kirv

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 Go to latest Published: May 15, 2024 License: CC0-1.0, CC0-1.0 Imports: 3 Imported by: 4

README

Package kirv

import "gitlab.com/fospathi/maf/kirv"

Go package kirv provides functions for working with curves and some common curves.

Development status

Package kirv is unstable; expect breaking changes with every commit.

License

CC0
To the extent possible under law, Christian Stewart has waived all copyright and related or neighboring rights to kirv. This work is published from: United Kingdom.

Documentation

Overview

Package kirv is a library of elementary curves.

Index

Constants

View Source 
const (
	// H is the small variable increment for numerical approximations.
	//
	// The value of H is a balance between increment size (smaller is better)
	// and significant digits (more is better).
	//
	// Determined through trial and error on the sin function.
	H = 1e-6
)

Variables

View Source 
var Sin = NewParametric(
	func(t float64) maf.Vec {
		return maf.Vec{X: t, Y: math.Sin(t)}
	},
	ParSpec{
		Tangent: func(t float64) maf.Vec {
			return maf.Vec{X: 1, Y: math.Cos(t)}.Unit()
		},
		Normal: func(t float64) maf.Vec {

			s, c := math.Sincos(t)
			kn := maf.Vec{X: c, Y: -1}.Scale(s / ((1 + c*c) * (1 + c*c)))
			return kn.Unit()
		},
		Radius: func(t float64) float64 {
			s, c := math.Sincos(t)
			kn := maf.Vec{X: c, Y: -1}.Scale(s / ((1 + c*c) * (1 + c*c)))
			return 1.0 / kn.Mag()
		},
		PosTan: func(t float64) (maf.Vec, maf.Vec) {
			s, c := math.Sincos(t)
			return maf.Vec{X: t, Y: s},
				maf.Vec{X: 1, Y: c}.Unit()
		},
		NormRad: func(t float64) (maf.Vec, float64) {
			s, c := math.Sincos(t)
			kn := maf.Vec{X: c, Y: -1}.Scale(s / ((1 + c*c) * (1 + c*c)))
			return kn.Unit(), 1.0 / kn.Mag()
		},
		PosTanNormRad: func(t float64) (maf.Vec, maf.Vec, maf.Vec, float64) {
			s, c := math.Sincos(t)
			kn := maf.Vec{X: c, Y: -1}.Scale(s / ((1 + c*c) * (1 + c*c)))
			return maf.Vec{X: t, Y: s}, maf.Vec{X: 1, Y: c}.Unit(),
				kn.Unit(), 1.0 / kn.Mag()
		},
	},
)

Sin is a parametric sin curve.

The curve parameter is measured in units of radians.

View Source 
var SqrtH = math.Sqrt(H)

SqrtH is the square root of H.

Functions

func NewArchimedeanPosTan 

func NewArchimedeanPosTan(
	spec Archimedean,
) func(t float64) (maf.Vec2, maf.Vec2)

NewArchimedeanPosTan function finds the position and unit tangent vector for the given spiral curve.

The curve parameter is measured in units of radians.

func NewCirclePosTan 

func NewCirclePosTan(r float64) func(t float64) (maf.Vec2, maf.Vec2)

NewCirclePosTan function finds the position and unit tangent vector for the circle curve with the given radius.

The curve parameter is measured in units of radians.

func NewCycloidPosTan 

func NewCycloidPosTan(r float64) func(t float64) (maf.Vec2, maf.Vec2)

NewCycloidPosTan function finds the position and unit tangent vector for the cycloid curve generated by a rolling circle with the given radius.

The curve parameter is measured in units of radians.

func NewEllipsePosTan 

func NewEllipsePosTan(spec Ellipse) func(t float64) (maf.Vec2, maf.Vec2)

NewEllipsePosTan function finds the position and unit tangent vector for the ellipse curve with the given properties.

The curve parameter is measured in units of radians.

func NewHelixPosTan 

func NewHelixPosTan(spec Helix) func(t float64) (maf.Vec, maf.Vec)

NewHelixPosTan function finds the position and unit tangent vector for the given helix curve.

The curve parameter is measured in units of radians.

func NewSinusoid 

func NewSinusoid(spec Sinusoid, decay ExpDecay) func(t float64) Parametric

NewSinusoid returns a function which constructs new parametric sinusoid curves.

