geometry
reciprocation
find the reciprocation of a point or a line with respect to a circle
Calling Sequence
Parameters
Description
Examples
reciprocation(Q, P, c)
Q
-
the name of the object to be created
P
point or line
c
circle
Let c=O⁡r be a fixed circle and let P be any ordinary point other than the center O. Let P' be the inverse of P in circle O⁡r. Then the line Q through P' and perpendicular to OPP' is called the polar of P for the circle c. Note that when P is a line, then Q will be a point.
Note that this routine in particular, and the geometry package in general, does not encompass the extended plane, i.e., the polar of center O does not exist (though in the extended plane, it is the line at infinity) and the polar of an ideal point P does not exist either (it is the line through the center O perpendicular to the direction OP in the extended plane).
If line Q is the polar point P, then point P is called the pole of line Q.
The pole-polar transformation set up by circle c=O⁡r is called reciprocation in circle c
For a detailed description of Q (the object created), use the routine detail (i.e., detail(Q))
The command with(geometry,reciprocation) allows the use of the abbreviated form of this command.
with⁡geometry:
circle⁡c,point⁡OO,0,0,2,x,y:
point⁡P,1,0:
inversion⁡PP,P,c:
reciprocation⁡l,P,c
l
detail⁡l
name of the objectlform of the objectline2dequation of the line−4+x=0
draw⁡c,l,dsegment⁡dsg,P,PP,printtext=true,view=−3..5,−3..3,axes=NONE
reciprocation⁡A,l,c:
coordinates⁡A=coordinates⁡P
1,0=1,0
See Also
geometry/objects
geometry/transformation
geometry[draw]
geometry[inversion]
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