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For this question, we need some definitions. For a curve c(t) such that c' (t) 0 for any t, we define: T(t) = d' (t)/||c' (t)|| to be its unit tangent vector. We say that a curve is parametrized by arclength if ||c' (t)|| = 1 for all t. If a curve is parametrized by arclength, then its curvature at c(t) is defined to be k = ||T' (t)||. (a) Check that the helix c(t) = (cos(t), sin(t), t) is parametrized by arc length. (b) Compute the curvature of the helix c(t). (c) Find a constant c so that s(t) = (Rcos(ct), Rsin(ct)) has unit speed. And then, show that the curve s(t) has constant curvature.

For this question, we need some definitions. For a curve c(t) such that c' (t) 0 for any t, we define: T(t) = d' (t)/||c' (t)|| to be its unit tangent vector. We say that a curve is parametrized by arclength if ||c' (t)|| = 1 for all t. If a curve is parametrized by arclength, then its curvature at c(t) is defined to be k = ||T' (t)||. (a) Check that the helix c(t) = (cos(t), sin(t), t) is parametrized by arc length. (b) Compute the curvature of the helix c(t). (c) Find a constant c so that s(t) = (Rcos(ct), Rsin(ct)) has unit speed. And then, show that the curve s(t) has constant curvature.

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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For this question, we need some definitions. For a curve c(t) such that c' (t) 0 for any t, we define: T(t) = d' (t)/||c' (t)|| to be its unit tangent vector. We say that a curve is parametrized by arclength if ||c' (t)|| = 1 for all t. If a curve is parametrized by arclength, then its curvature at c(t) is defined to be k = ||T' (t)||. (a) Check that the helix c(t) = (cos(t), sin(t), t) is parametrized by arc length. (b) Compute the curvature of the helix c(t). (c) Find a constant c so that s(t) = (Rcos(ct), Rsin(ct)) has unit speed. And then, show that the curve s(t) has constant curvature.
Transcribed Image Text:For this question, we need some definitions. For a curve c(t) such that c' (t) 0 for any t, we define: T(t) = d' (t)/||c' (t)|| to be its unit tangent vector. We say that a curve is parametrized by arclength if ||c' (t)|| = 1 for all t. If a curve is parametrized by arclength, then its curvature at c(t) is defined to be k = ||T' (t)||. (a) Check that the helix c(t) = (cos(t), sin(t), t) is parametrized by arc length. (b) Compute the curvature of the helix c(t). (c) Find a constant c so that s(t) = (Rcos(ct), Rsin(ct)) has unit speed. And then, show that the curve s(t) has constant curvature.
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