The returned function accepts a time value and the curve parameter of the resulting parametric curve is measured in units of distance.

func NewSinusoidPosTan 

func NewSinusoidPosTan(
	spec Sinusoid, decay ExpDecay,
) func(t float64) func(x float64) (maf.Vec2, maf.Vec2)

NewSinusoidPosTan is a function that takes a given time value and produces, for that time value, a function that accepts distance values and produces the position and unit tangent vector for the sinusoid curve.

func NewStanding 

func NewStanding(spec Standing, decay ExpDecay) func(t float64) Parametric

NewStanding returns a function which constructs new parametric sinusoidal standing wave curves.

The returned function accepts a time value and the curve parameter of the resulting parametric curve is measured in units of distance.

func NewStandingPosTan 

func NewStandingPosTan(
	spec Standing, decay ExpDecay,
) func(t float64) func(x float64) (maf.Vec2, maf.Vec2)

NewStandingPosTan is a function that takes a given time value and produces, for that time value, a function that accepts distance values and produces the position and unit tangent vector for the standing wave curve.

func NewTimeDepArchimedeanPosTan 

func NewTimeDepArchimedeanPosTan(
	spec kinem.Archimedean,
) func(t float64) (maf.Vec2, maf.Vec2)

NewTimeDepArchimedeanPosTan function finds the position and unit tangent vector for the given spiral curve.

The curve parameter is measured in units of time.

func NewTimeDepHelixPosTan 

func NewTimeDepHelixPosTan(
	spec kinem.Helical,
) func(t float64) (maf.Vec, maf.Vec)

NewTimeDepHelixPosTan function finds the position and unit tangent vector for the given helix curve.

The curve parameter is measured in units of time.

func NewTimeDepProjectilePosTan 

func NewTimeDepProjectilePosTan(
	spec kinem.Projectile,
) func(t float64) (maf.Vec2, maf.Vec2)

NewTimeDepProjectilePosTan function finds the position and unit tangent vector for the given projectile curve.

The curve parameter is measured in units of time.

func NumericalConc 

func NumericalConc(car Gradienter, x float64) float64

NumericalConc is a rough approximation of the given curve's concavity at the given abscissa.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalConcavity 

func NumericalConcavity(ord Ordinater, x float64) float64

NumericalConcavity is an approximation of the given curve's concavity at the given abscissa.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalGradient 

func NumericalGradient(ord Ordinater, x float64) float64

NumericalGradient is an approximation of the given curve's gradient at the given abscissa.

func NumericalNorm 

func NumericalNorm(par Tangenter, t float64) maf.Vec

NumericalNorm is a rough approximation of the given curve's unit principal normal vector at the given parameter value.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalNormal 

func NumericalNormal(par Positioner, t float64) maf.Vec

NumericalNormal is a rough approximation of the given curve's unit principal normal vector at the given parameter value.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalNormalRadius 

func NumericalNormalRadius(par Positioner, t float64) (maf.Vec, float64)

NumericalNormalRadius is a rough approximation of the given curve's unit principal normal vector and radius of curvature at the given parameter value.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalOrdGradConc 

func NumericalOrdGradConc(ord Ordinater, x float64) (
	float64, float64, float64)

NumericalOrdGradConc is the given curve's ordinate and an approximation of its gradient and concavity at the given abscissa.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalPosInwardsTanFunc 

func NumericalPosInwardsTanFunc(
	pos func(t float64) maf.Vec, cen float64,
) func(t float64) (maf.Vec, maf.Vec)

NumericalPosInwardsTanFunc returns a new function that returns both the position and a rough approximation of the given curve's unit tangent vector for a given parameter value.

The argument function defines the position of a curve. The argument float parameter value is the centre of the given function's valid domain interval.

The tangent is calculated only with positions whose parameter values are between the given parameter and the centre of the domain. This makes it safe to use at domain limits outside which the position is not defined.

func NumericalPosTan 

func NumericalPosTan(
	pos func(t float64) maf.Vec, t float64,
) (maf.Vec, maf.Vec)

NumericalPosTan is the position and a rough approximation of the given curve's unit tangent vector at the given parameter value.

The given function shall define the position of a curve.

func NumericalPosTanFunc 

func NumericalPosTanFunc(
	pos func(t float64) maf.Vec,
) func(t float64) (maf.Vec, maf.Vec)

NumericalPosTanFunc returns a new function that returns both the position and a rough approximation of the given curve's unit tangent vector for a given parameter value.

