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New Packages in Maple V Release 5

This file describes the new packages available in Release 5 of Maple.

Context package

Interface to Matlink

Codegen - Code generation package

algcurves - The algebraic curves package

diffalg - Package for manipulating systems of differential polynomial equations (ODE or PDE)

geom3d - Introduction to the geom3d package

Groebner - The Groebner package and Groebner bases.

Ore_algebra - The Ore_algebra package and Ore algebras.

PDEtools - Solving Partial Differential Equations

Context package

•

Context

Interface to Matlink

•

Matlab

Codegen - Code generation package

•	GRAD - See GRADIENT

•	GRADIENT - Compute the Gradient of a Maple procedure

>	with(codegen):

>	F := proc(x) local t; t := x; x+t*t; end;

$F := proc (x) local t &semi; t := x &semi; x + t &ast; t end proc$

(1)

F(x,y);

$x^{2} + x$

(2)

>	G := GRADIENT(F);

$G := proc (x) local df, t &semi; t := x &semi; df := array (1 .. 1) &semi; df [1] := 2 &ast; t &semi; return df [1] + 1 end proc$

(3)

•	HESSIAN - Compute the HESSIAN matrix of a Maple procedure

>	F := proc(x,y) local t; t := xy; x+t-yt; end;

$F := proc (x, y) local t &semi; t := y &ast; x &semi; x + t - y &ast; t end proc$

(4)

F(x,y);

$- y^{2} x + y x + x$

(5)

>	H := HESSIAN(F); # reverse mode

$H := proc (x, y) local df, df1, dfr0, grd, grd1, grd2, t1 &semi; df1 := - y + 1 &semi; t1 := df1 &semi; grd1 := t1 &ast; y + 1 &semi; grd2 := t1 &ast; x - y &ast; x &semi; df := array (1 .. 4) &semi; dfr0 := array (1 .. 4) &semi; df [3] := 1 &semi; df [2] := df [3] &ast; y &semi; df [1] := df [2] &semi; dfr0 [4] := 1 &semi; dfr0 [2] := dfr0 [4] &ast; x &semi; dfr0 [1] := dfr0 [2] &semi; grd := array (1 .. 2, 1 .. 2) &semi; grd [1, 1] := 0 &semi; grd [1, 2] := t1 &ast; df [3] - df [1] &semi; grd [2, 1] := dfr0 [4] &ast; (t1 - y) &semi; grd [2, 2] := - dfr0 [4] &ast; x - dfr0 [1] &semi; return grd end proc$

(6)

•	JACOBIAN - Compute the JACOBIAN matrix of a Maple procedure

>	f := proc(x,y) x*y^2; end;

$f := proc (x, y) x &ast; y &Hat; 2 end proc$

(7)

>	g := proc(x,y) x^2*y; end;

$g := proc (x, y) x &Hat; 2 &ast; y end proc$

(8)

>	J := JACOBIAN([f,g]);

$J := proc (x, y) local df, dfr0, grd, t1, t2 &semi; t1 := y &Hat; 2 &semi; t2 := x &Hat; 2 &semi; df := array (1 .. 2) &semi; dfr0 := array (1 .. 2) &semi; df [1] := x &semi; dfr0 [2] := y &semi; grd := array (1 .. 2, 1 .. 2) &semi; grd [1, 1] := t1 &semi; grd [1, 2] := 2 &ast; y &ast; df [1] &semi; grd [2, 1] := 2 &ast; dfr0 [2] &ast; x &semi; grd [2, 2] := t2 &semi; return grd end proc$

(9)

>	print(J(x,y));

$[\begin{array}{c} y^{2} & 2 y x \\ 2 y x & x^{2} \end{array}]$

(10)

•	declare - Declare the type of a parameter

>	g := proc(y) local A; A := array(1 .. 2); A[1] := y^2; A[2] := y^2; end:

>	declare(y::numeric,g);

$proc (y :: numeric) local A &semi; A := array (1 .. 2) &semi; A [1] := y &Hat; 2 &semi; A [2] := y &Hat; 2 end proc$

(11)

•	dontreturn - Do not return a value from a Maple procedure

>	f := proc(a,b) a := 3; b := array(1..2,1..2,[[1,2],[3,4]]); RETURN(a,b); end;

$f := proc (a, b) a := 3 &semi; b := array (1 .. 2, 1 .. 2, [[1, 2], [3, 4]]) &semi; RETURN (a, b) end proc$

(12)

>	dontreturn(a,f);

$proc (a, b) a := 3 &semi; b := array (1 .. 2, 1 .. 2, [[1, 2], [3, 4]]) &semi; return b end proc$

(13)

•	codegen,fortran - Generate fortran code

>	f := 12x+4-y+6x;

$f ≔ 18 x + 4 - y$

(14)

>	fortran(f);

t0 = 18*x+4-y

•	horner - Convert formulae in a procedure to horner form

>	h := proc(x,y,z) 1-2xy-xy^2z*(1-x); end;

$h := proc (x, y, z) 1 - 2 &ast; y &ast; x - x &ast; y &Hat; 2 &ast; z &ast; (1 - x) end proc$

(15)

>	horner(h,x);

$proc (x, y, z) 1 + (x &ast; y &Hat; 2 &ast; z - y &Hat; 2 &ast; z - 2 &ast; y) &ast; x end proc$

(16)

>	horner(h,[x,y]);

$proc (x, y, z) 1 + ((- y &ast; z - 2) &ast; y + x &ast; y &Hat; 2 &ast; z) &ast; x end proc$

(17)

•	intrep2maple - Converts an abstract syntax tree to a Maple procedure

>	f := proc(x) local t; t := sin(x); x*t; end:

>	tree := maple2intrep(f);

$tree ≔ Proc (Name (f .. float), Parameters (x .. float), Options (), Description (), Locals (t .. float), Globals (), StatSeq (Assign (t, \sin (x)), x t))$

(18)

>	intrep2maple(tree);

$proc (x) local t &semi; t := \sin (x) &semi; x &ast; t end proc$

(19)

•	joinprocs - Join the body of two or more Maple procedures together

>	f := proc(x,y) local e,t; e := exp(-x); t := x^2; t*e; end:

>	g := proc(x,y) local e,t; e := exp(-x); t := y^2; t*e; end:

>	J := joinprocs([f,g]);

$J := proc (x, y) local e, resultf, resultg, t &semi; e := \exp (- x) &semi; t := x &Hat; 2 &semi; resultf := e &ast; t &semi; e := \exp (- x) &semi; t := y &Hat; 2 &semi; resultg := e &ast; t &semi; return resultf, resultg end proc$

(20)

•	makeglobal - Make a variable to be a global variable

>	f := proc(x) local b; b := hfarray([[x*2,3],[4.5,32]]); end:

>	makeglobal(b,f);

$proc (x) global b &semi; b := hfarray ([[2 &ast; x, 3], [4.5, 32]]) end proc$

(21)

•	makeparam - Change the variable to be a parameter

>	f := proc() local x; x := 3*x; end:

>	g := makeparam(x,f);

$g := proc (x) x := 3 &ast; x end proc$

(22)

•	makeproc - Make a Maple procedure from a formula(e)

>	f := x-yxexp(-3);

$f ≔ x - y x {&ExponentialE;}^{−3}$

(23)

>	makeproc(f,[x,y]);

$proc (x, y) x - y &ast; x &ast; \exp (- 3) end proc$

(24)

•	makevoid - Do not return any values from a Maple procedure

>	f := proc(x,A::array(2..2)) A[1,1] := x^2; RETURN(x,A); end:

>	makevoid(f);

$proc (x, A :: (array (2 .. 2))) A [1, 1] := x &Hat; 2 &semi; return end proc$

(25)

•	maple2intrep - Converts a Maple procedure to an abstract syntax tree

>	f := proc(x) local t; t := sin(x); x*t; end:

>	tree := maple2intrep(f);

$tree ≔ Proc (Name (f .. float), Parameters (x .. float), Options (), Description (), Locals (t .. float), Globals (), StatSeq (Assign (t, \sin (x)), x t))$

(26)

•	optimize - Common subexpression optimization

>	f := 2^x-x^3;

$f ≔ 2^{x} - x^{3}$

(27)

>	optimize(f);

$t1 = 2^{x}, t2 = x^{2}, t4 = - t2 x + t1$

(28)

•	packlocals - Pack locals of a Maple procedure into an array

>	f := proc(x,y,z) local w,t; w := y^2; t := xwz; z-t^2; end:

>	packlocals(f,[w,t],A);

$proc (x, y, z) local A &semi; A := array (1 .. 2) &semi; A [1] := y &Hat; 2 &semi; A [2] := x &ast; A [1] &ast; z &semi; - A [2] &Hat; 2 + z end proc$

(29)

•	packparams or packargs - Pack parameters (or arguments) of a Maple procedure into an array

>	f := proc(x,y,z) local s,t; s := y^2; t := xsz; t^2; end;

$f := proc (x, y, z) local s, t &semi; s := y &Hat; 2 &semi; t := x &ast; s &ast; z &semi; t &Hat; 2 end proc$

