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at a unit-speed regular parametrised curve 3: (-7, 7)→ R² with T, N) has constant curvature k> 0. Show that its tangent vector n T(s) = cos(Ks) a + sin(Ks) b, for some a, b e R2.

at a unit-speed regular parametrised curve 3: (-7, 7)→ R² with T, N) has constant curvature k> 0. Show that its tangent vector n T(s) = cos(Ks) a + sin(Ks) b, for some a, b e R2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A1. (a) Give an example of parametrised curve y: (0, 1) → R' which is i. not regular; ii. regular but not unit-speed; iii. regular and unit-speed. (b) Assume that a unit-speed regular parametrised curve ß:(-7, 7) → R? with Frenet frame (T, N) has constant curvature k > 0. Show that its tangent vector T is of the form T(s) = cos(ks) a + sin(xs) b, for some a, b e R?. Deduce that |a||? = 1, ||b||2 = 1, while a b = 0. Furthermore, prove that 3 is (part of) a circle of radius K. %3D %3D
Transcribed Image Text:A1. (a) Give an example of parametrised curve y: (0, 1) → R' which is i. not regular; ii. regular but not unit-speed; iii. regular and unit-speed. (b) Assume that a unit-speed regular parametrised curve ß:(-7, 7) → R? with Frenet frame (T, N) has constant curvature k > 0. Show that its tangent vector T is of the form T(s) = cos(ks) a + sin(xs) b, for some a, b e R?. Deduce that |a||? = 1, ||b||2 = 1, while a b = 0. Furthermore, prove that 3 is (part of) a circle of radius K. %3D %3D
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