1. AND RADIUS OF CURVATURE

2. RADIUS OF CURVATURE
Let P be any point on the curve C. Draw the tangent at P to the
circle. The circle having the same curvature as the curve at P
touching the curve at P, is called the circle of curvature. It is also
called the osculating circle. The centre of the circle of the curvature
is called the centre of curvature. The radius of the circle
of curvature is called the radius of curvature and is denoted
by ‘ρ’
. Note : 1. If k (> 0) is the curvature of a curve at P, then the radius
of curvature of the curve of ρ is 1/k
. This follows from the definition
of radius of curvature and the result that the curvature of a circle is the
reciprocal of its radius
Note : 2. If for an arc of a curve, ψ decreases as s increases, then
dψ/ds is negative, i.e., k is negative. But the radius of a circle is non-negative. So to take ρ = 1/k(in magnitude)=ds/dψ some authors regard k also as non-negative i.e .,k= dψ/ds

3. The sign of dψ/ds indicates the convexity and concavity of the curve in the neighbourhood of the point. Many authors take ρ =ds/dψ and discard negative sign if computed value is negative.
∴ Radius of curvature ρ =1/ |k|

4. Radius of Curvature in Cartesian Form
Suppose the Cartesian equation of the curve C is given by y = f (x) and A be a fixed point on it. Let
P(x, y) be a given point on C such that arc AP = s.
Then we know that
dy /dx = tan ψ ...(1)
where ψ is the angle made by the tangent to the curve C at P with the x-axis and
ds/dx ={ 1+(dy/dx)^2}^1/2……(2)
Differentiating (1) w.r.t x, we get
d^2y/dx^2 = sec^2ψ ⋅ dψ /dx
= (1 + tan^2 ψ )dψ /ds.ds/dx
=[1+(dy/dx)^2] 1/ ρ [1+(dy/dx)^2] ^1/2

5. Radius of Curvature in Parametric Form
[By using the (1) and (2)]
Radius of Curvature in Parametric Form
Let x = f (t) and y = g (t) be the Parametric equations of a curve C and P (x, y) be a given point on it
Then dy/dx= dy/dt / dx/dt
And d^2y/dx^2 = d/dt{dy/dt / dx/dt}.dt/dx

6. =dx/dt.d^2y/dt^2-dy/dt.d^2x/dt^2
= _________________________
[dx/dt]^3
Substituting the values of dy/dx and d^2y/dx^2 in the Cartesian form of the radius of curvature of the curve y = f (x)
Therefore . .
=[1+(dy/dt /dx/dt)^2] ^3/2
____________________________
dx/dt.d^2y/dt^2-dy/dt.d^2x/dt^2/dx/dt^3

7. {(dx/dt)^2+(dy/dt)^2}^3/2
ρ= _____________________________
dx/dt.d^2y/dt^2-dy/dt.d^2x/dt^2
This is the cartesian form of the radius of curvature in parametric form.
Examples
Find the radius of curvature at any point on the curve y = a log sec x/a
2. For the curve y = c cos h (x /c), show that ρ =y^2/c
3. Find the radius of curvature at (1, –1) on the curve y = x2 – 3x + 1.
4. Find the radius of curvature at (a, 0) on y = x^3 (x – a).
5. Find the radius of curvature at x =πa/4 on y = a sec (x/a) .
6.Find ρ at any point on x = a (θ + sinθ) and y = a (1 – cosθ).

8. Radius of Curvature in Pedal Form
Let polar form of the equation of a curve be r = f (θ) and
P(r, θ) be a given point on it. Let the tangent to the curve
at P subtend an angle ψ with the initial side. If the angle
between the radius vector OP and the tangent at P is φ then
we have ψ = θ + φ
Let p denote the length of the perpendicular from the
pole O to the tangent at P. Then from the figure,
sin φ =OM/OP =p/r
Hence, p= r sin φ ………………(1)
Therefore
This is the Pedal form of the radius of curvature

9. Radius of Curvature in Polar Form
Let r = f (θ) be the equation of a curve in the polar form and p(r, θ) be a point on it. Then
This is the formula for the radius of curvature in the polar form.

10. Examples for pedal and polar forms
Find the radius of the curvature of each of the following curves:
(i) r^3 = 2ap^2 (Cardiod)
(ii) p^2 = ar
2. Find the radius of curvature of the cardiod r = a (1 + cos θ) at any point (r, θ) on it. Also prove that ρ^2/r is a constant
3. Show that for the curve rn = an cos nθ the radius of curvature is
A^n/ (n+1) r^n-1
4. Find the radii of curvature of the following curves
(i) r = a e^θ cot α
(ii) r (1 + cos θ) = a

11. CENTER OF CURVATURE
Let P(x, y) be any point on the curve and let PT be the tangent at P making an angle ψ with the positive direction of x-axis. Let c(α, β) be the center of curvature corresponding to P(x, y).
Then eqation of circle of curvature is
(X-α)^2 +(y- β)^2 = p^2

12. Length of chords of curvature parallel to x-axis and y- axis
= 2dy/dx[1+(dy/dx)2]
__________________
d^2y/dx^2
Parellel to y-axis
2[1+(dy/dx)2]
________________ =
d^2y/dx^2
Example- If y =a log sec x/a ,then prove that the chord of curvature parallel tp y – axis is of constant length. -

13. THANKS