(30)

>	packparams(f,[x,y,z],A);

$proc (A :: (array (1 .. 3))) local s, t &semi; s := A [2] &Hat; 2 &semi; t := A [1] &ast; s &ast; A [3] &semi; t &Hat; 2 end proc$

(31)

•	prep2trans - Prepare a Maple procedure for translation

>	f := proc(x) piecewise(x<0,0,x<1,x,x>1,2-x,0); end;

$f := proc (x) piecewise (x < 0, 0, x < 1, x, 1 < x, 2 - x, 0) end proc$

(32)

>	prep2trans(f);

$proc (x) if x < 0 then 0 elif x < 1 then x elif 1 < x then 2 - x else 0 end if end proc$

(33)

•	split - Prepare a Maple procedure for automatic differentiation

>	h := proc(a,b) 324exp(a)/56b3a; end;

$h := proc (a, b) 243 / 14 &ast; \exp (a) &ast; b &ast; a end proc$

(34)

split(h);

$proc (a, b) local s0 &semi; s0 := b &ast; a &semi; return 243 / 14 &ast; s0 &ast; \exp (a) end proc$

(35)

•	swapargs - Interchange two arguments in a Maple procedure

>	f := proc(x,y,z) local s,t; s := y^2; t := x/60-z; t^2; end:

>	swapargs(f,1=2);

$proc (y, x, z) local s, t &semi; s := y &Hat; 2 &semi; t := 1 / 60 &ast; x - z &semi; t &Hat; 2 end proc$

(36)

algcurves - The algebraic curves package

•	Weierstrassform - Find an isomorphism for function fields with genus 1

>	with(algcurves):

>	f:=x^4+y^4-2x^3+x^2y-2y^3+x^2-xy+y^2:

>	v:=Weierstrassform(f,x,y,x0,y0);

$v ≔ [{x0}^{3} + \frac{2}{3} x0 + \frac{1}{108} + {y0}^{2}, \frac{- 3 y + 2 x}{3 x}, - \frac{x^{3} - 2 x y^{2} + 2 y^{3} - x^{2} + 2 y x - 2 y^{2}}{2 x^{2} (x - 1)}, \frac{- 162 {x0}^{3} + 324 {x0}^{2} - 162 x0 y0 - 135 x0 + 108 y0 + 156}{162 {x0}^{4} - 432 {x0}^{3} + 432 {x0}^{2} - 192 x0 + 194}, \frac{162 {x0}^{4} - 432 {x0}^{3} + 162 {x0}^{2} y0 + 351 {x0}^{2} - 216 x0 y0 - 246 x0 + 72 y0 + 104}{162 {x0}^{4} - 432 {x0}^{3} + 432 {x0}^{2} - 192 x0 + 194}]$

(37)

•	genus - Find the genus of an algebraic curve

>	f:=x^4-y^3-y;

$f ≔ x^{4} - y^{3} - y$

(38)

>	factor(f);

$x^{4} - y^{3} - y$

(39)

>	genus(f,x,y);

$3$

(40)

•	homogeneous - Make a polynomial in two variables homogeneous in three variables

>	f:=a^2-b^3;

$f ≔ - b^{3} + a^{2}$

(41)

>	homogeneous(f,b,a,c);

$a^{2} c - b^{3}$

(42)

•	integral_basis - Find integral bases of algebraic function fields

>	alpha:=RootOf(x^3+7,x);

$α ≔ RootOf ({_Z}^{3} + 7)$

(43)

>	f:=y^8+3xy^5alpha+5x^4+x^6*alpha;

$f ≔ y^{8} + RootOf ({_Z}^{3} + 7) x^{6} + 3 RootOf ({_Z}^{3} + 7) x y^{5} + 5 x^{4}$

(44)

>	integral_basis(f,x,y);

$[1, y, y^{2}, \frac{y^{3}}{x}, \frac{y^{4}}{x}, \frac{y^{2} (y^{3} + 3 x RootOf ({_Z}^{3} + 7))}{x^{2}}, \frac{y^{3} (y^{3} + 3 x RootOf ({_Z}^{3} + 7))}{x^{2}}, \frac{y^{4} (y^{3} + 3 x RootOf ({_Z}^{3} + 7))}{x^{3}}]$

(45)

•	j_invariant - Find the j invariant of an elliptic curve

>	f:=y^5+4/3-23/3y^2+11y^3-17/3y^4-16/3x^2+16/3x^3-4/3x^4:

check to see if genus is 1 first

>	genus(f,x,y);

$1$

(46)

>	j_invariant(f,x,y);

$- \frac{1404928}{171}$

(47)

•	parametrization - Calculate parametrization for a curve with genus 0

>	f:=y^5+2xy^2+2xy^3+x^2y-4x^3y+2x^5:

>	v:=parametrization(f,x,y,t);

$v ≔ [\frac{- 24192 t^{5} - 6048 t^{4} + 2520 t^{3} - 238 t^{2} + 7 t}{181604 t^{5} - 103680 t^{4} + 17280 t^{3} - 1440 t^{2} + 60 t - 1}, \frac{16464 t^{5} + 6860 t^{4} - 686 t^{3}}{181604 t^{5} - 103680 t^{4} + 17280 t^{3} - 1440 t^{2} + 60 t - 1}]$

(48)

•	plot_knot - Make a tubeplot for a singularity knot

>	plot_knot(x^2+y^2,x,y,axes=normal);

•	puiseux - Determine the Puiseux expansion of an algebraic function

>	Root:=RootOf(x^2-5);

$Root ≔ RootOf ({_Z}^{2} - 5)$

(49)

>	f := 4+Root-y^4+x*5;

$f ≔ - y^{4} + RootOf ({_Z}^{2} - 5) + 5 x + 4$

(50)

>	puiseux(f,x=0,y,3);

$\{(\frac{75 RootOf ({_Z}^{4} - RootOf ({_Z}^{2} - 5) - 4) RootOf ({_Z}^{2} - 5)}{484} - \frac{1575 RootOf ({_Z}^{4} - RootOf ({_Z}^{2} - 5) - 4)}{3872}) x^{2} + (- \frac{5 RootOf ({_Z}^{4} - RootOf ({_Z}^{2} - 5) - 4) RootOf ({_Z}^{2} - 5)}{44} + \frac{5 RootOf ({_Z}^{4} - RootOf ({_Z}^{2} - 5) - 4)}{11}) x + RootOf ({_Z}^{4} - RootOf ({_Z}^{2} - 5) - 4)\}$

(51)

•	singularities - Determine the singularities of an algebraic curve

>	f := x^3y+63y-4y^2+6xy-8x;

$f ≔ y x^{3} + 6 y x - 4 y^{2} - 8 x + 63 y$

(52)

>	singularities(f,x,y);

$\{[[0, 1, 0], 2, 1, 1]\}$

(53)

diffalg - Package for manipulating systems of differential polynomial equations (ODE or PDE)

For examples of differential polynomials, please see the Differential example worksheet.

•	Rosenfeld_Groebner - Find the representation of radical ideal

•	belongs_to - Test if a differential polynomial belongs to a radical differential ideal

•	delta_leader - Return the difference of the derivation operators of the leaders of two differential polynomials

•	delta_polynomial - Return the delta-polynomial generated by two differential polynomials

•	denote - Convert a differential polynomial from an external form to another one

•	derivatives - Return the derivatives occurring in a differential polynomial

•	differential_ring - Define a differential polynomial ring endowed with a ranking and notations

•	differential_sprem - Return the sparse pseudo remainder of a differential polynomial

•	differentiate - Differentiate a differential polynomial

•	equations - Return the equations of a regular differential ideal

•	field_extension - Define a field extension of the field of the rational numbers

•	formal_power_series - Compute a formal power series from a regular differential ideal

•	greater - Compare the rank of two differential polynomials

•	inequations - Return the inequations of a regular differential ideal

•	initial - Return the initial of a differential polynomial

•	initial_conditions - Return the list of the initial conditions of a regular differential ideal

•	is_orthonomic - Test if a regular differential ideal is presented by an orthonomic system of equations

•	leader - Return the leader of a differential polynomial

•	power_series_solution a solution of a regular differential ideal whose components are formal power series

•	print_ranking - Print a message describing the ranking of a differential polynomial ring

•	rank - Return the rank of a differential polynomial

•	reduced - Test if a differential polynomial is reduced w.r.t. differential polynomials

•	reduced_form - Find a reduced form of a differential polynomial modulo a radical differential ideal

•	rewrite_rules - Display the equations of a regular differential ideal using a special syntax

•	separant - Display the separant of a differential polynomial

geom3d - Introduction to the geom3d package

•	AreCollinear - Test if three points are collinear

restart;

>	with(geom3d):

>	point(A,0,0,0):

>	point(B,1,2,1):

>	point(C,2,4,2):

>	geom3d[AreCollinear](A,B,C);

$true$

(54)

•	AreConcurrent - Test if three lines are concurrent

>	point(A,1,1,1);

$A$

(55)

>	point(B,-1,-1,-1);

$B$

(56)

>	point(C,10,-10,10);

$C$

(57)

>	point(D,-10,10,-10);

Warning, a geometry object has been assigned to the protected name D. Use of protected names for geometry objects is not recommended and may break Maple functionality.

