Curvature and Radius of Curvature

15/5/2020 Curvature and Radius of Curvature
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Calculus
Applications of the Derivative
Curvature and Radius of Curvature
Consider a plane curve defined by the equation y = f (x) . Suppose that the tangent
line is drawn to the curve at a point M (x, y) . The tangent forms an angle α with the
horizontal axis (Figure 1).
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Figure 1.
At the displacement Δs along the arc of the curve, the point M moves to the point M 1.
The position of the tangent line also changes: the angle of inclination of the tangent to
the positive x − axis at the point M will be α + Δα. Thus, as the point moves by the
1
distance Δs, the tangent rotates by the angle Δα. (The angle α is supposed to be
increasing when rotating counterclockwise.)
Δα
The absolute value of the ratio is called the mean curvature of the arc M M 1. In the
Δs
limit as Δs → 0, we obtain the curvature of the curve at the point M :
∣ Δα ∣
K = lim ∣ ∣.
Δs→0 ∣ Δs ∣
From this definition it follows that the curvature at a point of a curve characterizes the
speed of rotation of the tangent of the curve at this point.
For a plane curve given by the equation y = f (x) , the curvature at a point M (x, y) is
expressed in terms of the first and second derivatives of the function f (x) by the
formula
′′
|y (x)|
K = .
3
2 2
′
[1 + (y (x)) ]
If a curve is defined in parametric form by the equations x = x (t) , y = y (t) , then its
curvature at any point M (x, y) is given by
′ ′′ ′ ′′
|x y − y x |
K = .
3
2 2 2
[(x ′ ) + (y ′ ) ]
If a curve is given by the polar equation r = r (θ) , the curvature is calculated by the
formula
2
∣ r2 + 2(r′ ) ′′
− rr ∣
∣ ∣
K = .
3
2 2
[r2 + (r′ ) ]
The radius of curvature of a curve at a point M (x, y) is called the inverse of the
curvature K of the curve at this point:
1
R = .
K
Hence for plane curves given by the explicit equation y = f (x) , the radius of curvature
at a point M (x, y) is given by the following expression:
′ 2 2
[1 + (y (x)) ]
R = .
′′
|y (x)|
Solved
This Problems
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Click or tap a problem to see the solution.
Example 1
Calculate the curvature of the ellipse
2 2
x y
+ = 1
2 2
a b
at its vertices.
Example 2
Find the curvature and radius of curvature of the parabola y = x
2
at the origin.
Example 3
Find the curvature and radius of curvature of the curve y = cos mx at a maximum
point.
Example 4
Calculate the curvature and radius of curvature of the graph of the function y = √x
at x = 1.
Example 5
Consider the curve given by the equation y 2
+ x
3
= 0. Find its curvature at the point
(−1, 1) .
Example 6
Find the curvature of the cardioid r = a (1 + cos θ) at θ = 0.
Example 7
Find the radius of curvature of the cycloid
x = a (t − sin t) , y = a (1 − cos t) .

Example 8
Determine the curvature of the curve y = arctan x at x = 0 and at infinity.
Example 9
Determine the least radius of curvature of the exponential function y x
= e .
Example 10
Find the least radius of curvature of the cubic parabola y = x .
3
Example 1.
Calculate the curvature of the ellipse
2 2
x y
+ = 1
2 2
a b
at its vertices.
Solution.
Obviously, it suffices to find the curvature of the ellipse at points A (a, 0) and B (0, b)
(Figure 2), because due to the symmetry of the curve, the curvature at the two opposite
vertices of the ellipse will be the same.

Figure 2.
To calculate the curvature, it is convenient to pass from the canonical equation of the
ellipse to the equation in parametric form:
x = a cos t, y = b sin t,
where t is a parameter. The parameter has the value t = 0 at the point A (a, 0) and is
π
equal to t = at the point B (0, b) .
2
Find the first and second derivatives:
′ ′ ′ ′′ ′′ ′
x = x = (a cos t) = −a sin t, x = x = (−a sin t) = −a cos t;
t tt
′ ′ ′′ ′
′ ′′
y = yt = (b sin t) = b cos t, x = x tt = (b cos t) = −b sin t.
The curvature of a parametrically defined curve is expressed by the formulas
′ ′′ ′ ′′
|x y − y x |
K = .
3
2 2 2
′ ′
[(x ) + (y ) ]
Substituting the above derivatives, we get:
22 2 2
∣
∣ ab sin t + ab cos t∣
∣ ∣
∣ ab (sin t + cos t)∣
∣ ab
K = = = .
3 3 3
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
(a sin t + b cos t) (a sin t + b cos t) (a sin t + b cos t)
Now we calculate the values of the curvature at the vertices A (a, 0) and B (0, b) :
ab ab ab a
K (A) = K (t = 0) = = = = ;
3 3 3 2
2 2
b b
2 2 2 2 2
(a sin 0 + b cos 0) (b )
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ab
We'll assume you're
ab
ok with
ab
this, but
b
you can
K (B) = K (t = ) = = = = .
opt-out
2 if you wish.
2
Accept
2 π 2
Reject
2 π
3
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2
3
a
3
a
2
(a sin + b cos )
2 (a ) 2
2 2
Page 1 Page 2
Problem 1 Problems 2-10
Recommended Pages
Increasing and Decreasing Functions
Rolle’s Theorem
Lagrange’s Mean Value Theorem
Cauchy’s Mean Value Theorem
Curvature and Radius of Curvature
Evolute and Involute
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15/5/2020 Curvature and Radius of Curvature

Curvature and Radius of Curvature

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limit as Δs → 0, we obtain the curvature of the curve at the point M :

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Determine the curvature of the curve y = arctan x at x = 0 and at infinity.

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Find the first and second derivatives:

The curvature of a parametrically defined curve is expressed by the formulas

Substituting the above derivatives, we get:

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Increasing and Decreasing Functions

Lagrange’s Mean Value Theorem

Cauchy’s Mean Value Theorem

Curvature and Radius of Curvature

Evolute and Involute

opt-out if you wish. Accept Reject Read More

opt-out if you wish. Accept Reject Read More

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