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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.