A conic p can be defined as follows:

from five distinct points. The input is a list of five points. Note that a set of five distinct points does not necessarily define a conic.

from the directrix, focus, and eccentricity. The input is a list of the form [dir, fou, ecc] where dir, fou, and ecc are explained above.

from its internal representation eqn. The input is an equation or a polynomial. If the optional argument n is not given, then:

if the two environment variables _EnvHorizontalName and _EnvVerticalName are assigned two names, these two names will be used as the names of the horizontal-axis and vertical-axis respectively.

if not, Maple will prompt for input of the names of the axes.

The routine returns a conic which includes the degenerate cases for the given input. The output is one of the following object: (or list of objects)

a parabola

an ellipse

a hyperbola

a circle

a point (ellipse: degenerate case)

two parallel lines or a "double" line (parabola: degenerate case)

a list of two intersecting lines (hyperbola: degenerate case)

The information relating to the output conic p depends on the type of output. Use the routine geometry[form] to check for the type of output. For a detailed description of the conic p, use the routine detail (i.e., detail(p))

The command with(geometry,conic) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{geometry}\right)\:$

$\mathrm{\_EnvHorizontalName}≔'x'\:$$\mathrm{\_EnvVerticalName}≔'y'\:$

$\mathrm{conic}\left(\mathrm{c1}\,x^2-2xy+y^2-6x-10y+9=0\,\left[x\,y\right]\right)\:$

$\mathrm{form}\left(\mathrm{c1}\right)$

$\mathrm{parabola2d}$

$\mathrm{detail}\left(\mathrm{c1}\right)$

$\begin{array}{ll}\text{name of the object}& \mathrm{c1}\\ \text{form of the object}& \mathrm{parabola2d}\\ \text{vertex}& \left[0\,1\right]\\ \text{focus}& \left[1\,2\right]\\ \text{directrix}& \frac{\sqrt{2}x}{2}+\frac{\sqrt{2}y}{2}+\frac{\sqrt{2}}{2}=0\\ \text{equation of the parabola}& x^2-2yx+y^2-6x-10y+9=0\end{array}$

$\mathrm{line}\left(l\,x=-2\,\left[x\,y\right]\right)\:$$\mathrm{point}\left(f\,1\,0\right)\:$$e≔\frac{1}{2}\:$

$\mathrm{conic}\left(\mathrm{c2}\,\left[l\,f\,e\right]\,\left[c\,d\right]\right)\:$

$\mathrm{form}\left(\mathrm{c2}\right)$

$\mathrm{ellipse2d}$

$\mathrm{point}\left(A\,1\,\frac{2}{3}\mathrm{sqrt}\left(10\right)\right),\mathrm{point}\left(B\,2\,-\frac{2}{3}\mathrm{sqrt}\left(13\right)\right),\mathrm{point}\left(C\,3\,2\mathrm{sqrt}\left(2\right)\right),\mathrm{point}\left(E\,4\,-\frac{10}{3}\right),\mathrm{point}\left(F\,5\,\frac{2}{3}\mathrm{sqrt}\left(34\right)\right)\:$

$\mathrm{conic}\left(\mathrm{c3}\,\left[A\,B\,C\,E\,F\right]\,\left[\mathrm{t1}\,\mathrm{t2}\right]\right)\:$

$\mathrm{form}\left(\mathrm{c3}\right)$

$\mathrm{hyperbola2d}$

$\mathrm{conic}\left(\mathrm{c4}\,x^2-6x+13+y^2-4y-9\,\left[x\,y\right]\right)\:$

ellipse:   "the given equation is indeed a circle"

$\mathrm{form}\left(\mathrm{c4}\right)$

$\mathrm{circle2d}$

$\mathrm{conic}\left(\mathrm{c5}\,x^2+y^2-4x-10y+29=0\,\left[x\,y\right]\right)\:$

conic:   "degenerate case: single point"

$\mathrm{detail}\left(\mathrm{c5}\right)$

$\begin{array}{ll}\text{name of the object}& \mathrm{c5}\\ \text{form of the object}& \mathrm{point2d}\\ \text{coordinates of the point}& \left[2\,5\right]\end{array}$

$\mathrm{conic}\left(\mathrm{c6}\,x^2-2xy+2x+y^2-2y+1\,\left[x\,y\right]\right)\:$

conic:   "degenerate case: a double line"

$\mathrm{detail}\left(\mathrm{c6}\right)$

$\begin{array}{ll}\text{name of the object}& \mathrm{c6}\\ \text{form of the object}& \mathrm{line2d}\\ \text{equation of the line}& -\frac{\sqrt{2}x}{2}+\frac{\sqrt{2}y}{2}=0\end{array}$

$\mathrm{conic}\left(\mathrm{c7}\,x^2-2xy-4x+y^2+4y-77\,\left[x\,y\right]\right)$

conic:   "degenerate case: two ParallelLine lines"

$\left[\mathrm{Line\_1\_c7}\,\mathrm{Line\_2\_c7}\right]$

$\mathrm{detail}\left(\mathrm{c7}\right)$

$\left[\begin{array}{ll}\text{name of the object}& \mathrm{Line\_1\_c7}\\ \text{form of the object}& \mathrm{line2d}\\ \text{equation of the line}& -\frac{\sqrt{2}x}{2}+\frac{\sqrt{2}y}{2}+\frac{11\sqrt{2}}{2}=0\end{array}\,\begin{array}{ll}\text{name of the object}& \mathrm{Line\_2\_c7}\\ \text{form of the object}& \mathrm{line2d}\\ \text{equation of the line}& -\frac{\sqrt{2}x}{2}+\frac{\sqrt{2}y}{2}-\frac{7\sqrt{2}}{2}=0\end{array}\right]$

$\mathrm{conic}\left(\mathrm{c8}\,11x^2+24xy+4y^2+26x+32y+15=0\,\left[x\,y\right]\right)$

conic:   "degenerate case: two intersecting lines"

$\left[\mathrm{Line\_1\_c8}\,\mathrm{Line\_2\_c8}\right]$

$\mathrm{detail}\left(\mathrm{c8}\right)$

$\left[\begin{array}{ll}\text{name of the object}& \mathrm{Line\_1\_c8}\\ \text{form of the object}& \mathrm{line2d}\\ \text{equation of the line}& x+2y+1=0\end{array}\,\begin{array}{ll}\text{name of the object}& \mathrm{Line\_2\_c8}\\ \text{form of the object}& \mathrm{line2d}\\ \text{equation of the line}& -\frac{11x}{5}-\frac{2y}{5}-3=0\end{array}\right]$