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Pedal Curves

Here is an example to show how to use Maple to calculate and draw
the pedal curve    ( q1(t), q2(t) )  associated to a given curve  ( p1(t), p2(t) ).


First we supply the data for   the   original curve   ( p1(t), p2(t) )

> p1 := cos(t)+1; p2 := sin(t); a := 0; b := 2*Pi; 


Now we calculate the  velocity  vector  ( p1'(t), p2'(t) )

> dp1 := diff(p1,t); dp2 := diff(p2,t);

Now we calculate the  scalar coefficient   coef   such that   the pedal curve
has the formula

       (q1(t), q2(t)) = ( p1(t), p2(t) ) - coef*( p1'(t), p2'(t) )


> coef := (p1*dp1 + p2*dp2)/(dp1^2+dp2^2);
Now we define the  components of the pedal curve position vector: 

> q1 := p1-coef*dp1; q2 := p2-coef*dp2;
Now we plot both the original curve and the pedal curve:   Note that
the units must be the same on both axes --- hence the ``scaling=constrained''.

> plot( {[p1,p2,t=a..b],[q1,q2,t=a..b]},scaling=constrained,title=`A circle and its pedal curve`);

By changing the data on the first input line, one can easily draw lots of pedal 
curves.   Try it.