###

DIFFERENTIAL GEOMETRY COURSE

10. *OLD EXAMINATIONS PART ONE*

*PROBLEM 88 :* K is a curve with parameter representation ((1/2)√3 sin(t) -(1/2)t, cos(t), (1/2)sin(t) + (1/2)√3).

a) Calculate the arc length function s(t) (starting point (0,1,0)) and the curvature function κ(t).

b) Prove that the curve is a circular helix by giving the matrix of a linear mapping (here a combination of a rotation and a reflection) that transforms K into the circular helix
(cos(t), sin(t), t).

*PROBLEM 89 :* Calculate an equation of the sphere that has a maximal contact with the curve (t, t^{4}, t^{2}) in (1,1,1). (Hint:
you don't need to calculate arc length, curvature and torsion.)

*PROBLEM 90 :*

a) Suppose we have a curve K_{1} and a curve K_{2} such that with each point P_{1} on K_{1} we have a point P_{2} on K_{2}
such that the binormal of K_{1} at P_{1} goes through P_{2} and lies in the osculating plane of K_{2} at P_{2}.

Find a differential equation that must hold for the curvature function κ(s) and the torsion function τ(s) of K_{1} and the function λ(s) representing the distance between P_{1} and
P_{2}.

b) Suppose in a) it says "rectifying plane" instead of "osculating plane". Explain which changes you have to make in your calculations. Elaborate as much as possible. What do we get if λ
is constant?

*PROBLEM 91 :* Let K_{1} be the curve with parameter representation: __x__(t) = (3 sin(t), 4t, 3 cos(t)).

a) Give a parametrisation of K_{1} with arc length s as parameter.

b) Give for each s the vectors __t__(s), __n__(s), __b__(s).

c) Give an equation of the plane that in __x__(t) has a maximal contact with K_{1}. Elaborate!

d) Give a parametrisation of the curve K_{2} that lies on the same cylinder as K_{1} and forms with K_{1} a pair of curves of Bertrand.

e) What is the angle between the tangents in corresponding partner points on K_{1} and K_{2}?

*PROBLEM 92 :* K_{1} is a curve with curvature κ(s) and torsion τ(s). Furthermore, we have a real number α ∈ (0,π/2).

Suppose we can choose for each s a line in the rectifying plane 'at s' making the angle α with the tangent at s and such that these lines are binormals of a curve K_{2}.
Then what do you know about K_{1}?

Explain your answer with calculation and reasoning. Also calculate for each s the distance between both partner points on K_{1} and K_{2}.

answers

HOME