pgl::Transformation< Number > Struct Template Reference

Affine transformation stored as a 2x3 matrix. More...

#include <transformation.hpp>

Inheritance diagram for pgl::Transformation< Number >:

Public Types
using	NumberType = Number

Public Member Functions
constexpr	Transformation ()=default
	Creates the identity transformation.
constexpr	Transformation (Number a, Number b, Number c, Number d, Number tx=Number{}, Number ty=Number{})
	Creates a transformation from its matrix entries.
template<class OtherNumber>
constexpr	Transformation (const Transformation< OtherNumber > &other)
	Converts a transformation with a different entry type.
constexpr Number	a () const
	Returns the row-0, column-0 matrix entry.
constexpr Number	b () const
	Returns the row-0, column-1 matrix entry.
constexpr Number	c () const
	Returns the row-1, column-0 matrix entry.
constexpr Number	d () const
	Returns the row-1, column-1 matrix entry.
constexpr Number	tx () const
	Returns the translation added to the first coordinate.
constexpr Number	ty () const
	Returns the translation added to the second coordinate.
constexpr auto	determinant () const
	Returns the determinant of the linear (non-translation) part.
constexpr bool	isInvertible () const
	Tests whether the transformation has an inverse.
template<class ResultNumber = division_result_t<NumberType>>
constexpr Transformation< ResultNumber >	inverse () const
	Returns the inverse transformation.
template<class OtherNumber>
constexpr auto	operator* (const Transformation< OtherNumber > &other) const
	Composes two transformations: (*this) * other applies other first, then *this.
constexpr bool	operator== (const Transformation &) const =default
	Compares the matrix entries for equality.

Static Public Member Functions
static constexpr Transformation	identity ()
	Returns the identity transformation.
static constexpr Transformation	translation (Number dx, Number dy)
	Returns a translation by (dx, dy).
static constexpr Transformation	scaling (Number sx, Number sy)
	Returns a non-uniform scaling around the origin.
static constexpr Transformation	scaling (Number s)
	Returns a uniform scaling around the origin.
static constexpr Transformation	rotation90 (int k=1)
	Returns a rotation around the origin by 90k degrees.
static constexpr Transformation	shearX (Number k)
	Returns a horizontal shear: (x, y) -> (x + k*y, y).
static constexpr Transformation	shearY (Number k)
	Returns a vertical shear: (x, y) -> (x, y + k*x).
static constexpr Transformation	reflectionX ()
	Returns a reflection across the x-axis.
static constexpr Transformation	reflectionY ()
	Returns a reflection across the y-axis.
template<std::floating_point ResultNumber = double>
static Transformation< ResultNumber >	rotation (ResultNumber radians)
	Returns a rotation around the origin by an arbitrary angle.

Detailed Description

template<class Number>
struct pgl::Transformation< Number >

Affine transformation stored as a 2x3 matrix.

Affine transformation of the plane, stored as a 2x3 matrix.

Applied to a point or shape with operator*; composed with another transformation with operator* as well.

Template Parameters

Number	Matrix entry / coordinate type.
Number	Matrix entry / coordinate type.

Member Typedef Documentation

NumberType

template<class Number>

using pgl::Transformation< Number >::NumberType = Number

Type of the matrix entries.

Constructor & Destructor Documentation

Transformation() [1/3]

template<class Number>

pgl::Transformation< Number >::Transformation ( )

constexprdefault

Creates the identity transformation.

Transformation() [2/3]

template<class Number>

pgl::Transformation< Number >::Transformation ( Number a, Number b, Number c, Number d, Number tx = Number{}, Number ty = Number{} )

inlineconstexpr

Creates a transformation from its matrix entries.

Parameters

a	Row 0, column 0 of the linear part.
b	Row 0, column 1 of the linear part.
c	Row 1, column 0 of the linear part.
d	Row 1, column 1 of the linear part.
tx	Translation added to the first coordinate.
ty	Translation added to the second coordinate.

Transformation() [3/3]

template<class Number>

template<class OtherNumber>

pgl::Transformation< Number >::Transformation ( const Transformation< OtherNumber > & other )

inlineconstexpr

Converts a transformation with a different entry type.

Template Parameters

OtherNumber

Entry type of the source transformation.

Parameters

other

Source transformation.

