A sphere is the set of all points in space that have the same distance from the center.

A sphere s can be defined as follows:

from four points A, B, C, and D.

from the two points which make a diameter of the sphere s.

from the center of s and its radius. The input is a list of two elements where the first element is a point, the second element is a number.

from a point A and a plane p. The sphere defined is the one with center A and is tangent to the plane p.

from its algebraic representation eqn. The input is an equation or a polynomial. In case the optional argument n is not given, if the environment variables _EnvXName, _EnvYName and _EnvZName are assigned to three names, these three names will be used as the names of the axes. If not, Maple will prompt for input of the names of the axes.

To access the information relating to a sphere s, use the following function calls:

The command with(geom3d,sphere) allows the use of the abbreviated form of this command.

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

$\mathrm{\_EnvXName}≔x\:$$\mathrm{\_EnvYName}≔y\:$$\mathrm{\_EnvZName}≔z\:$

$\mathrm{assume}\left(a\,\mathrm{real}\,a\ne 0\,b\,\mathrm{real}\,b\ne 0\,c\,\mathrm{real}\,c\ne 0\right)\:$

$\mathrm{point}\left(o\,0\,0\,0\right),\mathrm{point}\left(A\,a\,0\,0\right),\mathrm{point}\left(B\,0\,b\,0\right),\mathrm{point}\left(C\,0\,0\,c\right)\:$

$\mathrm{sphere}\left(s\,\left[o\,A\,B\,C\right]\right)$

$s$

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

$-x\mathrm{a~}-y\mathrm{b~}-z\mathrm{c~}+x^2+y^2+z^2=0$

$a≔'a'\:$$b≔'b'\:$$c≔'c'\:$

$\mathrm{assume}\left(\mathrm{t1}\,\mathrm{real}\,\mathrm{t2}\,\mathrm{real}\,\mathrm{t3}\,\mathrm{real}\,\mathrm{t1}\ne 0\,\mathrm{t2}\ne 0\,\mathrm{t3}\ne 0\,x\ne 0\,y\ne 0\,z\ne 0\right)\:$

$\mathrm{point}\left(C\,\left[0\,0\,\mathrm{t1}\right]\right),\mathrm{point}\left(U\,\left[\mathrm{t2}\,0\,0\right]\right),\mathrm{point}\left(V\,\left[0\,\mathrm{t3}\,0\right]\right)\:$

$\mathrm{point}\left(P\,\left[x\,y\,z\right]\right)\:$

$\mathrm{line}\left(\mathrm{PU}\,\left[P\,U\right]\right),\mathrm{line}\left(\mathrm{PV}\,\left[P\,V\right]\right),\mathrm{line}\left(\mathrm{PC}\,\left[P\,C\right]\right)\:$

$\mathrm{ArePerpendicular}\left(\mathrm{PU}\,\mathrm{PV}\,'\mathrm{cond1}'\right)\;$$\mathrm{ArePerpendicular}\left(\mathrm{PU}\,\mathrm{PC}\,'\mathrm{cond2}'\right)$

$\mathrm{FAIL}$

$\mathrm{FAIL}$

$\mathrm{ArePerpendicular}\left(\mathrm{PV}\,\mathrm{PC}\,'\mathrm{cond3}'\right)$

$\mathrm{FAIL}$

$\mathrm{cond1},\mathrm{cond2},\mathrm{cond3}$

$-\mathrm{x~}\left(-\mathrm{x~}+\mathrm{t2~}\right)-\left(\mathrm{t3~}-\mathrm{y~}\right)\mathrm{y~}+{\mathrm{z~}}^2=0,-\mathrm{x~}\left(-\mathrm{x~}+\mathrm{t2~}\right)+{\mathrm{y~}}^2-\left(\mathrm{t1~}-\mathrm{z~}\right)\mathrm{z~}=0,{\mathrm{x~}}^2-\left(\mathrm{t3~}-\mathrm{y~}\right)\mathrm{y~}-\left(\mathrm{t1~}-\mathrm{z~}\right)\mathrm{z~}=0$

$\mathrm{expand}\left(\mathrm{cond2}\right)-\mathrm{expand}\left(\mathrm{cond1}\right)$

$-\mathrm{z~}\mathrm{t1~}+\mathrm{y~}\mathrm{t3~}=0$

$\mathrm{subs}\left(y\mathrm{t3}=z\mathrm{t1}\,\mathrm{lhs}\left(\mathrm{expand}\left(\mathrm{cond3}\right)\right)\right)$

$-2\mathrm{z~}\mathrm{t1~}+{\mathrm{x~}}^2+{\mathrm{y~}}^2+{\mathrm{z~}}^2$

$\mathrm{sphere}\left(s\,\,\left[x\,y\,z\right]\right)$

$s$

$\mathrm{coordinates}\left(\mathrm{center}\left(s\right)\right)=\mathrm{coordinates}\left(C\right)$

$\left[0\,0\,\mathrm{t1~}\right]=\left[0\,0\,\mathrm{t1~}\right]$

$\mathrm{radius}\left(s\right)=\mathrm{distance}\left(\mathrm{center}\left(s\right)\,\mathrm{point}\left(o\,0\,0\,0\right)\right)$

$\sqrt{{\mathrm{t1~}}^2}=\sqrt{{\mathrm{t1~}}^2}$