$D$

(58)

>	point(E,-66,5,0);

$E$

(59)

>	point(F,66,-5,0);

$F$

(60)

>	line(line1,[A,B]);

$line1$

(61)

>	line(line2,[C,D]);

$line2$

(62)

>	line(line3,[E,F]);

$line3$

(63)

>	geom3d[AreConcurrent](line1,line2,line3);

$true$

(64)

•	AreConjugate - Test if a pair of points is conjugate with respect to a sphere

>	point(A,0,0,0):

>	point(B,3,-4,2):

>	point(C,5,3,4):

>	sphere(sph,[A,5],[x,y,z]):

>	geom3d[AreConjugate](B,C,sph);

$false$

(65)

•	AreCoplanar - Test if the given objects are on the same plane

>	point(A,0,0,0): point(B,34,2,0): point(C,2,-15,0):

>	plane(pl1,[A,B,C]):

>	randpoint(D,pl1):

>	geom3d[AreCoplanar](A,B,C,D);

$true$

(66)

•	AreDistinct - Test if given objects are distinct

>	sphere(sph1,[point(A,0,0,0),3]):

>	line(line1,[A,point(B,0,0,21)]):

>	rotation(sph2,sph1,Pi/4,line1):

>	geom3d[AreDistinct](sph1,sph2);

$false$

(67)

•	AreParallel - Test if two objects are parallel to each other

>	point(A,1,3,2): point(B,5,2,3):

>	line(line1,[A,B]);

$line1$

(68)

>	plane(pl,[point(C,6,2,43),point(D,6,23,3),point(E,56,2,31)]):

>	geom3d[AreParallel](line1,pl);

$false$

(69)

•	ArePerpendicular - Test if two objects are perpendicular to each other

>	point(A,[6,3,2]):

>	point(B,[5,2,3]):

>	point(C,[6,2,1]):

>	plane(pl,[A,B,C]):

>	line(line1,[point(D,24,26,-2),point(E,26,-61,4)]):

>	geom3d[ArePerpendicular](line1,pl);

$false$

(70)

•	AreSkewLines - Test if two lines are skew

>	line(line01,[point(A,34,2,-13),point(B,62,-21,5)]):

>	line(line02,[point(C,1,1,6),point(D,-26,12,-72)]):

>	AreSkewLines(line01,line02);

$true$

(71)

•	DefinedAs - Return the name of endpoints or vertices of segment, directed segment, triangle

>	point(A,6,3,2):

>	point(B,6,21,-2):

>	point(C,-25,2,6):

>	triangle(tr1,[A,B,C]);

$tr1$

(72)

>	DefinedAs(tr1);

$[A, B, C]$

(73)

•	DirectionRatios - Return the direction-ratios of a line

>	line(ln1,[point(A,6,2,65),point(B,-62,2,-46)]):

>	DirectionRatios(ln1);

$[- \frac{68 \sqrt{16945}}{16945}, 0, - \frac{111 \sqrt{16945}}{16945}]$

(74)

•	Equation - Find the equation of a geometric object

>	sphere(sph1,[point(A,0,0,5),4]):

>	Equation(sph1,[x,y,z]);

$x^{2} + y^{2} + z^{2} - 10 z + 9 = 0$

(75)

•	FindAngle - Find the angle between two given objects

>	plane(pl1,[point(A,3,5,2),point(B,-3,2,0),point(C,5,2,-5)]):

>	line(ln1,[point(D,5,3,-1),point(E,1,2,5)]):

>	FindAngle(ln1,pl1);

$\arcsin (\frac{130 \sqrt{154601}}{154601})$

(76)

•	FixedPoint - Return a fixed point on a line

>	plane(pl1,[point(A,5,2,1),[5,2,5]]):

>	plane(pl2,[point(B,2,1,-5),[2,5,-6]]):

>	line(ln1,[pl1,pl2]):

>	FixedPoint(ln1);

$pl1_pl2$

(77)

>	detail(pl1_pl2);

$\begin{array}{l} name of the object & pl1_pl2 \\ form of the object & point3d \\ coordinates of the point & [\frac{92}{21}, \frac{127}{21}, 0] \end{array}$

(78)

•	HarmonicConjugate - Find the harmonic conjugate of a point with respect to two other points

>	point(A,3,4,2):

>	point(B,2,6,1):

>	point(C,6,2,2):

>	HarmonicConjugate(HC,B,C,A);

$HC$

(79)

>	coordinates(HC);

$[0, 8, \frac{1}{2}]$

(80)

•	IsEquilateral - Test if a given triangle is equilateral

>	triangle(tri,[point(A,56,1,2),point(B,2,0,56),point(C,6,1,2)]):

>	IsEquilateral(tri);

$false$

(81)

•	IsFacetted - Check if the given polyhedron is of faceted form

>	facet(i1,cuboctahedron(c,point(o,0,0,0),2),1);

$i1$

(82)

>	IsFacetted(i1);

$true$

(83)

•	IsOnObject - Test if a point, a list or a set of points is on a given object

>	line(ln1,[point(B,0,0,0),point(C,10,4,12)]):

>	IsOnObject(point(A,5,2,6),ln1);

$true$

(84)

•	IsRightTriangle - Test if a given triangle is a right triangle

>	triangle(tri,[point(A,6,21,6),point(B,2,-7,5),point(C,-6,6,2)],[x,y,z]):

>	IsRightTriangle(tri);

$false$

(85)

•	IsStellated - Check if the given polyhedron is of stellated form

>	stellate(ste,icosahedron(i,point(o,1,1,1),2),22);

$ste$

(86)

>	IsStellated(ste);

$true$

(87)

•	IsTangent - Test if a plane is tangent to a sphere

>	sphere(sph,[point(A,0,0,0),3]):

>	plane(pl,[point(B,7,2,5),point(C,5,1,-2),point(D,-7,2,-3)]):

>	IsTangent(pl,sph);

$false$

(88)

•	NormalVector - Define the planes

>	plane(pl,[point(A,6,2,3),point(B,2,1,-5),point(C,-1,6,2)]):

>	Normvec := NormalVector(pl);

$Normvec ≔ [33, 52, −23]$

(89)

•	OnSegment - Find the point which divides the segment joining two given points by some ratio

>	OnSegment(pt,point(A,5,2,-6),point(B,-1,5,2),3);

$pt$

(90)

>	detail(pt);

$\begin{array}{l} name of the object & pt \\ form of the object & point3d \\ coordinates of the point & [\frac{1}{2}, \frac{17}{4}, 0] \end{array}$

(91)

•	ParallelVector - Return a direction-ratios of a line

>	line(ln1,[point(A,6,-6,2),point(B,-5,1,-61)]):

>	PVec := ParallelVector(ln1);

$PVec ≔ [−11, 7, −63]$

(92)

•	QuasiRegularPolyhedron - Define a quasi-regular polyhedron

>	QuasiRegularPolyhedron(QRpoly, [[3],[4]],point(A,6,6,2),3);

$QRpoly$

(93)

•	RadicalCenter - Find the radical center of four given spheres

>	sphere(sph1,[point(A,2,5,1),4]):

>	sphere(sph2,[point(B,6,-2,3),3]):

>	sphere(sph3,[point(C,-8,3,7),4]):

>	sphere(sph4,[point(D,-6,2,-1),6]):

>	RadicalCenter(pt,sph1,sph2,sph3,sph4);

$pt$

(94)

>	coordinates(pt);

$[\frac{141}{136}, - \frac{43}{34}, 0]$

(95)

•	RadicalLine - Find the radical line of three given spheres

>	sphere(sph1,[point(A,2,5,1),4]):

>	sphere(sph2,[point(B,6,-2,3),3]):

>	sphere(sph3,[point(C,-8,3,7),4]):

>	RadicalLine(ln1,sph1,sph2,sph3);

$ln1$

(96)

>	detail(ln1);

Warning, assume that the parameter in the parametric equations is _t

Warning, assuming that the names of the axes are _x, _y, and _z

$\begin{array}{l} name of the object & ln1 \\ form of the object & line3d \\ equation of the line & [_x = - \frac{148}{39} - 152_t,_y = - \frac{157}{39} - 176_t,_z = - 312_t] \end{array}$

(97)

•	RadicalPlane - Find the radical plane of two given spheres

>	sphere(sph1,[point(A,2,5,1),4]):

>	sphere(sph2,[point(B,6,-2,3),3]):

>	RadicalPlane(pl,sph1,sph2);

$pl$

(98)

>	detail(pl);

Warning, assuming that the names of the axes are _x, _y and _z

$\begin{array}{l} name of the object & pl \\ form of the object & plane3d \\ equation of the plane & - 26 + 8_x - 14_y + 4_z = 0 \end{array}$

(99)

•	StereographicProjection

•	TangentPlane - Find the tangent plane of a point on a sphere

>	sphere(sph,[point(A,0,0,0),1]):

