Overview
The degree of curvedness is used to determine the shape of a plane curve. Curvature is nothing but the measure of rate of change in of curvedness.
Here note that there is no bending in a straight line while in a circle there is constant bending.
Angle of contingence of the arc AB of a curve c is the angle between the tangents at A and B to the curve c
Given two arcs of the same length, the arc with greater angle of contingence is said to be more curved.
How do we find it?
Let the equation of the curve is given in Cartesian form y = f(x), then
Formula for polar form-
Radius of curvature
Radius of curvature is the reciprocal of curvature at any point. It is denoted by (rho)
Or
Parametric form:
Let the co-ordinates be defined in the form of functions depend on one independent variable t.
Suppose
, Then we will calculate the values in term of variables depend on t.
By chain rule we have
And
By
Newton’s formula
a. If x-axis is tangent to a curve at the origin, then
b. If y-axis is tangent to a curve at the origin, then
c. When the tangent is at the origin and neither on x-axis nor on y-axis then
Where .
Where
Centre of curvature-
Centre of curvature at any point P (x, y) on the curve y = f(x) is given by-
Circle of curvature at P-
Example: Find the radius of curvature at (0, c) of the catenary y = c cosh(x/c)?
Solution:
Given catenary
Differentiate (i) with respect to x.
Also consider
Again differentiating with respect to x , we get
We know that
Substituting values from (i),(ii) and (iii) we get
Hence
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