Algorithms.Geometry.SoS

Implementation of Simulation of Simplicity: A Technique to Cope with Degenerate Cases in Geometric Algorithms

By Herbert Edelsbrunner and Ernst Peter Mucke

Documentation

data Sign

The sign of an expression

Constructors

Negative

Positive

Instances

Bounded Sign

Defined in Algorithms.Geometry.SoS.Sign

Methods

minBound :: Sign

maxBound :: Sign

Enum Sign

Defined in Algorithms.Geometry.SoS.Sign

Methods

succ :: Sign -> Sign

pred :: Sign -> Sign

toEnum :: Int -> Sign

fromEnum :: Sign -> Int

enumFrom :: Sign -> [Sign]

enumFromThen :: Sign -> Sign -> [Sign]

enumFromTo :: Sign -> Sign -> [Sign]

enumFromThenTo :: Sign -> Sign -> Sign -> [Sign]

Eq Sign

Defined in Algorithms.Geometry.SoS.Sign

Methods

(==) :: Sign -> Sign -> Bool

(/=) :: Sign -> Sign -> Bool

Ord Sign

Defined in Algorithms.Geometry.SoS.Sign

Methods

compare :: Sign -> Sign -> Ordering

(<) :: Sign -> Sign -> Bool

(<=) :: Sign -> Sign -> Bool

(>) :: Sign -> Sign -> Bool

(>=) :: Sign -> Sign -> Bool

max :: Sign -> Sign -> Sign

min :: Sign -> Sign -> Sign

Show Sign

Defined in Algorithms.Geometry.SoS.Sign

Methods

showsPrec :: Int -> Sign -> ShowS

show :: Sign -> String

showList :: [Sign] -> ShowS

flipSign :: Sign -> Sign

Flip Positive = Negative.

signFromTerms :: (Num r, Eq r) => [r] -> Sign

Given the terms, in decreasing order of significance, computes the sign

i.e. expects a list of terms, we base the sign on the sign of the first non-zero term.

pre: the list contains at least one such a term.

type SoS d = (Arity d, HasDeterminant (d + 1))

A dimension d has support for SoS when we can: compute a dterminant of a d+1 by d+1 dimensional matrix.

sideTest :: (SoS d, Num r, Ord r, Ord i) => (Point d r :+ i) -> Vector d (Point d r :+ i) -> Sign

Given a query point q, and a vector of d points defining a hyperplane test if q lies above or below the hyperplane. Each point is assumed to have an unique index of type i that can be used to disambiguate it in case of degeneracies.

some 1D examples:

>>> sideTest (Point1 0 :+ 0) (Vector1 $ Point1 2 :+ 1)
Negative
>>> sideTest (Point1 10 :+ 0) (Vector1 $ Point1 2 :+ 1)
Positive
>>> sideTest (Point1 2 :+ 0) (Vector1 $ Point1 2 :+ 1)
Positive
>>> sideTest (Point1 2 :+ 3) (Vector1 $ Point1 2 :+ 1)
Negative

some 2D examples:

>>> sideTest (Point2 1 2 :+ 0) $ Vector2 (Point2 0 0 :+ 1) (Point2 2 2 :+ 3)
Positive
>>> sideTest (Point2 1 (-2) :+ 0) $ Vector2 (Point2 0 0 :+ 1) (Point2 2 2 :+ 3)
Negative
>>> sideTest (Point2 1 1 :+ 0) $ Vector2 (Point2 0 0 :+ 1) (Point2 2 2 :+ 3)
Positive
>>> sideTest (Point2 1 1 :+ 10) $ Vector2 (Point2 0 0 :+ 1) (Point2 2 2 :+ 3)
Negative
>>> sideTest (Point2 1 1 :+ 10) $ Vector2 (Point2 0 0 :+ 3) (Point2 2 2 :+ 1)
Negative

sideTest' :: (Num r, Ord r, Ord i, HasDeterminant (d + 1), Arity d, Arity (d + 1)) => Point d (Symbolic i r) -> Vector d (Point d (Symbolic i r)) -> Sign

Given a point q and a vector of d points defining a hyperplane, test on which side of the hyperplane q lies.

TODO: Specify what the sign means

toSymbolic :: (Ord i, Arity d) => (Point d r :+ i) -> Point d (Symbolic (i, Int) r)

Given an input point, transform its number type to include symbolic $varepsilon$ expressions so that we can use SoS.

signDet :: (HasDeterminant d, Ord i, Num r, Ord r) => Matrix d d (Symbolic i r) -> Sign

pre: computes the sign of the determinant