Prove the following theorem: If the curvature k(s) of a curve is a known function of the arclength parameter s, then a curve r(s)= with curvature function k(s) is given by x(s)=?cos[?k(t)dt]du and y(s)=?sin[?k(t)dt]du. (ie. t-bounds go from 0 to u and u-bounds go from 0 to s.) I am in a first year differential geometry course and just cant see the path on this proof. Thanks in advance for any explanation. I was given a hint to integrate the curvature function but still dont understand how the cos and sin functions arise.

Question

Prove the following theorem:

If the curvature k(s) of a curve is a known function of the arclength parameter s, then a curve r(s)=<x(s),y(s)> with curvature function k(s) is given by x(s)=?cos[?k(t)dt]du and y(s)=?sin[?k(t)dt]du.

(ie. t-bounds go from 0 to u and u-bounds go from 0 to s.)

I am in a first year differential geometry course and just cant see the path on this proof. Thanks in advance for any explanation.

I was given a hint to integrate the curvature function but still dont understand how the cos and sin functions arise.

anonymous · Answer

?=integral sign

anonymous · Answer

@ganeshie8  can you help please

anonymous · Answer

@Ashleyisakitty  helppp!!

anonymous · Answer

@satellite73  helpp!!! someoneee