>	point(B,0,0,1):

>	TangentPlane(pl,B,sph);

$pl$

(100)

>	detail(pl);

Warning, assuming that the names of the axes are _x, _y and _z

$\begin{array}{l} name of the object & pl \\ form of the object & plane3d \\ equation of the plane & 1 -_z = 0 \end{array}$

(101)

•	area - Calculate the area of a triangle, surface area of sphere, regular polyhedron

>	sphere(sph1,[point(A,2,5,1),4]):

>	area(sph1);

$64 π$

(102)

•	altitude - Find the altitude of a given triangle

>	triangle(tri,[point(A,3,5,1),point(B,-6,1,-4),point(C,7,3,0)]):

>	altitude(al,tri,B);

$al$

(103)

form(al);

$segment3d$

(104)

>	detail(al);

$\begin{array}{l} name of the object & al \\ form of the object & segment3d \\ the 2 ends of the segment & [[- \frac{29}{21}, \frac{151}{21}, \frac{44}{21}], [- \frac{29}{21}, \frac{151}{21}, \frac{44}{21}]] \end{array}$

(105)

•	center - Find the center of a given geometric object

>	tetrahedron(t,point(pt,1,2,3),3):

>	center(ctr,t);

$ctr$

(106)

>	form(ctr);

$point3d$

(107)

>	coordinates(ctr);

$[1, 2, 3]$

(108)

•	centroid - Find the centroid of a given general tetrahedron or a given triangle or a list of points in space

>	point(A,0,0,0),point(B,1,0,0),point(C,0,0,1),point(F,0,1,0):

>	gtetrahedron(t,[A,B,C,F]):

>	triangle(tri,[point(A,4,1,2),point(B,2,5,1),point(C,-5,1,-6)]);

$tri$

(109)

>	centroid(ctd,tri);

$ctd$

(110)

>	centroid(ctd1,t);

$ctd1$

(111)

>	form(ctd);

$point3d$

(112)

>	coordinates(ctd);

$[\frac{1}{3}, \frac{7}{3}, −1]$

(113)

•	circle - Create a circle

>	sphere(sph,[point(A,5,2,6),5]):

>	plane(pl,[point(B,-6,1,3),point(C,6,7,1),point(D,-2,2,5)]):

>	geom3d:-circle(cir,[sph,pl]);

$cir$

(114)

>	detail(cir);

Warning, assuming that the names of the axes are _x, _y and _z

$\begin{array}{l} name of the object & cir \\ form of the object & circle3d \\ name of the center & center_cir_1 \\ coordinates of the center & [\frac{1404}{341}, \frac{1370}{341}, \frac{2304}{341}] \\ radius of the center & \frac{2 \sqrt{569129}}{341} \\ the circle lies on the plane & pl \end{array}, \begin{array}{l} name of the object & pl \\ form of the object & plane3d \\ equation of the plane & 152 + 14_x - 32_y - 12_z = 0 \end{array}$

(115)

•	coordinates - Display the coordinates of a given point

>	point(A,5,2,5);

$A$

(116)

>	coordinates(A);

$[5, 2, 5]$

(117)

•	detail - Give a detailed description of an object, or a list or set of objects

>	sphere(sph,[point(A,5,2,6),4]);

$sph$

(118)

>	detail(sph);

Warning, assume that the name of the axes are _x, _y and _z

$\begin{array}{l} name of the object & sph \\ form of the object & sphere3d \\ name of the center & A \\ coordinates of the center & [5, 2, 6] \\ radius of the sphere & 4 \\ surface area of the sphere & 64 π \\ volume of the sphere & \frac{256 π}{3} \\ equation of the sphere & {_x}^{2} + {_y}^{2} + {_z}^{2} - 10_x - 4_y - 12_z + 49 = 0 \end{array}$

(119)

•	distance - Find the distance between two given objects

>	line(ln1,[point(A,6,2,1),point(B,-2,21,1)]):

>	line(ln2,[point(C,2,1,2),point(D,3,-5,3)]):

>	distance(ln1,ln2);

$\frac{55 \sqrt{1266}}{1266}$

(120)

•	draw - Create a three-dimensional plot

>	sphere(sph,[point(A,5,2,6),4]);

$sph$

(121)

>	draw(sph);

•	dsegment - Define a directed segment

>	dsegment(dseg,[point(A,5,2,-3),point(B,-5,2,1)]);

$dseg$

(122)

>	detail(dseg);

$\begin{array}{l} name of the object & dseg \\ form of the object & dsegment3d \\ the initial and end points of the directed segment & [[5, 2, −3], [−5, 2, 1]] \end{array}$

(123)

•	faces - Return the faces of a polyhedron

>	hexahedron(hexa,point(A,6,2,-3),3);

$hexa$

(124)

>	faces(hexa);

$[[[\sqrt{3} + 6, 2 + \sqrt{3}, \sqrt{3} - 3], [\sqrt{3} + 6, 2 - \sqrt{3}, \sqrt{3} - 3], [\sqrt{3} + 6, 2 - \sqrt{3}, - \sqrt{3} - 3], [\sqrt{3} + 6, 2 + \sqrt{3}, - \sqrt{3} - 3]], [[- \sqrt{3} + 6, 2 + \sqrt{3}, \sqrt{3} - 3], [- \sqrt{3} + 6, 2 + \sqrt{3}, - \sqrt{3} - 3], [- \sqrt{3} + 6, 2 - \sqrt{3}, - \sqrt{3} - 3], [- \sqrt{3} + 6, 2 - \sqrt{3}, \sqrt{3} - 3]], [[\sqrt{3} + 6, 2 + \sqrt{3}, \sqrt{3} - 3], [\sqrt{3} + 6, 2 + \sqrt{3}, - \sqrt{3} - 3], [- \sqrt{3} + 6, 2 + \sqrt{3}, - \sqrt{3} - 3], [- \sqrt{3} + 6, 2 + \sqrt{3}, \sqrt{3} - 3]], [[\sqrt{3} + 6, 2 - \sqrt{3}, \sqrt{3} - 3], [- \sqrt{3} + 6, 2 - \sqrt{3}, \sqrt{3} - 3], [- \sqrt{3} + 6, 2 - \sqrt{3}, - \sqrt{3} - 3], [\sqrt{3} + 6, 2 - \sqrt{3}, - \sqrt{3} - 3]], [[\sqrt{3} + 6, 2 + \sqrt{3}, \sqrt{3} - 3], [- \sqrt{3} + 6, 2 + \sqrt{3}, \sqrt{3} - 3], [- \sqrt{3} + 6, 2 - \sqrt{3}, \sqrt{3} - 3], [\sqrt{3} + 6, 2 - \sqrt{3}, \sqrt{3} - 3]], [[\sqrt{3} + 6, 2 - \sqrt{3}, - \sqrt{3} - 3], [- \sqrt{3} + 6, 2 - \sqrt{3}, - \sqrt{3} - 3], [- \sqrt{3} + 6, 2 + \sqrt{3}, - \sqrt{3} - 3], [\sqrt{3} + 6, 2 + \sqrt{3}, - \sqrt{3} - 3]]]$

(125)

•	facet - Define a faceting of a given polyhedron

>	facet(icofac,icosidodecahedron(ico,point(A,5,2,-4),2));

$icofac$

(126)

>	vertices(icofac);

$[[5, 2, −4], [5, \frac{2 5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, −4], [\frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [- \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [- \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [5, - \frac{2 5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, −4], [- \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [- \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [5, 2, \frac{2 5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [5, 2, - \frac{2 5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} + \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 \sqrt{5} (- \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, \frac{2 \sqrt{5} (\frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{3}{4}}}{20} - \frac{\sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}} 5^{\frac{1}{4}}}{4})}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 2, - \frac{5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} - 4], [\frac{2 5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, 2, −4], [- \frac{2 5^{\frac{1}{4}} \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}{\sqrt{\frac{5}{2} + \frac{\sqrt{5}}{2}}} + 5, 2, −4]]$

(127)

•	form - Return the form of the geometric object

>	tetrahedron(t,point(A,5,2,1),3);

$t$

(128)

form(t);

$tetrahedron3d$

(129)

•	gtetrahedron - Define a general tetrahedron

>	point(A,0,0,0),point(B,1,0,0),point(C,0,0,1),point(F,0,1,0):

>	gtetrahedron(gt, [A, B, C, F]);

$gt$

(130)

form(gt);

$gtetrahedron3d$

(131)

•	incident - Return the vertices incident to a given vertex of a polyhedron

>	tetrahedron(t,point(A,0,0,0),4):

>	incident(t,2);

$[[\frac{4 \sqrt{3}}{3}, \frac{4 \sqrt{3}}{3}, \frac{4 \sqrt{3}}{3}], [- \frac{4 \sqrt{3}}{3}, \frac{4 \sqrt{3}}{3}, - \frac{4 \sqrt{3}}{3}], [- \frac{4 \sqrt{3}}{3}, - \frac{4 \sqrt{3}}{3}, \frac{4 \sqrt{3}}{3}]]$

