4.4: Pedal Curves

In the study of plane curves, we defined the pedal curve as the locus of the foot of the perpendicular from the origin to the tangent line of the original curve. For space curves, we may define two analogous concepts, one for the feet of perpendiculars to the tangent lines and the other for feet of the perpendiculars to the osculating planes at various points.

A point of the first pedal curve Z_1,C(t) of a space curve X(t) is defined to be the foot of the perpendicular from C to the osculating plane to the curve at X(t) . We may express this by the formula

Z_1,C( t)=C+((X(t) -C)· B(t))B(t)

Z_1,C( t)=C-h_{1 ,C}(t)B(t)

A point of the second pedal curve Z_1,C(t) of a space curve X(t) is defined to be the foot of the perpendicular from C to the tangent line to the curve at X(t) . We may express this by the formula

Z_2,C( t)=C+((X(t) -C)· B(t))B(t) +((X( t)-C) ·P(t))P( t)

The control panel requires that you type in the pedal point Q . This is used here to let you determine Q more precisely. Finally, there are two checkboxes Draw Pedal 1 and Draw Pedal 2 which toggle their corresponding pedal curves. Often, the pedal curves will seem very small; that is why a type-in has been added that controls the domain of the pedal curves.

This demo allows you to put in a curve and visualize the first or second pedal curve as defined above.

What happens to the curve Z_1,C(t) as C changes? What is the nature of the first pedal curve if the original curve has a point where the curvature is zero? What if the original curve has a point where the torsion is zero?

What happens to the curve Z_1,C(t) as C changes? What is the nature of the second pedal curve if the original curve has a point where the curvature is zero? What if the original curve has a point where the torsion is zero?