Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
Example 2.7.7
With c=1 and a,b=3,−2 in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is κs=1/s.
Solution
Mathematical Solution
Figure 2.7.7(a) is a graph of the position vector Rs, whose components are obtained by the following calculations.
θu=∫1udtt = lnu
R
=3−2+∫0scos(ln(u))sin(ln(u))ⅆu
=126+coslnss+sinlnss−4− coslnss+sinlnss
Figure 2.7.7(a) Graph of Rs
By way of corroboration, obtain
T=R′s=cos(ln(s))sin(ln(s)) and T′=1s−sin(ln(s))cos(ln(s)
from which it follows that κ=∥T′∥ = 1/s.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Set the display format for vectors by executing the BasisFormat command to the right, or use the task template.
BasisFormatfalse:
Obtain θu=∫1sκu du
Calculus palette: Definite integral template
Context Panel: Simplify≻Assuming Positive
Context Panel: Assign to a Name≻theta
∫1u1t ⅆt→assuming positivelnu→assign to a nameθ
Obtain and display the position-vector form of the plane curve
Context Panel: Assign Name
R=3−2+∫0scos(θ)sin(θ)ⅆu→assign
Write R.
Context Panel: Evaluate and Display Inline
R =
Verify that T=R′s
Context Panel: Student Vector Calculus≻Differentiate≻With Respect To≻s
Context Panel: Assign to a Name≻T
R = →differentiate →assign to a nameT
Verify that T=1
Write T.
Context Panel: Student Vector Calculus≻ Norm≻Euclidean
T = →Euclidean-norm1
Verify that T′s=κ
Context Panel: Student Vector Calculus≻Norm≻Euclidean
T = →differentiate →2-normcsgn1ss→assuming positive1s
Graph the plane curve
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder Set s∈0,3
R = →to list3+12coslnss+12sinlnss,−2−12coslnss+12sinlnss→
Maple Solution - Coded
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Invoke the BasisFormat command.
Apply the int command to obtain θu.
θ≔int1t,t=1..u assuming u≥1:
Apply the int command to obtain Rs.
R≔intcosθ,sinθ,u=0..s+3,−2:
Verifications
Use the diff command to obtain T, and to it, apply the Norm command.
NormdiffR,s = 1
Use the diff command to obtain T′, and to it, apply the Norm and simplify commands.
simplifyNormdiffR,s,s assuming s>0 = 1s
Graph
Apply the ConvertVector command to R, making it a PositionVector.
Apply the PlotPositionVector command to obtain the graph.
PlotPositionVectorConvertVectorR,position,s=0..3
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