Math.Spline

Documentation

class (VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline s v where

A spline is a piecewise polynomial vector-valued function. The necessary and sufficient instance definition is toBSpline.

Methods

splineDomain :: s v -> Maybe (Scalar v, Scalar v)

Returns the domain of a spline. In the case of B-splines, this is the domain on which a spline with this degree and knot vector has a full basis set. In other cases, it should be no larger than splineDomain . toBSpline, but may be smaller. Within this domain, evalSpline should agree with evalSpline . toBSpline (not necessarily exactly, but up to reasonable expectations of numerical accuracy).

evalSpline :: s v -> Scalar v -> v

splineDegree :: s v -> Int

knotVector :: s v -> Knots (Scalar v)

toBSpline :: s v -> BSpline v

Instances

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline BSpline v

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline BezierCurve v

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline MSpline v

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline ISpline v

class Spline s v => ControlPoints s v where

Methods

controlPoints :: s v -> Vector v

Instances

Spline BSpline v => ControlPoints BSpline v

Spline BezierCurve v => ControlPoints BezierCurve v

Spline MSpline v => ControlPoints MSpline v

Spline ISpline v => ControlPoints ISpline v

data Knots a

Knot vectors - multisets of points in a 1-dimensional space.

Instances

Foldable Knots

Eq a => Eq (Knots a)

Ord a => Ord (Knots a)

Show a => Show (Knots a)

Ord a => Monoid (Knots a)

mkKnots :: Ord a => [a] -> Knots a

Create a knot vector consisting of all the knots in a list.

knots :: Knots t -> [t]

Returns a list of all knots (not necessarily distinct) of a knot vector in ascending order

data BezierCurve t

A BezierCurve curve on 0 <= x <= 1.

Instances

Spline BezierCurve v => ControlPoints BezierCurve v

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline BezierCurve v

Eq t => Eq (BezierCurve t)

Ord t => Ord (BezierCurve t)

Show v => Show (BezierCurve v)

bezierCurve :: Vector t -> BezierCurve t

Construct a Bezier curve from a list of control points. The degree of the curve is one less than the number of control points.

data BSpline t

Instances

Spline BSpline v => ControlPoints BSpline v

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline BSpline v

(Eq (Scalar v), Eq v) => Eq (BSpline v)

(Ord (Scalar v), Ord v) => Ord (BSpline v)

(Show (Scalar v), Show v) => Show (BSpline v)

bSpline :: Knots (Scalar a) -> Vector a -> BSpline a

bSpline kts cps creates a B-spline with the given knot vector and control points. The degree is automatically inferred as the difference between the number of spans in the knot vector (numKnots kts - 1) and the number of control points (length cps).

data MSpline v

M-Splines are B-splines normalized so that the integral of each basis function over the spline domain is 1.

Instances

Spline MSpline v => ControlPoints MSpline v

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline MSpline v

(Eq (Scalar v), Eq v) => Eq (MSpline v)

(Ord (Scalar v), Ord v) => Ord (MSpline v)

(Show (Scalar v), Show v) => Show (MSpline v)

mSpline :: Knots (Scalar a) -> Vector a -> MSpline a

mSpline kts cps creates a M-spline with the given knot vector and control points. The degree is automatically inferred as the difference between the number of spans in the knot vector (numKnots kts - 1) and the number of control points (length cps).

toMSpline :: Spline s v => s v -> MSpline v

data ISpline v

The I-Spline basis functions are the integrals of the M-splines, or alternatively the integrals of the B-splines normalized to the range [0,1]. Every I-spline basis function increases monotonically from 0 to 1, thus it is useful as a basis for monotone functions. An I-Spline curve is monotone if and only if every non-zero control point has the same sign.

Instances

Spline ISpline v => ControlPoints ISpline v

(VectorSpace v, Fractional (Scalar v), Ord (Scalar v)) => Spline ISpline v

(Eq (Scalar v), Eq v) => Eq (ISpline v)

(Ord (Scalar v), Ord v) => Ord (ISpline v)

(Show (Scalar v), Show v) => Show (ISpline v)

iSpline :: Knots (Scalar a) -> Vector a -> ISpline a

iSpline kts cps creates an I-spline with the given knot vector and control points. The degree is automatically inferred as the difference between the number of spans in the knot vector (numKnots kts - 1) and the number of control points (length cps).

toISpline :: (Spline s v, Eq v) => s v -> ISpline v