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geom3d

  

line

  

define a line

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

line(l, [A, B] )

line(l, [A, v] )

line(l, [A, dseg] )

line(l, [A, p1] )

line(l, [p1, p2] )

line(l, [b1t+a1, b2t+a2, b3t+a3 ], t)

Parameters

l

-

the name of the line

A, B

-

points

v

-

vector

dseg

-

directed line segment

p1, p2

-

planes

a1, a2, a3, b1, b2, b3

-

algebraic expressions

t

-

(optional) a name denoting the parameter in the parametric equations of the line.

Description

• 

A line l can be defined as follows:

  

+ from two given points A and B

  

+ from a given point A and a vector v of dimension 3 or a directed segment seg. The line defined is the line that passes through A and has v as its direction-ratios.

  

+ from a given point A and a plane p1. The line defined is the line that passes through A and perpendicular to the plane p1.

  

+ from two given planes p1 and p2. The line defined is the line of intersection of two planes p1 and p2 (if exists).

  

+ from the parametric equations of the line b1t+a1,b2t+a2,b3t+a3. If the third optional argument t is not given, and if a name is assigned to the environment variable _EnvTName, then this name will be used as the name of the parameter in the parametric equations of the line. Otherwise, Maple will prompt the user to input the name of the parameter.

• 

To access the information relating to a line l, use the following function calls:

form(l)

returns the form of the geometric object

 

(i.e., line3d if l is a line).

FixedPoint(l)

returns a fixed point on l.

ParallelVector(l)

returns a direction-ratios of l.

DirectionRatios(l)

returns the direction-ratios of l.

Equation(l)

returns the parametric equations

 

that represents the line l.

xname(l), yname(l), or zname(l)

returns the name of

 

the x-axis; y-axis, z-axis or FAIL if

 

the axis is not assigned to any name.

detail(l)

returns a detailed description of

 

the line l.

• 

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

Examples

withgeom3d:

Find the equation of the line through [1,2,-1] perpendicular to the plane 3x5y+4z=5,the length of the perpendicular, and the coordinates of its foot

pointA,1,2,1,planep,3x5y+4z=5,x,y,z:

the parametric equations of the line l with parameter t is

linel,A,p

l

(1)

Equationl,'t'

1+3t,25t,1+4t

(2)

the projection of A on the plane is

projectionB,A,p

B

(3)

coordinatesB

4925,25,725

(4)

the length of the perpendicular is

distanceA,B

852

(5)

Let the straight line pass through the point A=x1,y1,z1 and has direction-cosines (or ratios) l,m,n

assumel0:

pointA,x1,y1,z1:vl,m,n:

define the line l1 that passes through A and has l,m,n as its direction-ratios

linel1,A,v:

detaill1

Warning, assume that the parameter in the parametric equations is _t
Warning, assuming that the names of the axes are _x, _y, and _z

name of the objectl1form of the objectline3dequation of the line_x=_tl~+x1,_y=_tm+y1,_z=_tn+z1

(6)

l'l':

define the line l which is the intersection of two given planes p1, p2

planep1,4x+4y5z=12,x,y,z:

planep2,8x+12y13z=32,x,y,z:

linel,p1,p2

l

(7)

Equationl,'t'

1+8t,2+12t,16t

(8)

See Also

geom3d[altitude]

geom3d[intersection]

geom3d[objects]

geom3d[transformation]

 


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