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The commands in the geom3d package enable you to work in three-dimensional Euclidean geometry. Note that the package does not support the extended space, that is, it does not handle points at infinity and lines at infinity.

Each command in the geom3d package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The geometric objects supported in this package are: point, segment, directed segment, line, plane, triangle, sphere, and polyhedra. To create the basic geometric objects, use the following commands.

For basic geometric objects, the following commands are supported.

Point:

Segment/Directed Segment:

Line:

Plane:

Triangle:

Sphere:

The study of polyhedra has been one of the most enduring, fascinating topics of geometry. The geom3d package supports various types of polyhedra.

Regular Polyhedra:

Quasi-regular Polyhedra:

General Polyhedra:

Archimedean Solids:

Duality:

The two processes to construct new polyhedra from given polyhedra are also supported.

Various types of geom3d[transformation] are supported.

The geom3d[draw] command provides the graphical visualization of all objects supported in the package.

Other commands are also implemented.

To display the help page for a particular geom3d command, see Getting Help with a Function in a Package .

When an object is defined using its algebraic representation (that is, an equation or a polynomial), you can use any name for the x-axis, the y-axis, and the z-axis. In general, the names of the axes must be included when you define an object. A simple way to set the names without being prompted is to set the environment variables _EnvXName, _EnvYName, and _EnvZName to the axis names (or _EnvTName for the case of line) that you prefer.

For commands in the package that create a geometric object, or a list of geometric objects, the calling sequence is of the form command_call(obj,...) where obj is either a name of the geometric object to be created, or a list of geometric objects to be created.

Note that you must make explicit assumptions for the symbolic names in an object (such as, real, positive, ...) when you want to apply a test (for example, AreOnObject) to an object. In this case, the power of the geom3d package is dependent on the power of the Maple assume command.

For commands where output is a boolean value (true, false, FAIL), the calling sequence is of the form command_call(..., cond), where cond is a an optional name. If the output is FAIL, and this optional argument is given, then the condition that makes the output true is assigned to cond.

The value of cond might be a Maple expression (use $\mathrm{assume}\left(\mathrm{cond}\right)$), or of the form cond = &or(expr_1,...,expr_n) or cond = &and(expr_1,...,expr_n) (use assume(op(i,cond)) for the former case where i is from 1 to n; and assume(op(cond)) for the latter case.

$\mathrm{with}\left(\mathrm{geom3d}\right)\:$

$\mathrm{sphere}\left(s\,x^2+y^2+z^2=1\,\left[x\,y\,z\right]\,'\mathrm{centername}'=o\right)$

$s$

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

$\begin{array}{cc}\text{name of the object}& s\\ \text{form of the object}& \mathrm{sphere3d}\\ \text{name of the center}& o\\ \text{coordinates of the center}& \left[0\,0\,0\right]\\ \text{radius of the sphere}& 1\\ \text{surface area of the sphere}& 4\mathrm{\pi }\\ \text{volume of the sphere}& \frac{4\mathrm{\pi }}{3}\\ \text{equation of the sphere}& x^2+y^2+z^2-1=0\end{array}$

$\mathrm{sphere}\left(c\,m^2+n^2+p^2=1\right)$

$c$

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

$\begin{array}{cc}\text{name of the object}& c\\ \text{form of the object}& \mathrm{sphere3d}\\ \text{name of the center}& \mathrm{center\_c\_1}\\ \text{coordinates of the center}& \left[0\,0\,0\right]\\ \text{radius of the sphere}& 1\\ \text{surface area of the sphere}& 4\mathrm{\pi }\\ \text{volume of the sphere}& \frac{4\mathrm{\pi }}{3}\\ \text{equation of the sphere}& m^2+n^2+p^2-1=0\end{array}$

$\mathrm{\_EnvXName}≔'m'\:$$\mathrm{\_EnvYName}≔'n'\:$$\mathrm{\_EnvZName}≔'p'\:$

$\mathrm{sphere}\left(c\,\left[\mathrm{point}\left(\mathrm{oo}\,0\,0\,0\right)\,1\right]\right)$

$c$

$\mathrm{Equation}\left(c\right)$

$m^2+n^2+p^2-1=0$

$\mathrm{line}\left(\mathrm{l2}\,\left[\mathrm{point}\left(A\,0\,0\,0\right)\,\mathrm{point}\left(B\,1\,0\,0\right)\right]\right),\mathrm{sphere}\left(c\,x^2+y^2+z^2=1\,\left[x\,y\,z\right]\right)\:$

$\mathrm{intersection}\left(H\,\mathrm{l2}\,c\right)$

intersection: "two points of intersection"

$H$

$H$

$\left[\mathrm{l2\_intersect1\_c}\,\mathrm{l2\_intersect2\_c}\right]$

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

$\left[\begin{array}{cc}\text{name of the object}& \mathrm{l2\_intersect1\_c}\\ \text{form of the object}& \mathrm{point3d}\\ \text{coordinates of the point}& \left[1\,0\,0\right]\end{array}\,\begin{array}{cc}\text{name of the object}& \mathrm{l2\_intersect2\_c}\\ \text{form of the object}& \mathrm{point3d}\\ \text{coordinates of the point}& \left[\mathrm{-1}\,0\,0\right]\end{array}\right]$

$\mathrm{sphere}\left(s\,\left[\mathrm{point}\left(o\,0\,0\,0\right)\,1\right]\right)\:$

$\mathrm{IsOnObject}\left(\mathrm{point}\left(B\,1\,1\,a\right)\,s\,'\mathrm{cond}'\right)$

IsOnObject: "hint: unable to determine if a^2+1 is zero"

$\mathrm{FAIL}$

$\mathrm{assume}\left(\mathrm{cond}\right)$

$\mathrm{IsOnObject}\left(\mathrm{point}\left(B\,1\,1\,a\right)\,s\right)$

$\mathrm{true}$