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''to a curve''
 
''to a curve''
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$
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\newcommand{\vect}[1]{\mathbf{#1}}
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$
  
A straight line representing the limiting position of the secants. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921701.png" /> be a point on a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921702.png" /> (Fig. a). A second point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921703.png" /> is chosen on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921704.png" /> and the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921705.png" /> is drawn. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921706.png" /> is regarded as fixed, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921707.png" /> approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921708.png" /> along the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t0921709.png" />. If, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217010.png" /> goes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217011.png" />, the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217012.png" /> tends to a limiting line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217014.png" /> is called the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217016.png" />.
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A straight line representing the limiting position of the secants. Let $M$ be a point on a curve $L$ (Fig.&nbsp;1). A second point $M_1$ is chosen on $L$ and the straight line $M_M_1$ is drawn. The point $M$ is regarded as fixed, and $M_1$ approaches $M$ along the curve $L$. If, as $M_1$ goes to $M$, the line $MM_1$ tends to a limiting line $MT$, then $MT$ is called the tangent to $L$ at $M$.
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092170a.gif" />
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Not every continuous curve has a tangent, since $MM_1$ need not tend to a limiting position at all, or it may tend to two distinct limiting positions as $M_1$ tends to $M$ from different sides of $L$ (Fig.&nbsp;2). If a curve in the plane with rectangular coordinates is defined by the equation $y=f(x)$ and $f$ is differentiable at the point $x_0$, then the slope of the tangent at $M$ is equal to the value of the derivative $f'(x_0)$ at $x_0$; the equation of the tangent at this point has the form
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$$
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y - f(x_0) = f'(x_0)(x - x_0).
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$$
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The equation of the tangent to a curve $\vect{r} = \vect{r}(t)$ in space is
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$$
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\vect{t}(\lambda) =
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\vect{r} + \lambda \frac{\mathrm{d}\vect{r}}{\mathrm{d}t}, \quad
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\lambda \in \R.
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$$
  
Figure: t092170a
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By a tangent to a surface $S$ at a point $M$ one means a straight line passing through $M$ and lying in the [[Tangent plane|tangent plane]]to $S$ at $M$.
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t092170b.gif" />
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====References====
  
Figure: t092170b
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)</TD></TR></table>
 
 
Not every continuous curve has a tangent, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217017.png" /> need not tend to a limiting position at all, or it may tend to two distinct limiting positions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217018.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217019.png" /> from different sides of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217020.png" /> (Fig. b). If a curve in the plane with rectangular coordinates is defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217022.png" /> is differentiable at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217023.png" />, then the slope of the tangent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217024.png" /> is equal to the value of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217025.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217026.png" />; the equation of the tangent at this point has the form
 
 
 
<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/t/t092/t092170/t09217027.png" /></td> </tr></table>
 
 
 
The equation of the tangent to a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217028.png" /> in space is
 
 
 
<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/t/t092/t092170/t09217029.png" /></td> </tr></table>
 
 
 
By a tangent to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217030.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217031.png" /> one means a straight line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217032.png" /> and lying in the [[Tangent plane|tangent plane]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217033.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092170/t09217034.png" />.
 
 
 
 
 
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter,   "Introduction to geometry" , Wiley (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.W. Guggenheimer,   "Differential geometry" , McGraw-Hill (1963)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Hilbert,   S.E. Cohn-Vossen,   "Geometry and the imagination" , Chelsea (1952) (Translated from German)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> B. O'Neill,   "Elementary differential geometry" , Acad. Press (1966)</TD></TR></table>
 

Revision as of 19:28, 24 April 2012

to a curve $ \newcommand{\vect}[1]{\mathbf{#1}} $

A straight line representing the limiting position of the secants. Let $M$ be a point on a curve $L$ (Fig. 1). A second point $M_1$ is chosen on $L$ and the straight line $M_M_1$ is drawn. The point $M$ is regarded as fixed, and $M_1$ approaches $M$ along the curve $L$. If, as $M_1$ goes to $M$, the line $MM_1$ tends to a limiting line $MT$, then $MT$ is called the tangent to $L$ at $M$.

Not every continuous curve has a tangent, since $MM_1$ need not tend to a limiting position at all, or it may tend to two distinct limiting positions as $M_1$ tends to $M$ from different sides of $L$ (Fig. 2). If a curve in the plane with rectangular coordinates is defined by the equation $y=f(x)$ and $f$ is differentiable at the point $x_0$, then the slope of the tangent at $M$ is equal to the value of the derivative $f'(x_0)$ at $x_0$; the equation of the tangent at this point has the form $$ y - f(x_0) = f'(x_0)(x - x_0). $$ The equation of the tangent to a curve $\vect{r} = \vect{r}(t)$ in space is $$ \vect{t}(\lambda) = \vect{r} + \lambda \frac{\mathrm{d}\vect{r}}{\mathrm{d}t}, \quad \lambda \in \R. $$

By a tangent to a surface $S$ at a point $M$ one means a straight line passing through $M$ and lying in the tangent planeto $S$ at $M$.

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a3] H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963)
[a4] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)
[a5] B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
How to Cite This Entry:
Tangent line. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_line&oldid=25287
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article