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.