(132)

•	intersection - Find the intersections between two or three given objects

>	plane(pl1,[point(A,5,2,6),point(B,-6,1,2),point(C,5,-6,1)]):

>	plane(pl2,[point(D,26,-7,2),point(E,1,2,9),point(F,2,1,6)]):

>	intersection(ln1,pl1,pl2);

$ln1$

(133)

>	Equation(ln1,x);

$[- \frac{622}{23} + 5104 x, \frac{187}{23} - 1328 x, 736 x]$

(134)

•	line - Define the lines

>	line(ln1,[point(A,56,2,6),point(B,2,-6,2)]);

$ln1$

(135)

>	detail(ln1);

Warning, assume that the parameter in the parametric equations is _t

Warning, assuming that the names of the axes are _x, _y, and _z

$\begin{array}{l} name of the object & ln1 \\ form of the object & line3d \\ equation of the line & [_x = 56 - 54_t,_y = 2 - 8_t,_z = 6 - 4_t] \end{array}$

(136)

•	midpoint - Find the midpoint of segment joining two points

>	segment(seg,[point(A,5,2,3),point(B,-3,2,1)]):

>	midpoint(C,seg);

$C$

(137)

>	detail(C);

$\begin{array}{l} name of the object & C \\ form of the object & point3d \\ coordinates of the point & [1, 2, 2] \end{array}$

(138)

•	parallel - Find a plane or line going through a given point or line, and parallel to a given line or plane

>	point(A,5,2,1):

>	plane(pl,[point(C,-2,3,1),point(D,7,34,6),point(E,6,-5,21)]):

>	parallel(parpl,A,pl);

$parpl$

(139)

>	Equation(parpl,[x,y,z]);

$- 2700 + 660 x - 140 y - 320 z = 0$

(140)

•	plane - Define the planes

>	plane(pl,[point(A,6,2,1),[7,4,-3]]);

$pl$

(141)

>	detail(pl);

Warning, assuming that the names of the axes are _x, _y and _z

$\begin{array}{l} name of the object & pl \\ form of the object & plane3d \\ equation of the plane & - 47 + 7_x + 4_y - 3_z = 0 \end{array}$

(142)

•	point - Define the points

>	point(A,[6,-43,2]);

$A$

(143)

>	detail(A);

$\begin{array}{l} name of the object & A \\ form of the object & point3d \\ coordinates of the point & [6, −43, 2] \end{array}$

(144)

•	polar - Find the polar of a given point with respect to a given sphere

>	sphere(sph,[point(A,-2,1,5),3]):

>	point(B,5,2,3):

>	polar(pol,B,sph);

$pol$

(145)

>	detail(pol);

Warning, assuming that the names of the axes are _x, _y and _z

$\begin{array}{l} name of the object & pol \\ form of the object & plane3d \\ equation of the plane & 14 + 7_x +_y - 2_z = 0 \end{array}$

(146)

•	pole - Find the pole of a given plane with respect to a given sphere

>	sphere(sph,[point(A,-2,1,5),3]):

>	plane(pl,[point(B,3,5,1),point(C,6,-4,2),point(D,-2,3,5)]):

>	pole(pt,pl,sph);

$pt$

(147)

>	detail(pt);

$\begin{array}{l} name of the object & pt \\ form of the object & point3d \\ coordinates of the point & [- \frac{5}{7}, \frac{23}{14}, \frac{97}{14}] \end{array}$

(148)

•	powerps - Calculate the power of a given point with respect to a given sphere

>	point(A,3,5,2):

>	sphere(sph,[point(B,6,2,1),3]):

>	powerps(A,sph);

$10$

(149)

•	radius - Find the radius of a given object

>	hexahedron(hexa,point(A,6,2,-6),3):

>	radius(hexa);

$3$

(150)

•	randpoint - Generate a random point in a given range, on a line, a plane, or a sphere

>	line(ln1,[point(A,6,2,2),point(B,-6,2,1)]):

>	randpoint(C,ln1);

$C$

(151)

•	schlafli - Return the schlafli symbol of a given polyhedron

>	tetrahedron(t,point(A,-8,9,3),3);

$t$

(152)

>	schlafli(t);

$[3, 3]$

(153)

•	segment - Define a segment

>	segment(seg,point(A,5,2,6),point(B,-6,2,-9));

$seg$

(154)

>	detail(seg);

$\begin{array}{l} name of the object & seg \\ form of the object & segment3d \\ the 2 ends of the segment & [[5, 2, 6], [−6, 2, −9]] \end{array}$

(155)

•	sides - The sides of a given triangle, a regular polyhedron, an Archimedean solid

>	triangle(tri,[point(A,5,2,6),point(B,-6,2,1),point(C,3,-9,0)]):

>	sides(tri);

$[\sqrt{146}, \sqrt{203}, \sqrt{161}]$

(156)

•	sphere - Define the sphere

>	sphere(sph,[point(A,6,6,7),3]);

$sph$

(157)

>	detail(sph);

Warning, assume that the name of the axes are _x, _y and _z

$\begin{array}{l} name of the object & sph \\ form of the object & sphere3d \\ name of the center & A \\ coordinates of the center & [6, 6, 7] \\ radius of the sphere & 3 \\ surface area of the sphere & 36 π \\ volume of the sphere & 36 π \\ equation of the sphere & {_x}^{2} + {_y}^{2} + {_z}^{2} - 12_x - 12_y - 14_z + 112 = 0 \end{array}$

(158)

•	stellate - Define a stellation of a given polyhedron

>	stellate(stelicos,icosahedron(icos,point(A,6,2,2),3),25);

$stelicos$

(159)

•	tname - Find the name of the parameter used in the parametric equations of the given line

>	_EnvTName := 't';

$_EnvTName ≔ t$

(160)

>	line(ln1,[point(A,5,2,1),point(B,-2,1,9)]):

>	detail(ln1);

Warning, assuming that the names of the axes are _x, _y, and _z

$\begin{array}{l} name of the object & ln1 \\ form of the object & line3d \\ equation of the line & [_x = 5 - 7 t,_y = 2 - t,_z = 1 + 8 t] \end{array}$

(161)

>	tname(ln1);

$t$

(162)

•	triangle - Define the triangles

>	triangle(tri,[point(A,65,2,1),point(B,-2,1,5),point(C,-6,-2,-1)]);

$tri$

(163)

>	detail(tri);

$\begin{array}{l} name of the object & tri \\ form of the object & triangle3d \\ the 3 vertices & [[65, 2, 1], [−2, 1, 5], [−6, −2, −1]] \end{array}$

(164)

•	vertices - Return the vertices of a polyhedron, parallelepiped

>	triangle(tri,[point(A,5,2,6),point(B,8,-3,2),point(C,-6,2,-4)]):

>	vertices(tri);

$[[5, 2, 6], [8, −3, 2], [−6, 2, −4]]$

(165)

•	volume - Calculate the volume of a sphere, regular polyhedron

>	sphere(sph,[point(A,6,2,3),3]);

$sph$

(166)

>	volume(sph);

$36 π$

(167)

•	xcoord - Find the ``x''-coordinate of a given point

>	point(A,5,2,1);

$A$

(168)

>	xcoord(A);

$5$

(169)

•	xname - Find the name of the x-axis used to define an object

>	sphere(sph,[point(A,6,2,-5),4]):

>	Equation(sph,[x,y,z]);

$x^{2} + y^{2} + z^{2} - 12 x - 4 y + 10 z + 49 = 0$

(170)

>	xname(sph);

$x$

(171)

•	ycoord - Find the ``y''-coordinate of a given point

>	point(A,5,2,1);

$A$

(172)

>	ycoord(A);

$2$

(173)

•	yname - Find the name of the y-axis used to define an object

>	sphere(sph,[point(A,6,2,-5),4]):

>	Equation(sph,[x,y,z]);

$x^{2} + y^{2} + z^{2} - 12 x - 4 y + 10 z + 49 = 0$

(174)

>	yname(sph);

$y$

(175)

•	zcoord - Find the ``z''-coordinate of a given point

>	point(A,5,2,1);

$A$

(176)

>	zcoord(A);

$1$

(177)

•	zname - Find the name of the z-axis used to define an object

>	sphere(sph,[point(A,6,2,-5),4]):

>	Equation(sph,[x,y,z]);

$x^{2} + y^{2} + z^{2} - 12 x - 4 y + 10 z + 49 = 0$

(178)

>	zname(sph);

$z$

(179)

Archimedean solids

•	Archimedean - Define an Archimedean solid

>	Archimedean(argon,_t([5,3]),point(A,5,2,6),3);

$argon$

(180)

>	form(argon);

$TruncatedDodecahedron3d$

(181)

•	GreatRhombicuboctahedron - Define an Archimedean solid

>	GreatRhombiicosidodecahedron(gon,point(A,-5,2,1),3);

$gon$

(182)

>	form(gon);

$RhombitruncatedIcosidodecahedron3d$

(183)

•	GreatRhombiicosidodecahedron - Define an Archimedean solid

>	GreatRhombiicosidodecahedron(gon,point(A,-5,2,1),3);

$gon$

(184)

>	form(gon);

$RhombitruncatedIcosidodecahedron3d$

(185)

•	IsArchimedean - Check if the given polyhedron is an Archimedean solid

>	Archimedean(argon,_t([5,3]),point(A,5,2,6),3);

$argon$

(186)

>	IsArchimedean(argon);

$true$

(187)

•	SmallRhombicuboctahedron - Define an Archimedean solid

>	SmallRhombicuboctahedron(gon,point(A,5,21,-9),2);

$gon$

(188)

>	form(gon);

$SmallRhombicuboctahedron3d$

(189)

•	SmallRhombiicosidodecahedron - Define an Archimedean solid

>	SmallRhombiicosidodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(190)

>	form(gon);

$SmallRhombiicosidodecahedron3d$

(191)

•	SnubCube - Define an Archimedean solid

>	SnubCube(gon,point(A,-6,1,-8),3);

$gon$

(192)

>	form(gon);

$SnubCube3d$

(193)

•	SnubDodecahedron - Define an Archimedean solid

>	SnubDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(194)

>	form(gon);

$SnubDodecahedron3d$

(195)

•	TruncatedCuboctahedron - Define an Archimedean solid

>	TruncatedCuboctahedron(gon,point(A,-6,1,-8),3);

$gon$

(196)

>	form(gon);

$RhombitruncatedCuboctahedron3d$

(197)

•	TruncatedDodecahedron - Define an Archimedean solid

>	TruncatedDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(198)

>	form(gon);

$TruncatedDodecahedron3d$

(199)

•	TruncatedHexahedron - Define an Archimedean solid

>	TruncatedHexahedron(gon,point(A,-6,1,-8),3);

$gon$

(200)

>	form(gon);

$TruncatedHexahedron3d$

(201)

•	TruncatedIcosahedron - Define an Archimedean solid

>	TruncatedIcosahedron(gon,point(A,-6,1,-8),3);

$gon$

(202)

>	form(gon);

$TruncatedIcosahedron3d$

(203)

•	TruncatedIcosidodecahedron - Define an Archimedean solid

>	TruncatedIcosidodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(204)

>	form(gon);

$RhombitruncatedIcosidodecahedron3d$

(205)

•	TruncatedTetrahedron - Define an Archimedean solid

>	TruncatedTetrahedron(gon,point(A,-6,1,-8),3);

$gon$

(206)

>	form(gon);

$TruncatedTetrahedron3d$

(207)

•	cuboctahedron - Define an Archimedean solid

>	TruncatedTetrahedron(gon,point(A,-6,1,-8),3);

$gon$

(208)

>	form(gon);

$TruncatedTetrahedron3d$

(209)

•	icosidodecahedron - Define an Archimedean solid

>	icosidodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(210)

>	form(gon);

$icosidodecahedron3d$

(211)

Quasi-regular Polyhedra

•	IsQuasi - Check if the given polyhedron is quasi-regular

>	QuasiRegularPolyhedron(QRpoly,[[3],[4]],point(A,5,2,1),3):

>	IsQuasi(QRpoly);

$true$

(212)

Regular Polyhedra (geom3d cont'd...)

•	GreatDodecahedron - Define a regular polyhedron

>	GreatDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(213)

>	form(gon);

$GreatDodecahedron3d$

(214)

•	GreatIcosahedron - Define a regular polyhedron

>	GreatIcosahedron(gon,point(A,-6,1,-8),3);

$gon$

(215)

>	form(gon);

$GreatIcosahedron3d$

(216)

•	GreatStellatedDodecahedron - Define a regular polyhedron

>	GreatStellatedDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(217)

>	form(gon);

$GreatStellatedDodecahedron3d$

(218)

•	InRadius - Return the in-radius of a regular polyhedron

>	GreatStellatedDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(219)

>	InRadius(gon);

$\frac{5^{\frac{3}{4}} \sqrt{3}}{5 {(\frac{1}{2} + \frac{\sqrt{5}}{2})}^{\frac{3}{2}}}$

(220)

•	IsRegular - Check if the given polyhedron is regular

>	GreatStellatedDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(221)

>	IsRegular(gon);

$true$

(222)

•	MidRadius - Return the mid-radius of a regular polyhedron

>	GreatIcosahedron(gon,point(A,-6,1,-8),3);

$gon$

(223)

>	MidRadius(gon);

$\frac{3 5^{\frac{3}{4}}}{5 \sqrt{\frac{1}{2} + \frac{\sqrt{5}}{2}}}$

(224)

•	RegularPolyhedron - Define a regular polyhedron

>	RegularPolyhedron(gon,[5/2,5],point(A,-6,1,-8),3);

$gon$

(225)

>	IsRegular(gon);

$true$

(226)

•	SmallStellatedDodecahedron - Define a regular polyhedron

>	SmallStellatedDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(227)

>	form(gon);

$SmallStellatedDodecahedron3d$

(228)

•	cube - Define a regular polyhedron

>	cube(gon,point(A,-6,1,-8),3);

$gon$

(229)

>	form(gon);

$hexahedron3d$

(230)

•	dodecahedron - Define a regular polyhedron

>	dodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(231)

>	form(gon);

$dodecahedron3d$

(232)

•	hexahedron - Define a regular polyhedron

>	hexahedron(gon,point(A,-6,1,-8),3);

$gon$

(233)

>	form(gon);

$hexahedron3d$

(234)

•	icosahedron - Define a regular polyhedron

>	icosahedron(gon,point(A,-6,1,-8),3);

$gon$

(235)

>	form(gon);

$icosahedron3d$

(236)

•	octahedron - Define a regular polyhedron

>	octahedron(gon,point(A,-6,1,-8),3);

$gon$

(237)

>	form(gon);

$octahedron3d$

(238)

•	tetrahedron - Define a regular tetrahedron

>	tetrahedron(gon,point(A,-6,1,-8),3);

$gon$

(239)

>	form(gon);

$tetrahedron3d$

(240)

Duality

•	HexakisIcosahedron Define the reciprocal polyhedra of the icosahedron

>	HexakisIcosahedron(gon,point(A,-6,1,-8),3);

$gon$

(241)

>	form(gon);

$HexakisIcosahedron3d$

(242)

•	HexakisOctahedron - Define the reciprocal polyhedra of the octahedron

>	HexakisOctahedron(gon,point(A,-6,1,-8),3);

$gon$

(243)

>	form(gon);

$HexakisOctahedron3d$

(244)

•	PentagonalHexacontahedron - Define the reciprocal polyhedra of the hexacontahedron

>	PentagonalHexacontahedron(gon,point(A,-6,1,-8),3);

$gon$

(245)

>	form(gon);

$PentagonalHexacontahedron3d$

(246)

•	PentagonalIcositetrahedron - Define the reciprocal polyhedra of the icositetrahedron

>	PentagonalIcositetrahedron(gon,point(A,-6,1,-8),3);

$gon$

(247)

>	form(gon);

$PentagonalIcositetrahedron3d$

(248)

•	PentakisDodecahedron - Define the reciprocal polyhedra of the dodecahedron

>	PentakisDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(249)

>	form(gon);

$PentakisDodecahedron3d$

(250)

•	RhombicDodecahedron - Define the reciprocal polyhedra of the dodecahedron

>	RhombicDodecahedron(gon,point(A,-6,1,-8),3);

$gon$

(251)

>	form(gon);

$RhombicDodecahedron3d$

(252)

•	RhombicTriacontahedron - Define the reciprocal polyhedra of the triacontahedron

>	RhombicTriacontahedron(gon,point(A,-6,1,-8),3);

$gon$

(253)

>	form(gon);

$RhombicTriacontahedron3d$

(254)

•	TetrakisHexahedron - Define the reciprocal polyhedra of the hexahedron

>	TetrakisHexahedron(gon,point(A,-6,1,-8),3);

$gon$

(255)

>	form(gon);

$TetrakisHexahedron3d$

(256)

•	TrapezoidalHexecontahedron - Define the reciprocal polyhedra of the hexacontahedron

>	TrapezoidalHexecontahedron(gon,point(A,-6,1,-8),3);

$gon$

(257)

>	form(gon);

$TrapezoidalHexecontahedron3d$

(258)

•	TrapezoidalIcositetrahedron - Define the reciprocal polyhedra of the icositetrahedron

>	TrapezoidalIcositetrahedron(gon,point(A,-6,1,-8),3);

$gon$

(259)

>	form(gon);

$TrapezoidalIcositetrahedron3d$

(260)

•	TriakisIcosahedron - Define the reciprocal polyhedra of the icosahedron

>	TriakisIcosahedron(gon,point(A,-6,1,-8),3);

$gon$

(261)

>	form(gon);

$TriakisIcosahedron3d$

(262)

•	TriakisOctahedron - Define the reciprocal polyhedra of the octahedron

>	TriakisOctahedron(gon,point(A,-6,1,-8),3);

$gon$

(263)

>	form(gon);

$TriakisOctahedron3d$

(264)

