geom3d
polar
find the polar of a given point with respect to a given sphere
Calling Sequence
Parameters
Description
Examples
polar(p, A, s)
p
-
the name of the polar
A
point
s
sphere
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.
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+z2−3x+y+z−5=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
spheres,x2+y2+z2−3x+y+z−5=0,x,y,z:
polarp1,A,s
p1
polarp2,B,s
p2
polarp3,C,s
p3
detailp3
name of the objectp3form of the objectplane3dequation of the plane−5−3x2+y=0
intersectionl1,p1,p2
l1
intersectionl2,p1,p3
l2
intersectionl3,p2,p3
l3
AreDistinctl1,l2,l3
false
The answers show that three planes p1, p2, and p3 form a pencil of planes.
See Also
geom3d[pole]
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