2DOF systems)
𝐷0
𝐷1,2
𝐷1,1
 Positivity/Negativity
test – purge?
 Topological test –
terminate subdivision
and trace numerically?
 Subdivide (tighter
coefficients!)
 Solve and unite
solutions
Center for Graphics and Geometric Computing, Technion
15
Motivation – the 2D case
surfaces in ℝ3
 Bisector surfaces
 Sweeping (implicit) surfaces
 2-DOF kinematic systems
 Medical iso-surfaces
 Minkowski sums
 Implicit
Center for Graphics and Geometric Computing, Technion
16
Related Work I
 Fully
determined systems (0-DOF)
Lane-Reisenfeld 1981
Sherbrooke-Patrikalakis 1993
Elber-Kim 2001
Hanniel-Elber 2007
Wei-Feng-Lin 2011
Center for Graphics and Geometric Computing, Technion
17
Related Work II
 Under
determined systems: 1-DOF
Sederberg et-al 1988, 1996, 2007…
Grandine-Klein 1997
Mourrain-Pavone 2008
Burr-Choi-Galehouse-Yap 2008
Bartoň-Elber-Hanniel 2011
Center for Graphics and Geometric Computing, Technion
18
Related Work III
 Under
determined systems: 2-DOF
Lorensen-Cline 1987 (“Marching cubes”)
Plantinga-Vegter 2004
Bloomenthal et al 1988, 1995
Stadner-Hart 2005
Lin-Yap-Yu 2013
Center for Graphics and Geometric Computing, Technion
19
Related Work IV
Previous results frequently used (0D):
 Cone Test (CT): root isolation in fully
determined systems.
(Figure) Elber and Kim, 2001: Geometric Constraint Solver using
Multivariate Rational Spline Functions
Center for Graphics and Geometric Computing, Technion
20
Related Work V
Previous results frequently used (1D):
 No Loop Test (NLT)
 Single Component Test (SCT)
(Figure) Barton, Hanniel & Elber, 2011: Topologically Guaranteed Solution of
Underconstrained Polynomial Systems via No-loop and Single Component
Tests.
Center for Graphics and Geometric Computing, Technion
21
Our Contribution
 Topologically
guaranteed subdivision
termination criteria for ℝ𝑛 , 𝑛 > 3:
Terminate when the unknown solution is
homeomorphic to a closed disk.
A
triangulation method in ℝ𝑛 :
Yielding a mesh with disc topology
Constrained to a known boundary loop
Vertices on the solution surface.
Center for Graphics and Geometric Computing, Technion
22
Topologically Guaranteed
Subdivision Termination
Step 1: Solve the boundary problem, chain
and count loops (subdivide if two or more)
Step 2: Attempt to ensure existence
of a 2D planar homeomorphic image
of the unknown solution.
If 2D planar homeomorphic image is possible:
Empty boundary ⇒ empty solution
Single loop ⇒ disc topology
(terminate)
Center for Graphics and Geometric Computing, Technion
23
The Single Loop Boundary Test
 Recursion
invariant: boundary problem
is solved (chain all curves on faces)
 More than one loop - subdivide
 One (or no) loop:
Exactly one (or no)
component with boundary
Other components if exist: must be closed
surfaces (resolved in next phase test)
Center for Graphics and Geometric Computing, Technion
24
The Injective Projection Test
 Check
sufficient conditions for
projecting the (unknown, perhaps
empty) solution on a 2D plane
 Such
a mapping (if exists) is a
homeomorphism onto its image.
Center for Graphics and Geometric Computing, Technion
25
The Injective Projection Test
 If
such an injective projection exists,
the solution cannot contain any closed
surfaces.
 If possible – the solution cannot
contain closed surfaces!
 Previous ambiguity resolved:
Zero loops – empty solution
One loop – homeomorphic to a disk
Center for Graphics and Geometric Computing, Technion
26
The actual IPT procedure
1
𝐹 𝑥 = 0 ∈ ℝ𝑛−2
𝑥𝑘 = 𝑢 ∈ [𝑎𝑘 , 𝑏𝑘 ]
𝑥𝑙 = 𝑣 ∈ [𝑎𝑙 , 𝑏𝑙 ]
Theorem:
An injective projection is possible on
coordinates 𝑘, 𝑙 if Eq. (1) has at most a
single solution for any
𝑢, 𝑣 ∈ 𝑎𝑘 , 𝑏𝑘 × [𝑎𝑙 , 𝑏𝑙 ].
