Show that a paramtrized curve C is a line iff k = 0.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1. Show that a paramtrized curve C is a line iff k = 0.

2. Show that a paramtrized curve C (with non-zero curvature k) lies on a plane iff τ = 0.

3. Let t(s) be the unit tangent vector of a curve with respect to the ar- clength parameter. Fix some s0 and define ϕ(s) as the angle between t(s) and t(s0). Show that ϕ′(s0) = k(s0) .

4. Suppose a curve C is defined as the intersection of two surfaces given by F (x, y, z) = 0 and G(x, y, z) = 0. Express the curvature k at a point (x, y, z) of C in terms of the functions F, G (assuming that the gradients of F, G do not vanish along C)

5. Show that if all normal lines to a curve C pass through a common point, then the C lies on a circle.

6. Let C be a curve (with non zero curvature k) and let C be its locus of centers of curvature. Show that the tangents along C and C are perpendicular.

7. Show that a curve C (with non zero curvature k and torsion (τ )) lies on a sphere iff [R2 + (TR′)2]′ = 0 along C where R(s) := 1/k(s) and T (s) := 1/τ (s) and s is the arclength parameter.

 

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