The argument function defines the position of a curve.

func NumericalPosTanNormRad 

func NumericalPosTanNormRad(par Positioner, t float64) (
	maf.Vec, maf.Vec, maf.Vec, float64)

NumericalPosTanNormRad is a numerical approximation of the given curve's position, unit tangent, unit principal normal, and radius of curvature at the given parameter value.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalRadius 

func NumericalRadius(par Positioner, t float64) float64

NumericalRadius is a numerical approximation of the given curve's radius of curvature at the given parameter value.

At the parameter value the curve shall not be an inflection point. Nor shall the curve be linear.

func NumericalTangent 

func NumericalTangent(par Positioner, t float64) maf.Vec

NumericalTangent is a rough approximation of the given curve's unit tangent vector at the given parameter value.

func Radius 

func Radius(v, a maf.Vec) float64

Radius of curvature at a non-inflecting point of a curve with the given non-parallel velocity and acceleration.

func UnitNormal 

func UnitNormal(v, a maf.Vec) maf.Vec

UnitNormal vector at a non-inflecting point of a curve with the given non-parallel velocity and acceleration.

func XYCartesianNumericalPosTanNormRad 

func XYCartesianNumericalPosTanNormRad(ord Ordinater, x float64) (
	maf.Vec2, maf.Vec2, maf.Vec2, float64)

XYCartesianNumericalPosTanNormRad is the given curve's position and a rough numerical approximation of its unit tangent, unit principal normal, and radius of curvature at the given abscissa.

At the abscissa the curve shall not be an inflection point. Nor shall the curve be linear.

func XYCartesianNumericalRadius 

func XYCartesianNumericalRadius(ord Ordinater, x float64) float64

XYCartesianNumericalRadius is

Types

type Archimedean 

type Archimedean struct {
	// Initial displacement from the origin along the X-axis.
	A float64
	// The ratio of the radial speed to the angular speed. The distance between
	// the loops is 2πB assuming the polar angle is measured in radians.
	B float64
}

Archimedean specifies the variables of a polar coordinates equation describing an Archimedean spiral.

r = a + bθ

To infer b from an equivalent spiral trace described with physical velocities v and w:

b = v / w

type Ellipse 

type Ellipse struct {
	// A is the semi-major axis length.
	A float64
	// B is the semi-minor axis length.
	B float64
}

Ellipse specifies the properties of an ellipse.

type ExpDecay 

type ExpDecay struct {
	// An exponential decay constant for a space dimension.
	Space float64
	// An exponential decay constant in time.
	Time float64
}

ExpDecay contains positive decay constants for an exponential decay.

func (ExpDecay) Decays 

func (d ExpDecay) Decays() bool

Decays returns true if the receiver decay has any non zero decay constants.

type Gradienter 

type Gradienter interface {
	Gradient(x float64) float64
}

A Gradienter is a Cartesian curve with a known gradient at a given abscissa.

type HalfLife 

type HalfLife float64

HalfLife is an exponential decay half-life value.

func (HalfLife) Decay 

func (hl HalfLife) Decay() float64

Decay is the decay constant which corresponds to the receiver half-life value.

type Helix 

type Helix struct {
	// The pitch of the helix, that is, the distance travelled along the Z-axis
	// in one complete helix turn.
	P float64
	// The radius of the enclosing cylinder of the helix.
	R float64
}

Helix specifies the properties of a helix.

type Ordinater 

type Ordinater interface {
	Ordinate(x float64) float64
}

An Ordinater is a Cartesian curve with a known ordinate at a given abscissa.

type ParSpec 

type ParSpec struct {
	// Optional. Defines the unit tangent direction vector of the curve.
	Tangent func(t float64) maf.Vec

	// Optional. Defines the unit principal normal vector of the curve.
	Normal func(t float64) maf.Vec

	// Optional. Defines the radius of curvature of the curve.
	Radius func(t float64) float64

	// Optional. Defines the position of the curve and the unit tangent
	// direction vector of the curve. May be more efficient than calling
	// position and tangent separately.
	PosTan func(t float64) (maf.Vec, maf.Vec)

	// Optional. Defines the unit principal normal vector and the radius of
	// curvature of the curve. May be more efficient than calling normal and
	// radius separately.
	NormRad func(t float64) (maf.Vec, float64)

	// Optional. Defines the position, unit tangent direction vector, unit
	// principal normal vector and radius of curvature of the curve. May be more
	// efficient than calling position, tangent, normal, and radius separately.
	PosTanNormRad func(t float64) (maf.Vec, maf.Vec, maf.Vec, float64)
}

ParSpec specifies some extra function(s) that define a parametric curve.