Center for Graphics and Geometric Computing, Technion
27
The actual IPT procedure
1
𝐹 𝑥 = 0 ∈ ℝ𝑛−2
𝑥𝑘 = 𝑢 ∈ [𝑎𝑘 , 𝑏𝑘 ]
𝑥𝑙 = 𝑣 ∈ [𝑎𝑙 , 𝑏𝑙 ]

The CT (Cone Test) for
uniqueness of 𝑛 × 𝑛
systems uses gradients only
 Independent of 𝑢, 𝑣
 The normal cones are the
original 𝑛 − 2 along with 𝑒𝑘 and 𝑒𝑙 (the standard
basis elements)
Center for Graphics and Geometric Computing, Technion
28
The actual IPT procedure
 An
example for 𝑛 = 3:
Center for Graphics and Geometric Computing, Technion
29
Tessellation (The Numeric Step)
 The
back-projection idea:
3. Vertices in ℝ𝑛 with the
same 𝑘, 𝑙 coordinates:
2. The 𝑛 − 2 dim’ normal
space where the back
projection problem is solved
1. Vertices in ℝ2 :
Center for Graphics and Geometric Computing, Technion
30
Test Results
𝑥 2 + 𝑦 2 + 𝑧 2 − 0.52 𝑧 − 0.75 = 0
Center for Graphics and Geometric Computing, Technion
31
Test Results
6
2
2
𝑖 2
2
𝑥 + 𝑦 + 𝑧 − 0.8 × 0.5
=0
𝑖=0
(Degree 14 with seven
disjoint components)
Center for Graphics and Geometric Computing, Technion
32
Test Results
𝑥 2 𝑦 2 + 𝑦 2 𝑧 2 + 𝑧 2 𝑥 2 + 0.32 𝑥𝑦𝑧 = 0
Subdivision tolerance: 0.05 (left) and 0.005 (middle and
right).
Polyline loops only (left and middle) and tessellation
(right).
33
Center for Graphics and Geometric Computing, Technion
Test Results
 Let 𝑆1
𝑢, 𝑣 , 𝑆2 𝑟, 𝑠 be parametric surfaces.
 The bisector surface is the locus of points for
which the (locally) shortest distance to each of
the input surfaces is equal.
 Two equations and four unknowns (solved in
𝑢, 𝑣, 𝑟, 𝑠 space and mapped to 𝑥, 𝑦, 𝑧 :
𝑆1 𝑢, 𝑣 − 𝑆2 𝑟, 𝑠 × 𝑛1 𝑢, 𝑣 , 𝑛2 𝑟, 𝑠 = 0
𝑛2 𝑟, 𝑠 2 𝑆1 𝑢, 𝑣 − 𝑆2 𝑟, 𝑠 , 𝑛1 𝑢, 𝑣 2 −
𝑛1 𝑢, 𝑣 2 𝑆1 𝑢, 𝑣 − 𝑆2 𝑟, 𝑠 , 𝑛2 𝑟, 𝑠 2 = 0
Center for Graphics and Geometric Computing, Technion
34
Test Results
 Bisector
examples:
Center for Graphics and Geometric Computing, Technion
35
Test Results
 Bisector
examples (contd’)
Center for Graphics and Geometric Computing, Technion
36
Test Results
 Sweep
surfaces:
A section curve 𝐶 𝑢 is swept along a
trajectory curve 𝑇 𝑣
The surface is defined by:
𝑆 𝑢, 𝑣 = 𝑇 𝑣 + 𝑀 𝑣 𝐶 𝑢
Here consider 𝑀 𝑣 = 𝐼 but can be adopted to
any matrix (used for rotating and non-uniform
scaling – a different problem)
Center for Graphics and Geometric Computing, Technion
37
Test Results
 Problem
- what if the section curve is given
implicitly:
𝑓1 𝑥, 𝑦, 𝑧 = 0
𝑓2 𝑥, 𝑦, 𝑧 = 0
 Solve
(in 𝑥, 𝑦, 𝑧, 𝑣 space and project):
𝑓1 𝑥, 𝑦, 𝑧 − 𝑇 𝑣
=0
𝑓2 𝑥, 𝑦, 𝑧 − 𝑇 𝑣
=0
Center for Graphics and Geometric Computing, Technion
38
Test Results
 Implicit sweep
examples: planar sections of
a torus (“Oval of Casini”) along cubic &
quadratic trajectories):
Center for Graphics and Geometric Computing, Technion
39
Test Results
 Implicit sweep
examples (sphere-torus
intersection curve swept along a cubic
trajectory):
Center for Graphics and Geometric Computing, Technion
40
Test Results
Minkowski sums:
𝑆1 ⨁𝑆2 = 𝑎 + 𝑏: 𝑎 ∈ 𝑆1 , 𝑏 ∈ 𝑆2
Envelope points satisfy the matched
normal vectors constraint:
𝜕𝑆2 𝑟,𝑠
𝑁1 𝑢, 𝑣 , 𝜕𝑟
=0
𝑁1 𝑢, 𝑣
𝜕𝑆2 𝑟,𝑠
, 𝜕𝑠
=0
Where:
𝑁1 =
𝜕𝑆1 𝜕𝑆1
×
𝜕𝑢 𝜕𝑣
Center for Graphics and Geometric Computing, Technion
41
Test results
Center for Graphics and Geometric Computing, Technion
42
Test results
Center for Graphics and Geometric Computing, Technion
43
Test results
Center for Graphics and Geometric Computing, Technion
44
Conclusion and Future Work
 Topological
guarantee:
Can be achieved assuming regularity
Up to subdivision tolerance
 Singular
sub-domains can be contained
 Framework can theoretically be
extended to higher dimensions with the
same topological arguments
Center for Graphics and Geometric Computing, Technion
45
Thank you
The research leading to this research has received partial
funding under the European Union's Seventh Framework
Programme FP7/2007-2013/ under REA grant agreement PIAPGA-2011-286426 - the GEMS project
http://www.geometrie.tuwien.ac.at/ig/gems
46