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The directed segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910402.png" /> which are the projections on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910403.png" />-axis of the segments of the [[Tangent line|tangent line]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910404.png" /> and the [[Normal|normal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910405.png" /> to a certain curve at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910406.png" /> (see Fig.).
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The directed segments  $  QT $
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and  $  QN $
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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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091040a.gif" />
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Figure: s091040a
 
Figure: s091040a
  
If the curve is the graph of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910407.png" />, the values of the subtangent and subnormal are equal to
+
If the curve is the graph of a function $  y = f( x) $,  
 +
the values of the subtangent and subnormal are equal to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910408.png" /></td> </tr></table>
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$$
 +
QT  = -  
 +
\frac{f( x) }{f ^ { \prime } ( x) }
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,\ \
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ON  = f( x) f ^ { \prime } ( x),
 +
$$
  
respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s0910409.png" /> is the abscissa of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s09104010.png" />. If the curve is given parametrically by
+
respectively, where $  x $
 +
is the abscissa of the point $  M $.  
 +
If the curve is given parametrically by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s09104011.png" /></td> </tr></table>
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$$
 +
= \phi ( t),\  y  = \psi ( t),
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s09104012.png" /></td> </tr></table>
+
$$
 
+
QT  = -  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s09104013.png" /> is the value of the parameter defining the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091040/s09104014.png" /> on the curve.
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\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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Revision as of 08:24, 6 June 2020


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

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

[a1] M. Berger, "Geometry" , II , Springer (1989)
[a2] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[a3] H. Lamb, "Infinitesimal calculus" , Cambridge (1924) pp. 118
How to Cite This Entry:
Subtangent and subnormal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subtangent_and_subnormal&oldid=48901
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article