Differential Geometry | Topology Answers

Calculate the normal and the geodesic curvatures of the following curves on
the given surfaces:
(b) The right circular helix γ(θ) = (a cos θ, a sin θ, bθ) on the right circular cylinder
σ(u, v) = (a cos u, a sin u, v), where a, b > 0 are constants.

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

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