Member Function Documentation

a()

template<class Number>

Number pgl::Transformation< Number >::a ( ) const

inlineconstexpr

Returns the row-0, column-0 matrix entry.

b()

template<class Number>

Number pgl::Transformation< Number >::b ( ) const

inlineconstexpr

Returns the row-0, column-1 matrix entry.

c()

template<class Number>

Number pgl::Transformation< Number >::c ( ) const

inlineconstexpr

Returns the row-1, column-0 matrix entry.

d()

template<class Number>

Number pgl::Transformation< Number >::d ( ) const

inlineconstexpr

Returns the row-1, column-1 matrix entry.

determinant()

template<class Number>

auto pgl::Transformation< Number >::determinant ( ) const

inlineconstexpr

Returns the determinant of the linear (non-translation) part.

Negative when the transformation reverses orientation (a reflection or an odd number of shears/reflections composed together), zero when it collapses the plane onto a line or point.

identity()

template<class Number>

constexpr Transformation pgl::Transformation< Number >::identity ( )

inlinestaticconstexpr

Returns the identity transformation.

inverse()

template<class Number>

template<class ResultNumber = division_result_t<NumberType>>

Transformation< ResultNumber > pgl::Transformation< Number >::inverse ( ) const

inlineconstexpr

Returns the inverse transformation.

Template Parameters

ResultNumber

Matrix entry type of the result (default: division_result_t).

Warning

Divides by determinant. For an integral NumberType this is inexact unless ResultNumber is a type such as Rational<Number> that represents the division exactly.

isInvertible()

template<class Number>

bool pgl::Transformation< Number >::isInvertible ( ) const

inlineconstexpr

Tests whether the transformation has an inverse.

Returns

true when determinant is nonzero.

operator*()

template<class Number>

template<class OtherNumber>

auto pgl::Transformation< Number >::operator* ( const Transformation< OtherNumber > & other ) const

inlineconstexpr

Composes two transformations: (*this) * other applies other first, then *this.

Template Parameters

OtherNumber

Entry type of other.

operator==()

template<class Number>

bool pgl::Transformation< Number >::operator== ( const Transformation< Number > & ) const

constexprdefault

Compares the matrix entries for equality.

reflectionX()

template<class Number>

constexpr Transformation pgl::Transformation< Number >::reflectionX ( )

inlinestaticconstexpr

Returns a reflection across the x-axis.

reflectionY()

template<class Number>

constexpr Transformation pgl::Transformation< Number >::reflectionY ( )

inlinestaticconstexpr

Returns a reflection across the y-axis.

rotation()

template<class Number>

template<std::floating_point ResultNumber = double>

Transformation< ResultNumber > pgl::Transformation< Number >::rotation ( ResultNumber radians )

inlinestatic

Returns a rotation around the origin by an arbitrary angle.

Unlike rotation90, an arbitrary angle is generally irrational, so this requires an explicit floating-point ResultNumber: there is no exact-by-default overload to silently fall back from.

Template Parameters

ResultNumber

Floating-point matrix entry type of the result.

Parameters

radians

Rotation angle in radians.

rotation90()

template<class Number>

constexpr Transformation pgl::Transformation< Number >::rotation90 ( int k = 1 )

inlinestaticconstexpr

Returns a rotation around the origin by 90k degrees.

Exact for every NumberType since the matrix entries are 0, 1, or -1.

Parameters

k	Number of 90-degree CCW rotations (may be negative).

scaling() [1/2]

template<class Number>

constexpr Transformation pgl::Transformation< Number >::scaling ( Number s )

inlinestaticconstexpr

Returns a uniform scaling around the origin.

Parameters

s	Factor applied to both coordinates.

scaling() [2/2]

template<class Number>

constexpr Transformation pgl::Transformation< Number >::scaling ( Number sx, Number sy )

inlinestaticconstexpr

Returns a non-uniform scaling around the origin.

Parameters

sx	Factor applied to the x coordinate.
sy	Factor applied to the y coordinate.

shearX()

template<class Number>

constexpr Transformation pgl::Transformation< Number >::shearX ( Number k )

inlinestaticconstexpr

Returns a horizontal shear: (x, y) -> (x + k*y, y).

shearY()

template<class Number>

constexpr Transformation pgl::Transformation< Number >::shearY ( Number k )

inlinestaticconstexpr

Returns a vertical shear: (x, y) -> (x, y + k*x).

translation()

template<class Number>

constexpr Transformation pgl::Transformation< Number >::translation ( Number dx, Number dy )

inlinestaticconstexpr

Returns a translation by (dx, dy).

tx()

template<class Number>

Number pgl::Transformation< Number >::tx ( ) const

inlineconstexpr

Returns the translation added to the first coordinate.

ty()

template<class Number>

Number pgl::Transformation< Number >::ty ( ) const

inlineconstexpr

Returns the translation added to the second coordinate.