fun::Conic Class Reference

A conic section represented by a symmetric 3x3 matrix. More...

#include <conic.hpp>

Public Types
using	Mat3x3 = std::array< std::array< Fraction, 3 >, 3 >

Public Member Functions
constexpr	Conic (Mat3x3 matrix)
	Construct a new Conic from a symmetric matrix.
constexpr auto	contains (const PgPoint &point) const -> bool
	Check whether a point lies on the conic.
constexpr auto	polar (const PgPoint &point) const -> PgLine
	Polar line of a point with respect to the conic.
constexpr auto	tangent (const PgPoint &point) const -> PgLine
	Tangent line at a point on the conic.
constexpr auto	pole (const PgLine &line) const -> PgPoint
	Pole of a line with respect to the conic (placeholder).
auto	intersect (const PgLine &line) const -> std::vector< PgPoint >
	Intersection of a line with the conic (placeholder).
constexpr auto	discriminant () const -> Fraction
	Discriminant of the conic (det of the 2x2 upper-left block).
constexpr auto	conic_type () const -> ConicType
	Determine the conic type from the discriminant.
constexpr auto	matrix () const -> const Mat3x3 &
	Access the matrix.
constexpr auto	operator== (const Conic &other) const -> bool

Static Public Member Functions
static constexpr auto	circle (std::int64_t center_x, std::int64_t center_y, std::int64_t radius_sq) -> Conic
	Create a circle with centre (cx, cy) and squared radius r².
static constexpr auto	unit_circle () -> Conic
	Create a unit circle centred at the origin.
static constexpr auto	parabola (const Fraction &a) -> Conic
	Create a parabola \(y = a x^2\).

Detailed Description

A conic section represented by a symmetric 3x3 matrix.

A point \(\mathbf{x}=(x:y:z)\) lies on the conic iff \(\mathbf{x}^T Q \mathbf{x}=0\).

Member Typedef Documentation

Mat3x3

using fun::Conic::Mat3x3 = std::array<std::array<Fraction, 3>, 3>

Constructor & Destructor Documentation

Conic()

constexpr fun::Conic::Conic ( Mat3x3  matrix )

inlineexplicitconstexpr

Construct a new Conic from a symmetric matrix.

Parameters

[in]

matrix

The 3x3 symmetric matrix Q.

Member Function Documentation

circle()

static constexpr auto fun::Conic::circle ( std::int64_t  center_x, std::int64_t  center_y, std::int64_t  radius_sq  ) -> Conic

inlinestaticconstexpr

Create a circle with centre (cx, cy) and squared radius r².

\[ (x-c_x)^2 + (y-c_y)^2 = r^2 \]

Parameters

[in]	center_x	X coordinate of centre.
[in]	center_y	Y coordinate of centre.
[in]	radius_sq	Squared radius.

Returns

constexpr Conic

conic_type()

constexpr auto fun::Conic::conic_type ( ) const -> ConicType

inlineconstexpr

Determine the conic type from the discriminant.

Returns

ConicType

contains()

constexpr auto fun::Conic::contains ( const PgPoint &  point ) const -> bool

inlineconstexpr

Check whether a point lies on the conic.

\[ \mathbf{x}^T Q \mathbf{x} = 0 \]

Parameters

[in]

pt

The point to test.

Returns

true if the point lies on the conic.

discriminant()

constexpr auto fun::Conic::discriminant ( ) const -> Fraction

inlineconstexpr

Discriminant of the conic (det of the 2x2 upper-left block).

\[ \Delta = a e - b d \]

>0 → ellipse, =0 → parabola, <0 → hyperbola.

Returns

Fraction

intersect()

auto fun::Conic::intersect ( const PgLine &  line ) const -> std::vector<PgPoint>

inline

Intersection of a line with the conic (placeholder).

Currently returns an empty vector.

Parameters

[in]

line

The line.

Returns

std::vector<PgPoint> (0, 1, or 2 points).

matrix()

constexpr auto fun::Conic::matrix ( ) const -> const Mat3x3&

inlineconstexpr

Access the matrix.

operator==()

constexpr auto fun::Conic::operator== ( const Conic &  other ) const -> bool

inlineconstexpr

parabola()

static constexpr auto fun::Conic::parabola ( const Fraction &  a ) -> Conic

inlinestaticconstexpr

Create a parabola \(y = a x^2\).

Parameters

[in]

a

Coefficient.

Returns

constexpr Conic

polar()

constexpr auto fun::Conic::polar ( const PgPoint &  point ) const -> PgLine

inlineconstexpr

Polar line of a point with respect to the conic.

\[ \mathbf{l} = Q \mathbf{x} \]

Parameters

[in]

point

The point.

Returns

PgLine

pole()

constexpr auto fun::Conic::pole ( const PgLine &  line ) const -> PgPoint

inlineconstexpr

Pole of a line with respect to the conic (placeholder).

\[ \mathbf{x} = Q^{-1} \mathbf{l} \]

Currently returns a point with the same coordinates as the line.

Parameters

[in]

line

The line.

Returns

PgPoint

tangent()

constexpr auto fun::Conic::tangent ( const PgPoint &  point ) const -> PgLine

inlineconstexpr

Tangent line at a point on the conic.

At a point \(\mathbf{p}\) on the conic, the tangent is the polar:

\[ \mathbf{t} = Q \mathbf{p} \]

Parameters

[in]

point

A point on the conic.

Returns

PgLine

unit_circle()

static constexpr auto fun::Conic::unit_circle ( ) -> Conic

inlinestaticconstexpr

Create a unit circle centred at the origin.

\[ x^2 + y^2 = 1 \]

Returns

constexpr Conic

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The documentation for this class was generated from the following file:

• projgeom/conic.hpp