Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 34E
See similar textbooks
icon
Related questions
Question
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.
Expert Solution
steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS