Overview of the geom3d Package - Maple Help

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Overview of the geom3d Package

 

Calling Sequence

Description

Examples

Calling Sequence

geom3d[command](arguments)

command(arguments)

Description

• 

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.

dsegment

line

plane

point

segment

sphere

triangle

 

• 

For basic geometric objects, the following commands are supported.

• 

Point:

AreCollinear

AreConjugate

AreCoplanar

centroid

coordinates

distance

HarmonicConjugate

IsOnObject

midpoint

OnSegment

randpoint

xcoord

ycoord

zcoord

 

 

• 

Segment/Directed Segment:

AreParallel

ArePerpendicular

DefinedAs

midpoint

• 

Line:

AreConcurrent

AreCoplanar

AreParallel

ArePerpendicular

AreSkewLines

distance

Equation

FindAngle

intersection

parallel

tname

 

• 

Plane:

AreParallel

ArePerpendicular

distance

Equation

FindAngle

intersection

IsTangent

parallel

• 

Triangle:

altitude

area

centroid

DefinedAs

FindAngle

IsEquilateral

IsRightTriangle

sides

• 

Sphere:

area

AreConjugate

ArePerpendicular

center

Equation

FindAngle

intersection

IsTangent

polar

pole

powerps

RadicalCenter

RadicalLine

RadicalPlane

radius

TangentPlane

volume

 

 

 

• 

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:

area

center

cube

dodecahedron

faces

GreatDodecahedron

GreatIcosahedron

GreatStellatedDodecahedron

hexahedron

icosahedron

incident

InRadius

IsRegular

MidRadius

octahedron

radius

RegularPolyhedron

schlafli

sides

SmallStellatedDodecahedron

tetrahedron

vertices

volume

 

• 

Quasi-regular Polyhedra:

cuboctahedron

icosidodecahedron

IsQuasi

schlafli

sides

vertices

 

 

• 

General Polyhedra:

gtetrahedron

parallelepiped

 

 

• 

Archimedean Solids:

Archimedean

center

cuboctahedron

faces

GreatRhombicuboctahedron

GreatRhombiicosidodecahedron

icosidodecahedron

incident

IsArchimedean

radius

schlafli

sides

SmallRhombicuboctahedron

SmallRhombiicosidodecahedron

SnubCube

SnubDodecahedron

TruncatedCuboctahedron

TruncatedDodecahedron

TruncatedHexahedron

TruncatedIcosahedron

TruncatedIcosidodecahedron

TruncatedOctahedron

TruncatedTetrahedron

vertices

• 

Duality:

center

duality

faces

HexakisIcosahedron

HexakisOctahedron

PentagonalHexacontahedron

PentagonalIcositetrahedron

PentakisDodecahedron

radius

RhombicDodecahedron

RhombicTriacontahedron

schlafli

TetrakisHexahedron

TrapezoidalHexecontahedron

TrapezoidalIcositetrahedron

TriakisIcosahedron

TriakisOctahedron

TriakisTetrahedron

vertices

 

• 

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

center

faces

facet

IsFacetted

IsStellated

schlafli

stellate

vertices

• 

Various types of geom3d[transformation] are supported.

GlideReflection

homology

homothety

inverse

inversion

projection

reflection

rotation

RotatoryReflection

ScrewDisplacement

StereographicProjection

transform

translation

transprod

 

 

• 

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

• 

Other commands are also implemented.

AreDistinct

detail

form

xname

yname

zname

 

 

• 

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 assumecond), 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.

Examples

withgeom3d:

spheres,x2+y2+z2=1,x,y,z,'centername'=o

s

(1)

details

name of the objectsform of the objectsphere3dname of the centerocoordinates of the center0,0,0radius of the sphere1surface area of the sphere4πvolume of the sphere4π3equation of the spherex2+y2+z21=0

(2)

Define the same sphere but without the names of the axes in the input; you will be prompted for them.

spherec,m2+n2+p2=1

c

(3)

detailc

name of the objectcform of the objectsphere3dname of the centercenter_c_1coordinates of the center0,0,0radius of the sphere1surface area of the sphere4πvolume of the sphere4π3equation of the spherem2+n2+p21=0

(4)

Define the same sphere where the names of the axes are assigned using the three environment variables.

_EnvXName'm':_EnvYName'n':_EnvZName'p':

spherec,pointoo,0,0,0,1

c

(5)

Equationc

m2+n2+p21=0

(6)

In the above examples, c is assigned to a geometric object (sphere), c can also be assigned to a list of objects.

linel2,pointA,0,0,0,pointB,1,0,0,spherec,x2+y2+z2=1,x,y,z:

intersectionH,l2,c

intersection:   "two points of intersection"

H

(7)

H

l2_intersect1_c,l2_intersect2_c

(8)

detailH

name of the objectl2_intersect1_cform of the objectpoint3dcoordinates of the point1,0,0,name of the objectl2_intersect2_cform of the objectpoint3dcoordinates of the point−1,0,0

(9)

The following is an example with unknown parameters, which returns the message FAIL.

spheres,pointo,0,0,0,1:

IsOnObjectpointB,1,1,a,s,'cond'

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

FAIL

(10)

assumecond

IsOnObjectpointB,1,1,a,s

true

(11)

See Also

assume

geom3d[draw]

geom3d[objects]

geom3d[transformation]

UsingPackages

 


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