Differential Geometry | Topology Answers

a) Calculate the principal, Gaussian, and mean curvatures of the surface of revolution
σ(u, v) = (f(u) cos v, f(u) sin v, g(u)), where f, g are smooth real-valued functions
with f > 0.
b) 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.

Calculate the normal and the geodesic curvatures of the following curves on
the given surfaces:
(a) The circle γ(t) = (cost, sin t, 1) on the paraboloid σ(u, v) = (u, v, u^2 + v^2).

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.
a) ) Show that FIII − 2HFII + KFI = 0, where K and H are the Gaussian and mean
curvatures, respectively, of σ.

Find the evolute of the four cusped hypocycloid
x=a cos^3θ y=a sin^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 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