Page 319 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Substituting and dropping unnecessary absolute values, get

x'y" y'x" 1.2 (2t)(0) 2


k(t)    .
2 3 /2
r2 3 /2
r2
(x  y ) (1 (2t)2) 3 /2 (1 4t )


25.2 Curvature of Space Curves
As in the case of curves in two dimensions, the curvature of a regular space curve C in three
dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit
speed along a curve. Thus if (s) is the arc length parametrization of C then the unit tangent
vector T(s) is given by
T(s) = ’(s)
and the curvature is the magnitude of the acceleration:

k(s)  T'(s)   "(s) .
The direction of the acceleration is the unit normal vector N(s), which is defined by
T'(s)
N(s)  .
T'(s)
The plane containing the two vectors T(s) and N(s) is called the osculating plane to the curve at
(s). The curvature has the following geometrical interpretation. There exists a circle in the
osculating plane tangent to (s) whose Taylor series to second order at the point of contact agrees
with that of (s). This is the osculating circle to the curve. The radius of the circle R(s) is called the
radius of curvature, and the curvature is the reciprocal of the radius of curvature:

1
k(s)  .
R(s)
The tangent, curvature, and normal vector together describe the second-order behavior of a
curve near a point. In three-dimensions, the third order behavior of a curve is described by a
related notion of torsion, which measures the extent to which a curve tends to perform a corkscrew
in space. The torsion and curvature are related by the Frenet–Serret formulas (in three dimensions)
and their generalization (in higher dimensions).

25.2.1 Local Expressions

For a parametrically defined space curve in three-dimensions given in Cartesian coordinates by
ã(t) = (x(t),y(t),z(t)), the curvature is

2
2



(z"y' y"z')  (x"z' z"x')  (y"x' x"y') 2
k  .
r2 3 /2
r2
r2
(x  y  z )
where the prime denotes differentiation with respect to time t. This can be expressed
independently of the coordinate system by means of the formula
'    "
k  3
' 
where × is the vector cross product. Equivalently,
t
det(( ', ") ( ', ")




k  3 .
' 
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