Optional functions should be specified if possible and they take precedence over their equivalent approximate numerical methods which are substituted in their absence.

Note that combination functions and dedicated functions can use different ways of calculating the same property and hence may return somewhat different values.

type Parametric 

type Parametric struct {
	// contains filtered or unexported fields
}

A Parametric curve in 3D space.

Note that the combination methods which compute multiple curve properties at once may use different methods than the dedicated methods and hence may return somewhat different values.

func NewArchimedean 

func NewArchimedean(spec Archimedean) Parametric

NewArchimedean constructs a new parametric Archimedean spiral curve.

The curve parameter is measured in units of radians.

The curve is constrained to the XY-plane and starts at the origin or on the X-axis.

func NewCircle 

func NewCircle(r float64) Parametric

NewCircle constructs a new parametric circle curve with the given radius.

The curve parameter is measured in units of radians.

func NewCycloid 

func NewCycloid(r float64) Parametric

NewCycloid constructs a new parametric cycloid curve generated by a rolling circle with the given radius.

The curve parameter is measured in units of radians.

The parametric equation of the curve with radius a is 

X = a (t − sin(t))
Y = a (1 − cos(t))
Z = 0

where 0 <= t <= 2π.

func NewEllipse 

func NewEllipse(spec Ellipse) Parametric

NewEllipse constructs a new parametric ellipse curve with the given properties.

The curve parameter is measured in units of radians.

func NewHelix 

func NewHelix(spec Helix) Parametric

NewHelix constructs a new parametric helix curve.

The curve parameter is measured in units of radians.

The helix rotates around the Z-axis and starts on the positive X-axis.

func NewParametric 

func NewParametric(pos func(t float64) maf.Vec, spec ParSpec) Parametric

NewParametric constructs a new Parametric.

The first, and always necessary, argument is the position function which defines the curve.

func NewTimeDepArchimedean 

func NewTimeDepArchimedean(spec kinem.Archimedean) Parametric

NewTimeDepArchimedean constructs a new parametric Archimedean spiral curve.

The curve parameter is measured in units of time.

The curve is constrained to the XY-plane and starts at the origin or on the X-axis.

func NewTimeDepHelix 

func NewTimeDepHelix(spec kinem.Helical) Parametric

NewTimeDepHelix constructs a new parametric helix curve.

The curve parameter is measured in units of time.

The helix rotates around the Z-axis and starts on the positive X-axis.

func NewTimeDepProjectile 

func NewTimeDepProjectile(spec kinem.Projectile) Parametric

NewTimeDepProjectile constructs a new parametric projectile curve.

The curve parameter is measured in units of time.

The motion is constrained to the XY-plane, starts at the origin, and is subject to a constant acceleration parallel to the Y-axis.

Beware of the degenerate case of a projectile with no horizontal velocity component. Such a straight-line-ish motion is not a very good subject for a parametric curve.

func (Parametric) CentreOfCurvature 

func (par Parametric) CentreOfCurvature(t float64) maf.Vec

CentreOfCurvature of the curve, if it exists, at the given curve parameter.

func (Parametric) NormRad 

func (par Parametric) NormRad(t float64) (maf.Vec, float64)

NormRad is the unit principal normal vector and radius of curvature of the curve at the given curve parameter.

May be more efficient than calling normal and radius separately.

func (Parametric) Normal 

func (par Parametric) Normal(t float64) maf.Vec

Normal is the unit principal normal vector of the curve at the given curve parameter.

func (Parametric) Osculating 

func (par Parametric) Osculating(t float64) maf.Circle

Osculating circle of the curve, if it exists, at the given curve parameter.

func (Parametric) PosTan 

func (par Parametric) PosTan(t float64) (maf.Vec, maf.Vec)

PosTan is the position and unit tangent direction vector of the curve at the given curve parameter.

May be more efficient than calling position and tangent separately.

func (Parametric) PosTanFunc 

func (par Parametric) PosTanFunc() func(t float64) (maf.Vec, maf.Vec)

PosTanFunc is the function used by PosTan.

func (Parametric) PosTanNormRad 

func (par Parametric) PosTanNormRad(t float64) (
	maf.Vec, maf.Vec, maf.Vec, float64)

PosTanNormRad is the position, unit tangent direction vector, unit principal normal vector, and radius of curvature of the curve at the given curve parameter.

