View source for Pedal curve

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''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|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

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071950a.gif" />

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

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

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''I''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Gauthier-Villars  (1887–1896)</TD></TR></table>