conic - Maple Help

geometry

 conic
 define a conic

 Calling Sequence conic(p, [A, B, C, E, F], n) conic(p, [dir, fou, ecc], n) conic(p, eqn, n)

Parameters

 p - the name of the conic A, B, C, E, F - five distinct points dir - the line which is the directrix of the conic fou - point which is the focus of the conic ecc - a positive number denoting the eccentricity of the conic eqn - the algebraic representation of the conic (i.e., a polynomial or an equation) n - (optional) list of two names representing the names of the horizontal-axis and vertical-axis

Description

 • 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.

Examples

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

define conic c1 from its algebraic representation:

 > $\mathrm{_EnvHorizontalName}≔'x':$$\mathrm{_EnvVerticalName}≔'y':$
 > $\mathrm{conic}\left(\mathrm{c1},{x}^{2}-2xy+{y}^{2}-6x-10y+9=0,\left[x,y\right]\right):$
 > $\mathrm{form}\left(\mathrm{c1}\right)$
 ${\mathrm{parabola2d}}$ (1)
 > $\mathrm{detail}\left(\mathrm{c1}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{c1}}\\ {\text{form of the object}}& {\mathrm{parabola2d}}\\ {\text{vertex}}& \left[{0}{,}{1}\right]\\ {\text{focus}}& \left[{1}{,}{2}\right]\\ {\text{directrix}}& \frac{\sqrt{{2}}{}{x}}{{2}}{+}\frac{\sqrt{{2}}{}{y}}{{2}}{+}\frac{\sqrt{{2}}}{{2}}{=}{0}\\ {\text{equation of the parabola}}& {{x}}^{{2}}{-}{2}{}{x}{}{y}{+}{{y}}^{{2}}{-}{6}{}{x}{-}{10}{}{y}{+}{9}{=}{0}\end{array}$ (2)
 > $\mathrm{line}\left(l,x=-2,\left[x,y\right]\right):$$\mathrm{point}\left(f,1,0\right):$$e≔\frac{1}{2}:$
 > $\mathrm{conic}\left(\mathrm{c2},\left[l,f,e\right],\left[c,d\right]\right):$
 > $\mathrm{form}\left(\mathrm{c2}\right)$
 ${\mathrm{ellipse2d}}$ (3)
 > $\mathrm{point}\left(A,1,\frac{2}{3}\mathrm{sqrt}\left(10\right)\right),\mathrm{point}\left(B,2,-\frac{2}{3}\mathrm{sqrt}\left(13\right)\right),\mathrm{point}\left(C,3,2\mathrm{sqrt}\left(2\right)\right),\mathrm{point}\left(E,4,-\frac{10}{3}\right),\mathrm{point}\left(F,5,\frac{2}{3}\mathrm{sqrt}\left(34\right)\right):$
 > $\mathrm{conic}\left(\mathrm{c3},\left[A,B,C,E,F\right],\left[\mathrm{t1},\mathrm{t2}\right]\right):$
 > $\mathrm{form}\left(\mathrm{c3}\right)$
 ${\mathrm{hyperbola2d}}$ (4)
 > $\mathrm{conic}\left(\mathrm{c4},{x}^{2}-6x+13+{y}^{2}-4y-9,\left[x,y\right]\right):$
 ellipse:   "the given equation is indeed a circle"
 > $\mathrm{form}\left(\mathrm{c4}\right)$
 ${\mathrm{circle2d}}$ (5)
 > $\mathrm{conic}\left(\mathrm{c5},{x}^{2}+{y}^{2}-4x-10y+29=0,\left[x,y\right]\right):$
 conic:   "degenerate case: single point"

degenerate case of an ellipse

 > $\mathrm{detail}\left(\mathrm{c5}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{c5}}\\ {\text{form of the object}}& {\mathrm{point2d}}\\ {\text{coordinates of the point}}& \left[{2}{,}{5}\right]\end{array}$ (6)
 > $\mathrm{conic}\left(\mathrm{c6},{x}^{2}-2xy+2x+{y}^{2}-2y+1,\left[x,y\right]\right):$
 conic:   "degenerate case: a double line"

degenerate case of a parabola

 > $\mathrm{detail}\left(\mathrm{c6}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{c6}}\\ {\text{form of the object}}& {\mathrm{line2d}}\\ {\text{equation of the line}}& {-}\frac{\sqrt{{2}}{}{x}}{{2}}{+}\frac{\sqrt{{2}}{}{y}}{{2}}{=}{0}\end{array}$ (7)
 > $\mathrm{conic}\left(\mathrm{c7},{x}^{2}-2xy-4x+{y}^{2}+4y-77,\left[x,y\right]\right)$
 conic:   "degenerate case: two ParallelLine lines"
 $\left[{\mathrm{Line_1_c7}}{,}{\mathrm{Line_2_c7}}\right]$ (8)

degenerate case of a parabola

 > $\mathrm{detail}\left(\mathrm{c7}\right)$
 $\left[\begin{array}{ll}{\text{name of the object}}& {\mathrm{Line_1_c7}}\\ {\text{form of the object}}& {\mathrm{line2d}}\\ {\text{equation of the line}}& {-}\frac{\sqrt{{2}}{}{x}}{{2}}{+}\frac{\sqrt{{2}}{}{y}}{{2}}{+}\frac{{11}{}\sqrt{{2}}}{{2}}{=}{0}\end{array}{,}\begin{array}{ll}{\text{name of the object}}& {\mathrm{Line_2_c7}}\\ {\text{form of the object}}& {\mathrm{line2d}}\\ {\text{equation of the line}}& {-}\frac{\sqrt{{2}}{}{x}}{{2}}{+}\frac{\sqrt{{2}}{}{y}}{{2}}{-}\frac{{7}{}\sqrt{{2}}}{{2}}{=}{0}\end{array}\right]$ (9)
 > $\mathrm{conic}\left(\mathrm{c8},11{x}^{2}+24xy+4{y}^{2}+26x+32y+15=0,\left[x,y\right]\right)$
 conic:   "degenerate case: two intersecting lines"
 $\left[{\mathrm{Line_1_c8}}{,}{\mathrm{Line_2_c8}}\right]$ (10)

the degenerate case of a hyperbola

 > $\mathrm{detail}\left(\mathrm{c8}\right)$
 $\left[\begin{array}{ll}{\text{name of the object}}& {\mathrm{Line_1_c8}}\\ {\text{form of the object}}& {\mathrm{line2d}}\\ {\text{equation of the line}}& {1}{+}{x}{+}{2}{}{y}{=}{0}\end{array}{,}\begin{array}{ll}{\text{name of the object}}& {\mathrm{Line_2_c8}}\\ {\text{form of the object}}& {\mathrm{line2d}}\\ {\text{equation of the line}}& {-}{3}{-}\frac{{11}{}{x}}{{5}}{-}\frac{{2}{}{y}}{{5}}{=}{0}\end{array}\right]$ (11)