May be more efficient than calling position, tangent, normal, and radius separately.

func (Parametric) Position 

func (par Parametric) Position(t float64) maf.Vec

Position of the curve at the given curve parameter.

func (Parametric) Radius 

func (par Parametric) Radius(t float64) float64

Radius of curvature of the curve at the given curve parameter.

func (Parametric) Tangent 

func (par Parametric) Tangent(t float64) maf.Vec

Tangent unit direction vector of the curve at the given curve parameter.

type Positioner 

type Positioner interface {
	Position(t float64) maf.Vec
}

A Positioner is a parametric curve with a known position at a given curve parameter value.

type Sinusoid 

type Sinusoid struct {
	// Amplitude. Maximum displacement from Y = 0.
	A float64
	// Wavelength.
	L float64
	// Phase. The angle of the oscillation at t = 0, measured in radians.
	P float64
	// Linear speed.
	V float64
}

Sinusoid specifies the properties of a sine wave travelling to the right.

type Standing 

type Standing struct {
	// Amplitude. Maximum displacement from Y = 0.
	//
	// This is twice the maximum amplitude of the two constituent waves that
	// form this resulting standing wave.
	A float64

	// Wavelength.
	L float64

	// Angular velocity, related to the frequency (f) by: w = 2πf.
	W float64
}

Standing specifies the properties of a sinusoidal standing wave.

type Tangenter 

type Tangenter interface {
	Tangent(t float64) maf.Vec
}

A Tangenter is a parametric curve with a known unit tangent at a given curve parameter value.

type XYCarSpec 

type XYCarSpec struct {
	// Optional. Defines the gradient (first derivative) of the curve.
	Gradient func(x float64) float64
	// Optional. Defines the concavity (second derivative) of the curve.
	Concavity func(x float64) float64
	// Optional. Defines the ordinate, gradient, and concavity of the curve. May
	// be more efficient than calling ordinate, gradient, and concavity
	// separately.
	OrdGradConc func(x float64) (float64, float64, float64)
	// Optional. Defines the radius of curvature of the curve.
	Radius func(x float64) float64
	// PosTanNormRad is the position, unit tangent direction vector, unit
	// principal normal vector, and radius of curvature of the curve at the
	// given curve parameter.
	//
	// May be more efficient than calling position, tangent, normal, and radius
	// separately.
	PosTanNormRad func(x float64) (maf.Vec2, maf.Vec2, maf.Vec2, float64)
}

XYCarSpec specifies some extra function(s) that define a Cartesian curve in the XY-plane.

Optional functions should be specified if possible and they take precedence over approximate numerical methods which are used as a last resort.

Note that the combination function may use different methods than the dedicated functions and hence may return somewhat different values.

type XYCartesian 

type XYCartesian struct {
	// contains filtered or unexported fields
}

An XYCartesian is a Cartesian curve in the XY plane.

Note that the combination methods which compute multiple curve properties at once may use different methods than the dedicated methods and hence may return somewhat different values.

func NewXYCartesian 

func NewXYCartesian(ord func(float64) float64, spec XYCarSpec) XYCartesian

NewXYCartesian constructs a new XYCartesian.

The first, and always necessary, argument is the ordinate function which defines the curve and maps the abscissa (X-coordinate) to the corresponding ordinate (Y-coordinate).

func (XYCartesian) Concavity 

func (car XYCartesian) Concavity(x float64) float64

Concavity is the value of the second derivative of the curve at the given abscissa.

func (XYCartesian) Gradient 

func (car XYCartesian) Gradient(x float64) float64

Gradient is the value of the first derivative of the curve at the given abscissa.

func (XYCartesian) Normal 

func (car XYCartesian) Normal(x float64) maf.Vec2

Normal is the unit principal normal vector of the curve at the given abscissa.

func (XYCartesian) OrdGradConc 

func (car XYCartesian) OrdGradConc(x float64) (float64, float64, float64)

OrdGradConc is the Y-coordinate, gradient, and concavity of the curve at the given abscissa.

func (XYCartesian) Ordinate 

func (car XYCartesian) Ordinate(x float64) float64

Ordinate is the Y-coordinate of the curve at the given abscissa.

func (XYCartesian) Position 

func (car XYCartesian) Position(x float64) maf.Vec2

Position of the curve at the given abscissa.

func (XYCartesian) Radius 

func (car XYCartesian) Radius(x float64) float64

Radius of curvature of the curve at the given abscissa.

func (XYCartesian) Tangent 

func (car XYCartesian) Tangent(x float64) maf.Vec2

Tangent unit direction vector of the curve at the given abscissa.

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