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Elementary Constructions in Spatial Constraint Solving
Christoph M. HoffmannChing-Shoei Chiang, Bo Yuan
Computer Science, Purdue University
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Graphics Vs. Constraints
n Traditional connections:n Constraint-based model creation (CAD)n Constraint-based scene creation (assembly)
n Other connections:n Constrained motion (actors, shadows, …)
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What is a GC Problem?n A set of geometric elements in some space
n Points, lines, arcs, spheres, cylinders, …
n A set of constraints on themn Distance, angle, tangency, incidence, …
n Solution:n Coordinate assignment such that the constraints
are satisfied, or notification that this cannot be done.
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Task StructureProblem preparation§ Underconstrained, fixed, etc.§ Certain transformations, reasoning
Decomposition of large problemsn Degree of freedom analysisn Graph analysis
Equation solvingn Numerical techniquesn Algebraic techniques
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2D Constraint Solving
n Fairly mature technology –n Efficient, robust and competent solvers
n Triangle decomposition of problems or other methodsn Points, lines, circular arcsn Distance, angle, tangency, perpendicularity, etc.n Under- and overconstrained casesn Variety of extensions
n Other techniques also succeed
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What Helps the Planar Case
1. Small vocabulary already useful2. Small catalogue of algebraic systems 3. Algebraic systems easy
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Example: Apollonius’ Problem
n Given 3 circles, find a circle tangent to all of them:n Degree 8 system – but it factors into
univariate quadratic equations by a suitable coordinate transformation
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3D Solvers and Issuesn Points and planesn Lines as well as points and planesn Graph decomposition is OK
n Hoffmann, Lomonosov, Sitharam. JSC 2001
n But equation solving is tricky:n Sequential case involving linesn Simultaneous casesn No compact subset that has good applicability
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Consequences
n Spatial constraint solvers are fairly limited in ability:n Technology limitations impair application
conceptsn Limited application concepts fail to make
the case for better technology
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Problem Subtypes
n Sequential:n Place a single geometric element by
constraints on other, known elements
n Simultaneous:n Place a group of geometric elements
simultaneously
n In 2D, sequential problems are easy, but in 3D…
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Equation Solving Techniques
1. Geometric reasoning plus elimination
2. Systematic algebraic manipulation
3. Parametric computation
4. Geometric analysis (of sequential line constructions)
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Octahedral Problems
n 6 points/planes, 12 constraints:
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6p Example
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Early Solutions (Vermeer)
n Mixture of geometric reasoning and algebraic simplification using resultants
n Univariate polynomial of degree 16 for 6p – tight bound
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Michelucci’s Solution
n Formulate the Cayley-Menger determinant for 2 subsets of 5 entities
n Yields two degree 4 equations in 2 unknowns
n Extensions for planes
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Systematic Framework (Durand)
n Process for 6p:1. Gaussian elimination2. Univariate equation solving3. Bilinear and biquadratic equation
parameterizationn 3 quartic equations in 3 variables (6p).
BKK bound is 16.n Homotopy tracking for 16 paths.
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Simultaneous 3p3L
n Complete graph K6
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Systematic Solution (Durand)
n Initially 21 equations, process as before
1. Gaussian elimination2. Univariate equation solving3. Bilinear and biquadratic equation
parameterization
n 6 equations in 6 variables, but total degree is 243 83
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Durand cont’d
n Homotopy techniques applied to special case of orthogonal lines (~4100 paths):
122Failure3031At Infinity
895Complex48Real
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3p3L Example (Ortho Lines)
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Simultaneous 5p1L
n Place 5 points and 1 line from distance constraints between the line and every point and between the points, in a square pyramid
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5p1L Problem, Systematic
n Systematic algebraic treatment yields a system of degree 512
n Coordinate system choicesn Heuristic: Choosing the line in a standard
position tends to yield simpler equations
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5p1L, Adding Reasoningn Approach:
n Line on x-axis, 1 point on z-axisn One point placed as function of z(t) = tn Other points yield constraint equations
n Result:n System (42,34,22) not resolving square
roots.n No significant algebraic simplifications
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5p1L, Computation (Yuan)
n The parameterized equations are numerically quite tractablen Trace the curve of the “missing dimension”
numericallyn Intersect with the nominal value
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Example Problem
7.1859d419.5569d23
9.1500d342.4992d12
0.8548r5
5.0478d544.4887r4
6.5569d535.1201r3
4.9275d523.4797r25.4039d515.1286r1
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Resulting Curve
12 real solutions found
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Sequential: L-pppp
n Given 4 fixed points and 4 distances, place a line
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Nota Bene
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L-pppp Solved
Coordinate system choice
),,,(:
),0,,(:),0,0,(:
),0,0,0(:),,;,,(:
44
33
22
11
rfedS
rcbSraS
rSwvuzyxL
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Constraint Equations on L
1
0
)()()()(
)()()(
)()(
222
23
2222
22
2222
21
2222
20
222
=++
=++
=++−−+−+−
=+−+−+−
=−++−
=++
wvu
zwyvxu
rfwevdufzeydx
rcvbuzcybx
rauzyax
rzyx
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Algebraic Simplifications
n Use equations (2), (3) and (4) to solve for x, y, and z
n Resulting system has three equations of degree 4, 3, and 2 (Bezout bound 24)
n But if (u,v,w) solves the system, then so does (-u,-v,-w)…
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Structure of surface of line tangents to 3 spheres on Gauss sphere
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L-Lppp Problemn Construct a line from another line and up to 3
pointsn Subcases, by LL constraints:
n L-Lpp: The lines are parallel; clearly 2 solutions maximum
n L-Lpp: A distance is required; need good understanding of a kinematic curve
n L-Lppp: No distance is required (includes perpendicular); intersect 3 of the L-Lpp curves
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Subcase LL Parallel
n Take a plane perpendicular to the fixed line
n Sphere silhouettes intersect in up to two points
n Up to two solutions
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Main Tool
n Given 2 spheres and a direction, find the two tangents in that direction
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Subcase LL Distance
n Only 2 spheres neededn Fix plane at complement angle to fixed
linen Rotate the 2 spheres around the fixed
axis yielding silhouette intersection curves
n Intersect with horizontal line
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Silhouette curve degree 18?
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No LL Distance
n Additional constraint from a third sphere (point with distance)
n Intersect the silhouette intersection pairs
n No degree estimates
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Silhouette intersection curves meet at intersections of sought tangents