Student[MultivariateCalculus] - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Education : Student Package : Multivariate Calculus : Lines and Planes : Student/MultivariateCalculus/Plane

Student[MultivariateCalculus]

  

Plane

  

Create and initialize a plane object

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

Plane(eqn, opts)

Plane(expr, opts)

Plane(p1, p2, p3, opts)

Plane(p1, v1, opts)

Plane(v1, p1, opts)

Plane(p1, p2, v1, opts)

Plane(p1, v1, p2, opts)

Plane(v1, p1, p2, opts)

Plane(p1, v1, v2, opts)

Plane(v1, p1, v2, opts)

Plane(v1, v2, p1, opts)

Plane(p1, l1, opts)

Plane(l1, p1, opts)

Plane(l1, l2, opts)

Plane(P, p1, opts)

Plane(P, opts)

Parameters

eqn

-

Linear equation defining the plane

expr

-

Expression linear in the coordinate variables, equated to 0 to define the plane

p1, p2, p3

-

Points on the plane, expressed as lists of coordinates

v1, v2

-

Vector used to define the plane; either a normal Vector, or directions lying within the plane - see below for details

l1, l2

-

Line objects used to define the plane; either in the plane or parallel to the plane - see below for details

P

-

Plane object defined earlier; the plane to be defined will be parallel (or equal) to this plane

opts

-

(optional) equation defining the variables to be used in the representation

Options

• 

The opts arguments can contain the following option.

• 

variables = list

  

The variables to be used in the plane's equation.  This is used in interpreting the equation and expression forms of the call to the Plane function, and when obtaining equations for the plane. The default is x,y,z.

• 

id = positive integer, name, or string

  

Planes display as << Plane n >>, where n is an identification for the plane that is by default a positive integer assigned in order of creation. The id option can be used to force the plane to be given a different identification. It is an error to use the same identification for two different planes that are both in use.

Description

• 

The Plane command creates a plane object that can be operated on or graphed.

• 

Planes can be specified in the following ways:

– 

A linear equation eqn, such as ax&plus;by&plus;cz&plus;d&equals;0.

– 

A linear expression expr of the form ax&plus;by&plus;cz&plus;d, which is set equal to 0 to obtain the plane.

– 

Three points p1, p2, p3, such as 1&comma;2&comma;3, 1&comma;0&comma;1, 0&comma;1&comma;2, lying in the plane. The points must not lie on a line.

– 

A point p1, such as 1&comma;2&comma;3, and a Vector v1, such as 2&comma;0&comma;1. The plane contains p1 and has v1 as its normal. The arguments can also be specified in the other order.

– 

Two points, p1 and p2, such as 1&comma;2&comma;3 and 1&comma;0&comma;1, lying in the plane, and a Vector v1, such as 1&comma;0&comma;2, the direction of which lies within the plane. The arguments can be specified in any order. The Vector v1 cannot be a multiple of the Vector from p1 to p2.

– 

A point p1, such as 1&comma;2&comma;3, and two Vectors v1 and v2, such as 0&comma;1&comma;2 and 1&comma;0&comma;2. The directions of both Vectors lie within the plane. The arguments can be specified in any order. The Vectors v1 and v2 must be linearly independent.

– 

A point p1, such as 1&comma;2&comma;3, lying in the plane, and a line l1 defined earlier, say Line0&comma;0&comma;0&comma;0&comma;1&comma;2, also lying in the plane. The arguments can be specified in any order. The line l1 cannot contain the point p1.

– 

Two Line objects l1 and l2 defined earlier, say Line1&comma;2&comma;3&comma;1&comma;0&comma;1 and Line0&comma;0&comma;0&comma;0&comma;1&comma;2. If the lines intersect or are parallel, then this defines the plane containing both lines. If the lines are skew, then it defines the plane containing l1 and parallel to l2. If the lines are equal, an error is returned.

– 

A Plane object P, defined earlier, such as Planex&plus;y&plus;z&equals;0, and a point p1, such as 1&comma;2&comma;3. This defines the plane through p1 that is parallel to P.

– 

A plane object P, defined earlier, such as Planex&plus;y&plus;z&equals;0. (This can be useful to change the names of the coordinate variables and the parameter, using the options explained below.)

• 

It is possible to specify two mathematically identical planes using different Plane commands.

• 

The coefficients and coordinates defining a Plane can contain parameters. For example, we can define a plane containing the points 2&comma;3&comma;a and 1&comma;a&plus;b&comma;3, and the direction 0&comma;0&comma;1, with the command Plane2&comma;3&comma;a&comma;1&comma;a&plus;b&comma;3&comma;0&comma;0&comma;1.

  

These parameters must be different from the coordinate variables (set by the variables option explained below; by default x&comma;y&comma;z).

