Any approximate circle's radius at any particular given point is called the radius of curvature of the curve. As we move along the curve the radius of curvature changes. The radius of curvature formula is denoted as 'R'. The amount by which a curve derivates itself from being flat to a curve and from a curve back to a line is called the curvature. It is a scalar quantity. The radius of curvature is the reciprocal of the curvature. The radius of curvature is not a real shape or figure rather it's an imaginary circle. Let us understand the radius of curvature formula in detail using solved examples in the following section.
What is the Radius of the Curvature Formula?
The distance from the vertex to the center of curvature is known as the radius of curvature (represented by R). Any approximate circle's radius at any particular given point is called the radius of curvature of the curve or the vector length of curvature is also called the radius of curvature. For any curve with equation y = f(x), with x as its parameter the radius of curvature can be given as:
Radius of Curvature, R =\(\dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx^{2}} |}\)
Radius of Curvature Formula
- In polar coordinates r=r(Θ), the radius of curvature formula is given as: \(\rho=\frac{1}{\mathrm{K}} \frac{\left[r^{2}+\left(\frac{d r}{d \theta}\right)^{2}\right]^{3 / 2}}{\left | r^{2}+2\left(\frac{d r}{d \theta}\right)^{2}-r \frac{d^{2} r}{d \theta^{2}}\right |}\)
- R= 1/K, where R is the radius of curvature and K is the curvature.
- R = \(\dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx^{2}} |}\)
where K is the curvature of the curve, K = dT/ds, (Tangent vector function)
R the radius of curvature
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Let's take a quick look at a couple of examples to understand the radius of curvature formula, better.
Solved Examples Using Radius of Curvature Formula
Example 1: Find the radius of curvature of for 3x2 + 2x - 5 at x = 1
Solution:
To find: The radius of curvature.
y = 3x2 +2x-5
\(\dfrac{dy}{dx} = 6x + 2\)
\(\dfrac{d^2y}{dx^2} = 6\) (given)
Using radius of curvature formula,
\(R = \dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx}|}\)
Put the values,
\(R = \dfrac{(1+(6x+2)^{2})^{3/2}}{|6|}\)
\(R = \dfrac{(1+36x^2+4+24x)^{3/2}}{6}\)
\(R = \dfrac{(36x^2+5+24x)^{3/2}}{6}\)
Putting x = 1
\(R = \dfrac{(36+5+24)^{3/2}}{6}\)
\(R = \dfrac{(65)^{3/2}}{6}\)
R = 87.34
Answer: Radius of curvature, R = 87.34 units
Example 2: Find the radius of curvature of for 3x3 + 2x - 5 at x = 2.
Solution:
To find: the radius of curvature of the given curve
y = 3x3 +2x-5
\(\dfrac{dy}{dx} = 9x^2 + 2\)
\(\dfrac{d^2y}{dx^2} = 18x\) (given)
Using radius of curvature formula,
\(R = \dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx}|}\)
Put the values,
\(R = \dfrac{(1+(9x^2+2)^{2})^{3/2}}{|18x|}\)
R = \(\dfrac{(1+81x^4+4+36x^2)^{3/2}}{18x}\)
R = \(\dfrac{(81x^4+5+36x^2)^{3/2}}{18x}\)
Putting x = 2
R = \(\dfrac{(1296+5+144)^{3/2}}{36}\)
R = \(\dfrac{(1373)^{3/2}}{36}\)
R = 1413.19
Answer: Radius of curvature, R = 1413.19 units
Example 3: Find the curvature of any point on the curve r = eθ
Solution:
To find R of r = eθ
We know that by radius of curvature formula, R =\(\dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx^{2}} |}\)
Thus we find the first and second derivatives of the curve and apply them to the formula.
Given : r = eθ
dr/dθ = eθ
dr2/dθ2 = eθ
\(\begin{align}R &=\dfrac{((e^{\theta})^2 + (e^{\theta})^2)^\dfrac{3}{2}}{(e^{\theta})^2 -(e^{\theta})^2+ 2(e^{\theta})^2}\\\\&= \dfrac{2 (e^{\theta})^3}{(2 e^{\theta})^2}\\\\&=2 e^{\theta}\\\\&= \sqrt {2}e^{\theta}\\\\&= \sqrt{2}r\end{align}\)
Answer: Thus the radius of curvature of the curve r = eθ is √2 r
FAQs on Radius of Curvature Formula
What is The Radius of the Curvature Formula?
The radius of curvature of a curve y= f(x) at a point is \(\dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx^{2}} |}\). It is the reciprocal of the curvature K of the curve at a point. R = 1/K, where K is the curvature of the curve and R = radius of curvature of the curve.
How Do You Determine the Radius of Curvature?
We find the curvature of the curve at a point and take the reciprocal of it. If y = f(x), then the curve is r(t) = (t, f(t), 0) where x'(t) = 1 and x"(t) = 0, which gives the curvature as K = \(\dfrac{y''(x)}{(1+(y'(x)^2)^{\dfrac{3}{2}}}\). Now radius of curvature, R = 1/ K
How Do You Measure Curvature of a path?
The curvature(K) of a path is measured using the radius of the curvature of the path at the given point. If y = f(x) is a curve at a particular point, then the formula for curvature is given as K = 1/R. Where R = the radius calculated using the radius of curvature formula \(\dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx^{2}} |}\).
How is The Curvature measured?
The Curvature tells how fast the direction is changing as a point moves along a curve. The curvature is measured in radians/meters or radians/miles or degrees/mile. The curvature is the reciprocal of the radius of curvature of the curve at a given point. The radius of curvature formula is \(R = \dfrac{(1+(\dfrac{dy}{dx})^{2})^{3/2}}{|\dfrac{d^{2}y}{dx^{2}} |}\).
