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Curvature



Let y = f (x) be a curve that does not intersect itself and having tangents at each point. Let
A be a fixed point on the curve from which arc length is measured. Let P be any point on a
given curve and Q a neighbouring points sothat AP = s and AQ = s + ∆s.
Therefore Length arc P Q = ∆s
Let the tangents at P and Q make an angles Ψ and Ψ + ∆Ψ respectively with positive direction
of x-axis, so that the angle between the tangents at P and Q = ∆Ψ. Thus for a change of ∆s
in the arcual length of the curve, the direction of the tangent to the curve changes by ∆Ψ.
∆Ψ
Hence is the average rate of bending of the curve (or average rate of change of direction of
∆s
the tangent to the
 curve
 in the arcual interval P Q) or average curvature of the arc P Q.
∆Ψ dΨ
Therefore lim = is the rate of bending of the curve with respect to arcual distance
∆s→0 ∆s ds
at P or the curvature of the curve at the point P. The curvature is denoted by κ.




dΨ
Therefore average Curvature of arc P Q =
ds
Find the curvature of a circle of radius at any point on it
Let the arcual distances of points on the circle be measured from A, the lowest point of the
circle and let the tangent at A be chosen as the x-axis. Let AP = s and let the tangent at P
make an angle Ψ with the x-axis.

1

,2




Then s = a.AĈP = aΨ [Since the angle between CA and CP equals the angle between the
respective perpendiculars AT and P T.]
1
or Ψ = s
a
dΨ 1
Therefore =
ds a
Thus the curvature of a circle at any point on it equals the reciprocal of its radius. Equivalently,
the radius of a circle equals the reciprocal of the curvature at any point on it.

Radius of Curvature

Radius of curvature of a curve at any point on it is defined as the reciprocal of the curvature of
1 ds
the curve at that point and denoted by ρ. Thus ρ = = .
κ dΨ
Note: To find ρ of a curve at any point on it, we should know the relation between s and
Ψ for that curve, which is not easily derivable in most cases. Generally curves will be defined
by means of their Cartesian, parametric or polar equations. Hence formulas for ρ in terms of
cartesian, parametric or polar co-ordinates are given below.

(1) For cartesian curve y = f (x) :

(1 + y12 )3/2
ρ=
y2

dy d2 y
where y1 = , y2 = .
dx dx2
(2) For parametric equations x = f (t), y = g(t) :

(x02 + y 02 )3/2 (ẋ2 + ẏ 2 )3/2
ρ = 0 00 or ρ =
x y − y 0 x00 ẋÿ − ẏẍ

dx 0 dy 00 d2 x 00 d2 y
where x0 = ,y = ,x = , y =
dt dt dt2 dt2

, 3

(3) For polar coordinates r = f (θ)
(r2 + r12 )3/2
ρ=
r2 + 2r12 − rr2
dr d2 r
where r1 = , r2 = .
dθ dθ2

Note:

• Curvature of a straight line is zero.
• Curvature of a circle is the reciprocal of its radius.
dy
• To calculate ρ when becomes infinite, we can use the formula
dx
"  2 #3/2
dx
1+
dy
ρ=
d2 y
dx2
√
 
1 1 √
Example 1: Find the radius of curvature at the point , on the curve x + y = 1.
4 4
Solution: Given
√ √
x+ y =1 (1)

Differentiate (1) w.r.to x, we get
1 1 dy
√ + √ . =0
2 x 2 y dx
√
dy y
⇒ = −√ (2)
dx x
 
dy 
⇒  = −1
dx  1 , 1 
44
Again Differentiate (2) w.r.to x, we get
√
1 dy √ 1

x. √ − y. √
d2 y  2 y dx 2 x
2
= − √ 2 
dx  ( x) 

√  √  √ 
1 x y y
=− √ . −√ −√
2x y x x
√ √
x+ y
= √
2x x