Page 318 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 25: Curvature

25.1.2 Local Expressions Notes

For a plane curve given parametrically in Cartesian coordinates as ã(t) = (x(t),y(t)), the curvature
is
x'y" y'x"

k  ,
'2
'2 3 /2
(x  y )
where primes refer to derivatives with respect to parameter t. The signed curvature k is
' "
"
'
x y  y x
k  .
'2
'2 3 /2
(x  y )
These can be expressed in a coordinate-independent manner via
det( ', ") det( ', ")




k  3 , k  3
'  ' 
25.1.3 Curvature of a Graph
For the less general case of a plane curve given explicitly as y = f(x), and now using primes for
derivatives with respect to coordinate x , the curvature is
y"
k  ,
2 3 /2
(1 y' )

and the signed curvature is
y"
k  .
(1 y' )
2 3 /2

This quantity is common in physics and engineering; for example, in the equations of bending
in beams, the 1D vibration of a tense string, approximations to the fluid flow around surfaces (in
aeronautics), and the free surface boundary conditions in ocean waves. In such applications, the
assumption is almost always made that the slope is small compared with unity, so that the
approximation:

d y
2
k 
dx 2
may be used. This approximation yields a straightforward linear equation describing the
phenomenon, which would otherwise remain intractable.
If a curve is defined in polar coordinates as r(), then its curvature is

r2
2
r  2r  rr"

k( ) 
2
r2 3 /2
(r  r )
where here the prime now refers to differentiation with respect to .
Example: Consider the parabola y = x . We can parametrize the curve simply as
2
(t) = (t,t ) = (x,y). If we use primes for derivatives with respect to parameter t , then
2
x' 1, x" 0, y' 2t, y" 2.



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