Exploring Analyic Geometry with Mathematica®

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Reciprocals of Points and Lines

recptln.html

Exploration

Show that the polar reciprocal of "recptln_1.gif" in the auxiliary conic "recptln_2.gif" is the point "recptln_3.gif", assuming that the line does not pass through the origin.  Also, show that the line x+y-1=0 is the polar reciprocal of the point (x,y) with respect to C.

Approach

Create the auxiliary conic, C. The pole point is the reciprocal of the line. The polar line is the reciprocal of the point.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Define the auxiliary conic (circle), C.

c1=Circle2D[{0,0},1];

The pole point is the reciprocal.

Clear[A1,B1,C1];
Point2D[Line2D[A1,B1,C1],c1]

"recptln_4.gif"

The polar line is the reciprocal.

Clear[x,y];
Line2D[Point2D[x,y],c1] //Simplify

"recptln_5.gif"


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