Pedal curve

of a curve $ l $ with respect to a point $ O $

The set of bases to the perpendiculars dropped from the point $ O $ to the tangents to the curve $ l $. For example, the Pascal limaçon is the pedal of a circle with respect to the point $ O $( see Fig.). The pedal (curve) of a plane curve $ x = x( t) , y = y( t) $ relative to the coordinate origin is

$$ X = x - x ^ \prime \frac{xx ^ \prime + yy ^ \prime }{x ^ \prime 2 + y ^ \prime 2 } ,\ \ Y = y - y ^ \prime \frac{xx ^ \prime + yy ^ \prime }{x ^ \prime 2 + y ^ \prime 2 } . $$

Figure: p071950a

The equation for the pedal of a curve $ x= x( t), y= y( t), z= z( t) $ in space relative to the origin is

$$ X = x - x ^ \prime \frac{xx ^ \prime + yy ^ \prime + zz ^ \prime }{x ^ \prime 2 + y ^ \prime 2 + z ^ \prime 2 } ,\ \ Y = y - y ^ \prime \frac{xx ^ \prime + yy ^ \prime + zz ^ \prime }{x ^ \prime 2 + y ^ \prime 2 + z ^ \prime 2 } , $$

$$ Z = z - z ^ \prime \frac{xx ^ \prime + yy ^ \prime + zz ^ \prime }{x ^ \prime 2 + y ^ \prime 2 + z ^ \prime 2 } . $$

The antipedal of a curve $ l $ with respect to a point $ O $ is the name given to the curve with as pedal, with respect to the point $ O $, the curve $ l $.

The pedal of a surface with respect to a point $ O $ is the set of bases to the perpendiculars dropped from the point $ O $ to the tangent planes to the surface. The equation for the pedal of a surface $ F( x, y, z) = 0 $ with respect to the coordinate origin is

$$ X = F _ {x} \Phi ,\ \ Y = F _ {y} \Phi ,\ \ Z = F _ {z} \Phi , $$

where

$$ \Phi = \frac{xF _ {x} + yF _ {y} + zF _ {z} }{F _ {x} ^ { 2 } + F _ {y} ^ { 2 } + F _ {z} ^ { 2 } } . $$

References

[a1]	M. Berger, "Geometry" , I , Springer (1987)
[a2]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1–4 , Gauthier-Villars (1887–1896)