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geom3d

  

polar

  

find the polar of a given point with respect to a given sphere

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

polar(p, A, s)

Parameters

p

-

the name of the polar

A

-

point

s

-

sphere

Description

• 

The polar of a point A with respect to a sphere is defined to be the locus of all points conjugate to A.

• 

For a detailed description of the polar of p, use the routine detail.

• 

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

Examples

withgeom3d:

Write down the equations of the polars of the points 1,2,1, 3,5,2 and 0,12,12 with respect to the sphere x2+y2+z23x+y+z5=0 and show that they form a pencil of planes.

pointA,1,2,1,pointB,3,5,2,pointC,0,12,12

A,B,C

(1)

spheres,x2+y2+z23x+y+z5=0,x,y,z:

polarp1,A,s

p1

(2)

polarp2,B,s

p2

(3)

polarp3,C,s

p3

(4)

detailp3

name of the objectp3form of the objectplane3dequation of the plane53x2+y=0

(5)

intersectionl1,p1,p2

l1

(6)

intersectionl2,p1,p3

l2

(7)

intersectionl3,p2,p3

l3

(8)

AreDistinctl1,l2,l3

false

(9)

The answers show that three planes p1, p2, and p3 form a pencil of planes.

See Also

geom3d[pole]

 


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