Chapter 2: Space Curves
Section 2.5: Principal Normal
Example 2.5.4
At x=1 on the graph of C, the catenary defined by y=coshx, compute N. Graph C, along with N1 and T1. Does N point towards the center of curvature? Hint: The curvature of C was obtained in Example 2.4.5.
Solution
Mathematical Solution
Write the position vector as R=xcosh(x) so that R′=1sinh(x) and ρ=coshx. Then
T=sech(x)tanh(x), dTds=−tanh(x)sech2(x)sech3(x), κ=sech2x, N=−tanh(x)sech(x)
Evaluating at x=1 gives T1=sech(1)tanh(1) and N1=−tanh(1)sech(1).
Note that N can be obtained from T by interchanging components and negating the first component to place N to the left of T.
The center of curvature for the point P:1,cosh1 is given by
R+N/κx=a|f(x)x=1 ≐ −0.813.1
Hence, the center of curvature is to the left of P.
In Figure 2.5.4(a), T1 is represented by the black arrow; and N1, by the green.
use Student:-VectorCalculus in module() local R,p1; R:=PositionVector([x,cosh(x)]); p1:=PlotPositionVector(R,x=0..2, points=[1],normal,tangent, curveoptions=[scaling= constrained,labels=[x,y], size=[300,300]], tangentoptions=[width=.1], normaloptions=[width=.1]); print(p1); end module: end use:
Figure 2.5.4(a) Graph of C, T1,N1
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Execute the BasisFormat command at the right, or use the task template.
BasisFormatfalse:
Define C as the position vector R
Enter the vector notation for C as per Table 1.1.1. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To Position Vector
Context Panel: Assign to a Name≻R
x,coshx = →to position Vector →assign to a nameR
Obtain T1 and N1
Write R and press the Enter key.
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Tangent Vector≻x
Context Panel: Student Vector Calculus≻ Normalize≻Euclidean
Context Panel: Simplify≻Assuming Positive
Context Panel: Evaluate at a Point≻x=1
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Principal Normal≻x
R
→tangent vector
→Euclidean-normalize
→assuming positive
→evaluate at point
→principal normal
→2-normalize
Construct Figure 2.5.4(a)
Control drag T1 and N1
Context Panel: Plots≻Arrow from point≻x=1,y=cosh1
Write R Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder Set x∈0,2 Options: Constrained Scaling
Copy and paste the arrows onto the graph of C
1cosh1sinh1cosh1→plot arrow
−sinh1cosh11cosh1→plot arrow
R = →to listx,coshx→
Maple Solution - Coded
Install the Student Vector Calculus package.
Use the BasisFormat command to set the display of vectors.
withStudent:-VectorCalculus:
Use the PositionVector command to define C as the position vector R.
R≔PositionVectorx,coshx:
Use the PrincipalNormal command with the normalized option to obtain the general principal normal vector.
Use the eval and simplify commands to obtain the principal normal vector at x=1.
N≔simplifyevalPrincipalNormalR,normalized,x=1:
Use the TangentVector command with the normalized option to obtain the general tangent vector along the curve.
Use the eval and simplify commands to obtain the tangent vector at x=1.
T≔simplifyevalTangentVectorR,normalized,x=1:
Use the PlotPositionVector command to graph C along with the tangent and principal normal vectors at the single point x=1.
PlotPositionVectorR,x=0..2,points=1,normal,tangent,curveoptions=scaling=constrained,labels=x,y,size=300,300,tangentoptions=width=.1,normaloptions=width=.1
The principal normal indeed points towards the center of curvature. The components of N could be obtained by interchanging the components of T and negating the first component so that N points to the left of T.
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