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Consider the plane curves ve : [0, 1] + R2, defined by formulae ve(t) =

(cos(2nlt), sin(2nlt)), if l is a non zero integer;

(cos(2nt), sin(4nt)) if l = 0.

(i) What are rotation indices of these curves? Briefly explain your answer.

(ii) Using (a)(i), deduce that any regular closed curve 7 : [0, 1] + R2 is regularly

homotopic to a curve ve for some integer l e Z.

(iii) Draw a plane curve that is regularly homotopic to 72 and has five self-intersection

points. Explain why your drawing satisfies these conditions.

Consider the family of closed curves

(cos(2nt), sin(4nt))+p(cos(2nlt), sin(2mlt)),

where l is a non-zero integer, p > 0 is a real number, and t€ [0, 1].

(i) Prove that there exists a constant C > 0 such that if pl > C, then y is a

regular closed curve.

(ii) Give one possible value for the constant C in (b)(i). Justify your answer.

(ii) What are possible values of the rotation index r() when p satisfies the con-

dition in (b)(i)? Justify your answer.

Let a : [0, 1] + R” and B : [0, 1] + R” be two unit speed curves such that

for all t € [0, 1] the angle between the vectors a'(t) and B'(t) is not greater

than 7/3. Prove that the curves a and B are regularly homotopic.

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Tags:

plane curves

rotation indices

homotopic curve

non zero integer

regular closed curve

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