View source for Subtangent and subnormal

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The directed segments  $  QT $
and  $  QN $
which are the projections on the  $  x $-
axis of the segments of the [[Tangent line|tangent line]]  $  MT $
and the [[Normal|normal]]  $  MN $
to a certain curve at a point  $  M $(
see Fig.).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091040a.gif" />

Figure: s091040a

If the curve is the graph of a function  $  y = f( x) $, 
the values of the subtangent and subnormal are equal to

$$ 
QT  =  - 
\frac{f( x) }{f ^ { \prime } ( x) }
 ,\ \ 
ON  =  f( x) f ^ { \prime } ( x),
$$

respectively, where  $  x $
is the abscissa of the point  $  M $. 
If the curve is given parametrically by

$$ 
x  =  \phi ( t),\  y  =  \psi ( t),
$$

then

$$ 
QT  =  - 
\frac{\psi ( t) \phi  ^  \prime  ( t) }{\psi  ^  \prime  ( t) }
 ,\ \ 
QN  =  
\frac{\psi ( t) \psi  ^  \prime  ( t) }{\psi  ^  \prime  ( t) }
 ,
$$

where  $  t $
is the value of the parameter defining the point  $  M $
on the curve.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''II''' , Springer  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,   "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Lamb,   "Infinitesimal calculus" , Cambridge  (1924)  pp. 118</TD></TR></table>