Differential Geometry | Topology Answers

Find the centre of curvature of astroid
x^2/3+y^2/3=a^2/3

Find the centre of curvature of the curve
x=3t y=t^2-6 at (x,y)

Find envelope of the family of straight line
y=mx+a/m

Find the evolute of the four cuspecl hypocycloid
x=acos^3θ y=asin^3θ

Prove that for cardiod r=a(1+cosθ)
ρ^2/r is constant

For curve r^m=a^m cosmθ Prove that

P=a^m÷(m+1)r^m-1

Find the centre of curvature of the astroid
x^2/3+y^2/3=a^2/3

The third fundamental form of a surface σ(u, v) is
||N̂u|| ^2 du^2 + 2N̂u.N̂v dudv + ||N̂v||^2 dv^2
where N̂ (u, v) is the standard unit normal to σ(u, v). Let FIII be the associated 2 × 2
symmetric matrix.
Show that FIII = FIIF^−1I FII , where FI and FII are the 2 × 2 symmetric matrices
associated with the first and the second fundamental forms, respectively

Hence find these curvatures in the following cases:
i. f(u) = e^u, g(u) = u.
ii. f(u) = 2 + sin u, g(u) = u.

Given that \\(R=sin ¡t i+cos ¡t j+tk\\), find \\( (d^2 R)/(dt^2 )\\).