### 2.6 Conics ¶

An ellipse is internally represented by its center, major axis, minor axis and direction of its major axis. The associated parametric representation in the coordinate system defined by its axis is :

x = a cos(t) and y = b sin(t)

where a is the major axis, b is the minor axis and t ranges from -pi to pi.

An hyperbola is internally represented by its center, real axis, imaginary axis and direction of its real axis. The associated parametric representation in the coordinate system defined by its axis is :

x = a/sin(t) and y = b/tan(t)

where a is the real axis, b is the imaginary axis and t ranges from -pi to pi except 0.

A parabola is represented by its focus, parameter and the direction of its axis. The associated parametric representation is :

x = - p cos(t)/(1 + cos(t)) and y = - p sin(t)/(1 + cos(t))

where p is the parameter and t ranges from -pi to pi. The corresponding coordinate system uses the focus as origin and the axis of the parabola as ordinate axis.

#### Conic related functions ¶

`ellipse(A, x, y, a)`

  Ellipse of center A, major axis x, minor axis y. The direction of the major axis is a.

`hyperbola(A, x, y, a)`

  Hyperbola of center A, real axis x, imaginary axis y. The direction of the real axis is a.

`parabola(A, x, a)`

  Parabola of summit A and parameter x. The direction of its axis is a.

`parabola(A, l)`

  Parabola of focus A and directrix l.

`conic(A, l, x)`

  Conic of focus A, directrix l and eccentricity x.

`conic(A, B, x)`

  Conic of foci A and B and eccentricity x.

`major(c)`

  Major axis of conic c if c is an ellipse, real axis if c is an hyperbola, parameter if c is a parabola.

`minor(c)`

  Minor axis of conic c if c is an ellipse, imaginary axis if c is an hyperbola, 0 if c is a parabola.

`eccentricity(c)`

  Eccentricity of conic c.

`arg(c)`

  Polar angle of the major axis of c if c is an ellipse, of the real axis if c is an hyperbola, of the axis if c is a parabola (in degrees).

`arg(A, c)`

  Argument of point A on conic c with respect to its parametric representation. If A is not on c this function uses the projection of A on c with respect to its center for centered conics or its focus for parabolas.

`point(c, a)`

  Point on conic c of argument a with respect to its parametric representation.

`center(c)`

  Center of centered conic c.

`foci(c)`

  Set containing the foci of conic c.

`line(c, a)`

  Tangent line to conic c. The contact point has argument a with respect to the parametric representation of c.