•	TriakisTetrahedron - Define the reciprocal polyhedra of the tetrahedron

>	TriakisTetrahedron(gon,point(A,-6,1,-8),3);

$gon$

(265)

>	form(gon);

$TriakisTetrahedron3d$

(266)

•	duality - Define the dual of a given polyhedron

>	RegularPolyhedron(gon,[5/2,5],point(A,-6,1,-8),3);

$gon$

(267)

>	duality(dgon,gon,sphere(sph,[A,MidRadius(gon)]));

$dgon$

(268)

Transformation

•	GlideReflect transformation that can be applied directly to a specific geometric object

•	GlideReflection

•	RotatoryReflect - A transformation without specifying the object to be applied

•	RotaryReflection

•	ScrewDisplace - A transformation without specifying the object to be applied

•	ScrewDisplacement

•	StretchRotate - A transformation without specifying the object to be applied

•	dilate - A transformation without specifying the object to be applied

Define t1 which is a homothety with ratio 3, center of homothety (0,0,0)

>	t1 := dilate(3,point(o,0,0,0));

$t1 ≔ dilate (3, o)$

(269)

•	homology - Find the space homology of a geometric object

>	plane(pl,[point(A,4,6,2),point(B,1,2,-3),point(C,-5,2,-9)]):

>	point(D,3,0,0):

>	line(ln1,[point(E,0,0,0),point(F,6,0,0)]):

>	homology(newform,pl,3,D,30,ln1);

$newform$

(270)

>	detail(newform);

Warning, assuming that the names of the axes are _x, _y and _z

$\begin{array}{l} name of the object & newform \\ form of the object & plane3d \\ equation of the plane & 8_x +_y (4 \cos (30) + 8 \sin (30)) +_z (- 8 \cos (30) + 4 \sin (30)) - \frac{8463631539}{1250000000} - \frac{81536368461 {\cos (30)}^{2}}{1250000000} - \frac{81536368461 {\sin (30)}^{2}}{1250000000} = 0 \end{array}$

(271)

•	homothety - Find the homothety of a geometric object

>	point(A,-5,2,0), point(B,2,-6,1):

>	line(ln1,[A,B]):

>	homothety(newform,ln1,3,point(C,7,8,4));

$newform$

(272)

>	detail(newform);

Warning, assuming that the names of the axes are _x, _y, and _z

$\begin{array}{l} name of the object & newform \\ form of the object & line3d \\ equation of the line & [_x = - 29 + \frac{5129391309 t}{400000000},_y = - 10 - \frac{732770187 t}{50000000},_z = - 8 + \frac{732770187 t}{400000000}] \end{array}$

(273)

•	inverse - Find the inverse of a given product of transformations

>	t1 := dilate(3,point(o,0,0,0));

$t1 ≔ dilate (3, o)$

(274)

>	point(A,1,0,0), point(B,0,0,1):

>	line(l1,[o,A]), line(l2,[o,B]):

>	plane(p,[l1,l2]):

>	dsegment(AB,[A,B]):

>	t2 := GlideReflect(p,AB);

$t2 ≔ GlideReflect (p, AB)$

(275)

>	t3 := ScrewDisplace(Pi/2,line(l3,[A,B]),AB);

$t3 ≔ ScrewDisplace (\frac{π}{2}, l3, AB)$

(276)

>	q1 := transprod(t2^t1,t3);

$q1 ≔ transprod (dilate (\frac{1}{3}, o), reflect (p), translate (AB), dilate (3, o), rotate (\frac{π}{2}, l3), translate (AB))$

(277)

>	q2 := inverse(q1);

$q2 ≔ transprod (translate (_AB), rotate (\frac{3 π}{2}, l3), dilate (\frac{1}{3}, o), translate (_AB), reflect (p), dilate (3, o))$

(278)

•	inversion - Find the inversion of a point, plane, or sphere with respect to a given sphere

>	plane(pl,[point(A,4,-6,2),[5,2,1]]):

>	sphere(sph,[point(B,6,2,1),5]):

>	geom3d[inversion](sph1,pl,sph);

$sph1$

(279)

>	detail(sph1);

Warning, assume that the name of the axes are _x, _y and _z

$\begin{array}{l} name of the object & sph1 \\ form of the object & sphere3d \\ name of the center & center_sph1_1 \\ coordinates of the center & [\frac{7}{2}, 1, \frac{1}{2}] \\ radius of the sphere & \frac{\sqrt{30}}{2} \\ surface area of the sphere & 30 π \\ volume of the sphere & 5 π \sqrt{30} \\ equation of the sphere & {_x}^{2} + {_y}^{2} + {_z}^{2} - 7_x - 2_y -_z + 6 = 0 \end{array}$

(280)

•	projection - Find the projection of an object on the other object

>	point(A,3,6,1), point(B,-7,6,-9):

>	plane(pl,[B,[5,2,1]]):

>	projection(projectpt,A,pl);

$projectpt$

(281)

>	detail(projectpt);

$\begin{array}{l} name of the object & projectpt \\ form of the object & point3d \\ coordinates of the point & [−7, 2, −1] \end{array}$

(282)

•	reflect - A transformation without specifying the object to be applied

>	tetrahedron(t,point(A,3,6,2),3):

>	ref := reflect(line(ln1,[point(B,3,2,1),point(C,7,-9,2)]));

$ref ≔ reflect (ln1)$

(283)

•	reflection - Transformations that can be applied directly to a specific geometric object

>	point(A,5,2,1), point(B,-7,9,2), point(C,-3,-7,-9):

>	sphere(sph,[C,3]):

>	line(ln1,[A,B]):

>	reflection(sph1,sph,ln1);

$sph1$

(284)

•	rotate - A transformation without specifying the object to be applied

>	point(A,-9,7,3):

>	point(B,6,3,7):

>	line(ln1,[A,B]):

>	rot := rotate(Pi,ln1);

$rot ≔ rotate (π, ln1)$

(285)

•	rotation - Transformations that can be applied directly to a specific geometric object

>	plane(pl,[point(A,-9,7,3),[3,-9,2]]):

>	point(B,6,3,7):

>	line(ln1,[A,B]):

>	rotation(pl1,pl,Pi,ln1);

$pl1$

(286)

>	detail(pl1);

Warning, assuming that the names of the axes are _x, _y and _z

$\begin{array}{l} name of the object & pl1 \\ form of the object & plane3d \\ equation of the plane & \frac{5290}{257} + \frac{1899_x}{257} + \frac{1601_y}{257} + \frac{198_z}{257} = 0 \end{array}$

(287)

•	transform - Apply the ``result'' transformation to a specific geometric object

>	tetrahedron(t,point(A,3,6,2),3):

>	ref := reflect(line(ln1,[point(B,3,2,1),point(C,7,-9,2)]));

$ref ≔ reflect (ln1)$

(288)

>	transform(t4,t,ref);

$t4$

(289)

form(t4);

$tetrahedron3d$

(290)

•	translate - A transformation without specifying the object to be applied

>	point(A,5,2,-8), point(B,3,5,-1):

>	dsegment(dseg,A,B):

>	trans := translate(dseg);

$trans ≔ translate (dseg)$

(291)

•	translation - Transformations that can be applied directly to a specific geometric object

>	point(A,5,2,-8), point(B,3,5,-1):

>	dsegment(dseg,A,B):

>	triangle(tri,[point(C,5,2,-9),point(D,3,2,1),point(E,-9,3,5)]);

$tri$

(292)

>	translation(tri1,tri,dseg);

$tri1$

(293)

>	detail(tri1);

$\begin{array}{l} name of the object & tri1 \\ form of the object & triangle3d \\ the 3 vertices & [[3, 5, −2], [1, 5, 8], [−11, 6, 12]] \end{array}$

(294)

•	transprod - Converts a given transformation or product of transformations into a product of three ``primitive'' transformations (translate, rotate and dilate)

>	point(A,5,2,-8), point(B,3,5,-1):

>	dsegment(dseg,A,B):

>	trans := translate(dseg):

>	line(ln1,[A,B]):

>	rot := rotate(Pi,ln1);

$rot ≔ rotate (π, ln1)$

(295)

>	prod := transprod(trans,rot);

$prod ≔ transprod (translate (dseg), rotate (π, ln1))$

(296)

>	tetrahedron(t,point(C,6,2,6),2);

$t$

(297)

>	transform(t5,t,prod);

$t5$

(298)

form(t5);

$tetrahedron3d$

(299)

Groebner - The Groebner package and Groebner bases.