• 

The following is a list of commands available to Plane

AreOrthogonal

AreParallel

Contains

Distance

Equal

GetDimension

GetIntersection

GetNormal

GetPlot

GetPoint

GetRepresentation

Intersects

Projection

 

 

 

• 

In addition, there is some special behavior for the eval command when applied to a Plane. If the plane contains parameters, as in the example Plane2&comma;3&comma;a&comma;1&comma;a&plus;b&comma;3&comma;0&comma;0&comma;1 above, then one can use eval to substitute values for those parameters. For example, we might evaluate the plane given above at a&equals;2 to obtain the line through the points 2&comma;3&comma;2 and 1&comma;b&plus;2&comma;3 and containing the direction 0&comma;0&comma;1.

  

Additionally, one can rename the coordinate variables using eval: if we evaluate the plane given above at x&equals;u&comma;z&equals;v, then we obtain an identical plane but with the coordinate variables u, y, and v.

Examples

withStudent&lsqb;MultivariateCalculus&rsqb;&colon;

p1Plane3xy&plus;z&plus;3&colon;GetRepresentationp1

3xy&plus;z&equals;3

(1)

p2Planeu&plus;v3w&equals;1&comma;&apos;variables&apos;&equals;u&comma;v&comma;w&colon;GetRepresentationp2

u&plus;v3w&equals;1

(2)

p3Plane1&comma;2&comma;3&comma;1&comma;0&comma;1&comma;0&comma;1&comma;2&colon;GetRepresentationp3

x&plus;2yz&equals;0

(3)

p4Plane1&comma;2&comma;3&comma;2&comma;0&comma;1&comma;&apos;id&equals;A&apos;&colon;GetRepresentationp4

2xz&equals;1

(4)

Note how planes p1 through p3 are displayed using their automatically assigned identification, but p4 has the explicitly specified identification A:

p1&comma;p2&comma;p3&comma;p4

<< Plane 1 >>&comma;<< Plane 2 >>&comma;<< Plane 3 >>&comma;<< Plane A >>

(5)

p5Plane1&comma;2&comma;3&comma;1&comma;0&comma;1&comma;1&comma;0&comma;2&colon;GetRepresentationp5

2x&plus;2yz&equals;3

(6)

p6Plane1&comma;2&comma;3&comma;0&comma;a&comma;2&comma;a&comma;0&comma;b&colon;GetRepresentationp6

az&plus;bx&plus;2y&equals;3a&plus;b&plus;4

(7)

l1Line1&comma;2&comma;3&comma;0&comma;1&comma;3&colon;GetRepresentationl1

t&period;110&plus;123

(8)

p7Plane2&comma;0&comma;1&comma;l1&colon;GetRepresentationp7

2x2y&plus;3z&equals;7

(9)

l2Line0&comma;0&comma;0&comma;0&comma;1&comma;2&colon;GetRepresentationl2

t&period;012&plus;000

(10)

l3Line0&comma;0&comma;1&comma;0&comma;1&comma;3&colon;GetRepresentationl3

t&period;012&plus;001

(11)

p8Planel1&comma;l2&colon;GetRepresentationp8

2x&plus;2yz&equals;1

(12)

p9Planel2&comma;l1&colon;GetRepresentationp9

2x2y&plus;z&equals;0

(13)

p10Planel2&comma;l3&colon;GetRepresentationp10

x&equals;0

(14)

p11Plane1&comma;2&comma;3&comma;p10&colon;GetRepresentationp11

x&equals;1

(15)

p12Planep11&comma;&apos;variables&apos;&equals;u&comma;v&comma;w&colon;GetRepresentationp12

u&equals;1

(16)

eval can be used to make substitutions for the parameters in the plane object.

p13p6a&equals;1|p6a&equals;1&colon;GetRepresentationp13

bx&plus;2yz&equals;b&plus;1

(17)

p14evalp6&comma;a&equals;1&comma;b&equals;3&colon;GetRepresentationp14

3x&plus;2yz&equals;4

(18)

Query the properties of individual planes.

GetPointp3

1&comma;2&comma;3

(19)

GetNormalp4

201

(20)

AreParallel, Intersects, Distance, and GetIntersection find the relationships between two planes.

AreParallelp8&comma;p9

true

(21)

Intersectsp8&comma;p9

false

(22)

Intersectsp8&comma;p10

true

(23)

The intersection between p8 and p10 is the line l3.

Equall3&comma;GetIntersectionp8&comma;p10

true

(24)

Distancep8&comma;p10

0

(25)

Distancep8&comma;p9

13

(26)

Compatibility

• 

The Student[MultivariateCalculus][Plane] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

Student

Student[MultivariateCalculus]

Student[MultivariateCalculus][Line]

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam