Robust Geometric Computation

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Robust Geometric Computation

• Eli Broadhurst

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Define Robust

• Full of health and strength vigorous.
• Powerfully built sturdy.
• Requiring or suited to physical strength or
  endurance robust labor.
• Rough or crude boisterous a robust tale.
• Marked by richness and fullness full-bodied a
  robust wine.

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First a typical example

• Polyhedral intersection

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Numerical Errors in floating point numbers

• Input conversion
• Round-off
• Digit-cancellation

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Formal Definition

• In geometric computations, logical facts such
  as incidence, separation, tangency, etc., are
  deduced based on numerical calculations. Further
  inferences are drawn from these deductions. The
  imprecision of floating-point arithmetic makes
  the conclusions drawn from the numerical
  calculations unreliable. In particular, the same
  logical question may arise repeatedly, in the
  form of different floating-point computations
  giving contradictory answers. Unless this
  problem is avoided, it is not possible to write
  fully correct/robust/reliable code for geometric
  operations using standard floating-point
  arithmetic.

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What does this mean?

• (approx.) numerical data vs. exact symbolic data
  (used for geometric reasoning).
• Without robustness geometric programs cannot be
  layered or folded.

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Solutions

• Symbolic Reasoning
• Give priority to one of the data structures
• Exact Arithmetic
• Reliable calculation
• Interval arithmetic

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What is a solution?

• Exact positions of all primitives?
• Consistent symbolic data?

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A typical CS approach

• Use an epsilon
• Results are often sequence-dependent, not
  transitive, inaccurate, or inconsistent
• Cannot have features smaller then 2 epsilon,
  good and bad

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Symbolic Reasoning

• Give priority to the symbolic data
• Ask every question only once or make sure the
  answers agree
• Symbolically reason, as part of the computation,
  which deductions have already been made
• Non-trivial to prove an algorithm is robust,
  cognizant of all possibilities

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Symbolic Reasoning

• May not be correct
• Sequence-dependent but consistent symbolic data
• Exact implementation is too complex
• Vertices of face are not all co-planar
• Combine with epsilon approach

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Symbolic Reasoning

• Example
• Voronoi diagram computation by Sugihara, 1994
• May be incorrect but it always computes correct
  topological data structures and will never fail
• www.mpi-sb.mpg.de/mehlhorn/ftp/ifip94.ps

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Solutions

• Symbolic Reasoning
• Give priority to one of the data structures
• Exact Arithmetic
• Reliable calculation
• Interval arithmetic

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Exact Arithmetic

• In 1969 MIT built Macsyma, uses exact rational
  arithmetic
• Floating-point numbers ! real numbers
• Proliferation
• Irrationality

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Proliferation

• If the input to a geometric operation has k-digit
  precision, the output may require more
• Compound Geometric Operations
• Exponential growth

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Irrationality

• Rigid body motions
• Translation
• Rotation

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Segment Intersection

• Given endpoints with single-precision
• Output Do they intersect?
• Can be done exactly if internal calculations are
  done with double-precision

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Computing the intersection

• Need roughly triple precision to compute the
  actual intersection points
• 3k 3

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Exact Polyhedral Intersection

• Only face (plane) information is given
  numerically, everything else symbolic
• Triangle is defined by a main face and 3 side
  faces
• Vertex is defined by the intersection of the main
  face with 2 other faces

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Exact Poly. Intersection

• Plane has equation
• ax by cz d 0
• Where L lt a,b,c lt L
• -L2 lt d lt L2
• L 2k
• a,b, and c are defined by k bits
• d by 2k bits

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Exact Poly. Intersection

• Why use double precision for d?

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Exact Poly. Intersection

• Boolean Operations
• Always representable
• All operations depend on intersection test of 4
  planes
• No proliferation!
• Vertices are always the at intersection of 3
  planes

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Exact Poly. Intersection

• Computing Vertices

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Exact Poly. Intersection

• Coordinates are rational
• (Ux/D, Uy/D, Uz/D)
• Needed precision?
• Multiplication
• ab lt a b 2k 2k 22k
• Requires twice as many bits to represent
• Addition
• ab lt max(a,b) 1 2k1
• Requires one more bit

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Exact Poly. Intersection

• For vertex calculation
• Need 4k3 bits to do calculation and to
  represent as a rational number
• 4k because d 22k
• Precision needed for intersection of 4 planes?

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Exact Poly. Intersection

• If J 0 the 4 planes intersect
• If J, D have same sign, planes 1,23 intersect
  above plane 4, else below
• Needed precision?
• J lt 24L5lt 25k5

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Rigid motions

• Translation
• x1 x tx, y1 y ty, z1 z tz
• Maps
• axbyczd 0 -gt ax1by1cz1e 0
• Where e atx bty ctz d
• Representable only if
• e lt L2

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Rigid motions

• Rotations
• Only rational angles
• And need to check a, b, c and make sure they are
  still lt L

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Rounding

• Can find the closest rational number to a real
  number by using continued fractions
• Finding the closest plane to an unrepresentable
  one reduces to this same problem
• No guarantee there exist representable grid lines
  that can bound all features (preserve topology)
• Require minimum feature separation

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Solutions

• Symbolic Reasoning
• Give priority to one of the data structures
• Exact Arithmetic
• Reliable calculation
• Interval arithmetic

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Interval arithmetic

• We pretend to use real numbers, but really use a
  finite approximation
• Definition of floating-point interval
• x a,b where a,b are floating-point numbers
  and a lt b
• x r ? R a lt x lt b
• The key objective is to come up with an interval
  enclosure that is as tight as possible

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First real intervals

• Diameter d(x) ba
• Midpoint m(x) (ba)/2
• Smallest abs value ltxgt minx x ? x
• Greatest abs value x maxx x ? x
• 0 ? x iff ltxgt 0
• Relative Diameter drel d(x)/ltxgt
• x y ac, bd
• x y a-d, b-c
• x y minac,ad,bc,bd,
• maxac,ad,bc,bd

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Common Functions

• Continuity -gt these are still intervals
• x2 ltxgt 2, x 2
• Sqrt(x) sqrt(a), sqrt(b)
• ex ea, eb
• Log(x) log(a), log(b)

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Functions

• f(x)?

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Floating-point Intervals

• The floating-point numbers must be rounded
  properly
• Nearest machine-representable number
• xy a ? c, b d
• xy a ? d, b c
• Some FPUs allow mode change to the appropriate
  rounding type

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Floating-point Intervals

• Do obtain automatic enclosure of a true result
• Usefulness depends on how tight the interval is

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Interval Bezier Curves

• Control points have circular tolerance regions
• Union of all possible control points

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Conclusions

• Exact computation
• Problem of getting exact input data to be correct
• Planarity of non-triangles
• Avoided exponential growth of precision
• Less special cases than when dealing with the
  looseness of floating-point
• Works for Boolean operations, so so for
  translations and rotations
• http//cs.nyu.edu/exact/

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Conclusions

• Symbolic Reasoning
• Tries to make sense of imprecise data to find
  consistent interpretations
• Hard to capture all cases, large programming
  effort
• Not proven that it can confer absolute robustness

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Conclusions

• Interval Arithmetic
• Encapsulates the correct answer
• But there are cases when the enclosure is not
  sufficiently tight, and so have uncertainty in
  interpreting the results again

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References

• Robustness in Geometric Computations by C.
  Hoffmann, 2001
• http//www.cs.purdue.edu/homes/cmh/distribution/pa
  pers/Robustness/JCISE01.pdf
• Geometric and Solid Modeling by C. Hoffmann, 1989
• http//www.cs.purdue.edu/homes/cmh/distribution/bo
  oks/geo.html

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Floating-point numbers

• Single-precision
• 8-bit exponent
• 23-bit mantissa
• Double-precision
• 11-bit exponent
• 52-bit mantissa