•	fglm - Generalized FGLM algorithm

•	gbasis - Reduced Groebner basis

•	gsolve - Preprocess an algebraic system for solving

•	hilbertdim - The Hilbert dimension of an ideal

•	hilbertpoly - The Hilbert polynomial of an ideal

•	hilbertseries - The Hilbert series of an ideal

•	inter_reduce - Fully inter-reduce a list of polynomials

•	is_finite - Decide if a given algebraic system has (at most) finitely many solutions

•	is_solvable - Decide if a given algebraic system is algebraically consistent

•	leadcoeff - The leading coefficient of a polynomial

•	leadmon - The leading monomial of a polynomial

•	leadterm - The leading term of a polynomial

•	normalf - Normal form of a polynomial modulo an ideal

•	pretend_gbasis - Add a Groebner basis to the list of known ones

•	reduce - Full reduction of a polynomial

•	spoly - The S-polynomial of two skew polynomials

•	termorder - Create a term order

•	testorder - Test whether two terms are in increasing order with respect to a given term order

•	univpoly - The univariate polynomial of lowest degree in an ideal

Ore_algebra - The Ore_algebra package and Ore algebras.

For examples, please see the Ore Algebra example worksheet.

•	Ore_to_DESol - Convert a differential operator to a DESol structure

•	Ore_to_RESol - Convert a shift operator to a DESol structure

•	Ore_to_diff - Convert a differential operator to a differential equation

•	Ore_to_shift - Convert a shift operator to a recurrence equation

•	annihilators - Skew lcm of a pair of operators

•	applyopr - Apply an operator to a function

•	diff_algebra - Create an algebra of linear differential operators

•	poly_algebra - Create an algebra of commutative polynomials

•	qshift_algebra - Create an algebra of linear q-difference operators

•	rand_skew_poly - Random skew polynomial generator

•	shift_algebra - Create an algebra of linear difference operators

•	skew_algebra - Declare an Ore algebra

•	skew_elim - Skew elimination of an indeterminate

•	skew_gcdex - Extend skew gcd computation

•	skew_pdiv - Skew pseudo-division

•	skew_power - Power of an Ore algebra

•	skew_prem - Skew pseudo-remainder

•	skew_product - Inner product of an Ore algebra

PDEtools - Solving Partial Differential Equations

•	charstrip - Find the characteristic strip corresponding to a given, first order PDE

restart;

>	PDE := xdiff(psi(x,y,z),z)-psi(x,y,z)+y^2diff(psi(x,y,z),y) = 0;

$PDE ≔ x (\frac{\partial}{\partial z} ψ (x, y, z)) - ψ (x, y, z) + y^{2} (\frac{\partial}{\partial y} ψ (x, y, z)) = 0$

(300)

>	PDEtools[charstrip](PDE,psi(x,y,z));

$\{\frac{&DifferentialD;}{&DifferentialD;_s} ψ (_s) = ψ (_s), \frac{&DifferentialD;}{&DifferentialD;_s} x (_s) = 0, \frac{&DifferentialD;}{&DifferentialD;_s} y (_s) = {y (_s)}^{2}, \frac{&DifferentialD;}{&DifferentialD;_s} z (_s) = x (_s)\}$

(301)

•	dchange - Change variables in mathematical expressions

>	PDE := 1+psi(x,y)^2+ydiff(psi(x,y),x)-xdiff(psi(x,y),y);

$PDE ≔ 1 + {ψ (x, y)}^{2} + y (\frac{\partial}{\partial x} ψ (x, y)) - x (\frac{\partial}{\partial y} ψ (x, y))$

(302)

>	TR := {x=rcos(phi), y=rsin(phi)};

$TR ≔ \{x = r \cos (φ), y = r \sin (φ)\}$

(303)

>	PDEtools[dchange](TR, PDE, simplify);

${ψ (φ, r)}^{2} + 1 - \frac{\partial}{\partial φ} ψ (φ, r)$

(304)

>	II := int(psi(r),r):

>	eq := useintat(II) = II;

$eq ≔ \int_{}^{r} ψ (_a) &DifferentialD;_a = \int ψ (r) &DifferentialD; r$

(305)

>	TR2 := {r=sqrt(y(x)^2+x^2),phi(r)=arctan(y(x)/x)};

$TR2 ≔ \{r = \sqrt{{y (x)}^{2} + x^{2}}, φ (r) = \arctan (\frac{y (x)}{x})\}$

(306)

>	PDEtools[dchange](TR2,eq,known=psi);

$\int_{}^{\sqrt{{y (x)}^{2} + x^{2}}} ψ (_a) &DifferentialD;_a = \int \frac{ψ (\sqrt{{y (x)}^{2} + x^{2}}) (2 y (x) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 2 x)}{2 \sqrt{{y (x)}^{2} + x^{2}}} &DifferentialD; x$

(307)

•	mapde - Map a PDE to another PDE for easier solving

>	PDE1 := x^2diff(psi(x,y),x,x)+2yxdiff(psi(x,y),x,y)+y^2*diff(psi(x,y),y,y);

$PDE1 ≔ x^{2} (\frac{\partial^{2}}{\partial x^{2}} ψ (x, y)) + 2 y x (\frac{\partial^{2}}{\partial x \partial y} ψ (x, y)) + y^{2} (\frac{\partial^{2}}{\partial y^{2}} ψ (x, y))$

(308)

>	PDEtools[mapde](PDE1, psi, canop);

${_ξ1}^{2} (\frac{\partial^{2}}{\partial {_ξ1}^{2}} ψ (_ξ1,_ξ2)) where \{_ξ1 = x,_ξ2 = \frac{y}{x}\}$

(309)

•	difforder - Evaluate the differential order of an expression

>	eq := diff(psi(x,y),x$2,y$3);

$eq ≔ \frac{\partial^{5}}{\partial x^{2} \partial y^{3}} ψ (x, y)$

(310)

>	PDEtools[difforder](eq), PDEtools[difforder](eq,x), PDEtools[difforder](eq,y);

$5, 2, 3$

(311)

•	pdsolve - Find analytical solutions for PDEs

>	PDE := Diff(r^2diff(Phi(r,theta,phi),r),r) + 1/sin(theta)Diff(sin(theta)diff(Phi(r,theta,phi),theta),theta) + 1/sin(theta)^2diff(diff(Phi(r,theta,phi),phi),phi) = 0;

$PDE ≔ \frac{\partial}{\partial r} (r^{2} (\frac{\partial}{\partial r} Φ (r, θ, φ))) + \frac{\frac{\partial}{\partial θ} (\sin (θ) (\frac{\partial}{\partial θ} Φ (r, θ, φ)))}{\sin (θ)} + \frac{\frac{\partial^{2}}{\partial φ^{2}} Φ (r, θ, φ)}{{\sin (θ)}^{2}} = 0$

(312)

>	pdsolve(PDE);

$Φ (r, θ, φ) = f__1 (r) f__2 (θ) f__3 (φ) where [\{\frac{{&DifferentialD;}^{2}}{&DifferentialD; r^{2}} f__1 (r) = \frac{f__1 (r) {_c}_{1}}{r^{2}} - \frac{2 (\frac{&DifferentialD;}{&DifferentialD; r} f__1 (r))}{r}, \frac{{&DifferentialD;}^{2}}{&DifferentialD; θ^{2}} f__2 (θ) = - f__2 (θ) {_c}_{1} + \frac{f__2 (θ) {_c}_{2}}{{\sin (θ)}^{2}} - \frac{\cos (θ) (\frac{&DifferentialD;}{&DifferentialD; θ} f__2 (θ))}{\sin (θ)}, \frac{{&DifferentialD;}^{2}}{&DifferentialD; φ^{2}} f__3 (φ) = - f__3 (φ) {_c}_{2}\}]$

(313)

•	splitstrip - See charstrip

>	PDE := xdiff(psi(x,y,z),z)-psi(x,y,z)+y^2diff(psi(x,y,z),y)=0;

$PDE ≔ x (\frac{\partial}{\partial z} ψ (x, y, z)) - ψ (x, y, z) + y^{2} (\frac{\partial}{\partial y} ψ (x, y, z)) = 0$

(314)

>	PDEtools[splitstrip](PDE,psi(x,y,z));

$\{\{\frac{&DifferentialD;}{&DifferentialD;_s} ψ (_s) = ψ (_s)\}, \{\frac{&DifferentialD;}{&DifferentialD;_s} y (_s) = {y (_s)}^{2}\}, \{\frac{&DifferentialD;}{&DifferentialD;_s} x (_s) = 0, \frac{&DifferentialD;}{&DifferentialD;_s} z (_s) = x (_s)\}\}$

(315)

•	splitsys - Split sets of algebraic equations

>	sys := {x = r(x)cos(phi(x)), y(x) = r(x)sin(phi(x)), diff(psi(x),x) = cos(eta(x)), diff(eta(x),x) = psi(x)};

$sys ≔ \{x = r (x) \cos (φ (x)), \frac{&DifferentialD;}{&DifferentialD; x} η (x) = ψ (x), \frac{&DifferentialD;}{&DifferentialD; x} ψ (x) = \cos (η (x)), y (x) = r (x) \sin (φ (x))\}$

(316)

>	PDEtools[splitsys](sys,{psi,eta,r,phi}(x));

$\{x = r (x) \cos (φ (x)), y (x) = r (x) \sin (φ (x))\}, \{\frac{&DifferentialD;}{&DifferentialD; x} η (x) = ψ (x), \frac{&DifferentialD;}{&DifferentialD; x} ψ (x) = \cos (η (x))\}$

(317)

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