Chapter 2: Space Curves
Section 2.5: Principal Normal
Example 2.5.9
At p=−1 on the graph of C, the curve defined by Rp=3 p−p3 i+3 p2 j+3 p+p3 k, compute N. Graph C, along with N−1 and T−1. Does N point towards the center of curvature? Hint: The curvature of C was obtained in Example 2.4.9.
Solution
Mathematical Solution
Write the position vector as R=3 p−p33 p23 p+ p3 so that R′=31−p22 p1− p2 and ρ=32p2+1, then obtain
T=−122p2−1p2+12pp2+11/2, dTds=−131+p232 pp2−10, κ=13p2+12, N=−11+p22 pp2−10
Evaluating at p=−1 gives T−1=120−11 and N−1=100.
The center of curvature for the point −2,3,−4 is given by
R+N/κx=a|f(x)p=−1= 103−4
In Figure 2.5.9(a), T−1 is represented by the black arrow; and N−1, by the green. The center of curvature is shown as the gold dot, a visual clue that N− points toward the center of curvature.
use plots, Student:-VectorCalculus in module() local R,p1,p2,p3; R:=PositionVector([3*p-p^3,3*p^2,3*p+p^3]); p1:=PlotPositionVector(R,p=-1.3..0, points=[-1],normal,tangent, curveoptions=[scaling= constrained,labels=[x,y,z],tickmarks=[2,2,6],axes=frame,orientation=[150, 65,0]],tangentoptions=[width=.1], normaloptions=[width=.1]); p2:=pointplot3d([10,3,-4],symbol=solidsphere,symbolsize=20,color=gold); p3:=display(p1,p2); print(p3); end module: end use:
Figure 2.5.9(a) Graph of C, T−1,N−1
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
3 p−p3,3 p2,3 p+p3 = →to position Vector →assign to a nameR
Obtain T−1 and N−1
Write R and press the Enter key.
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Tangent Vector≻p
Context Panel: Student Vector Calculus≻ Normalize≻Euclidean
Context Panel: Evaluate at a Point≻p= −1
Context Panel: Simplify≻Simplify
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Principal Normal≻x
Context Panel: Simplify≻Assuming Real
Context Panel: Evaluate at a Point≻p=−1
R
→tangent vector
→Euclidean-normalize
→evaluate at point
= simplify
→principal normal
→assuming real
→2-normalize
Construct Figure 2.5.9(a)
Control drag T−1 and N−1
Context Panel: Plots≻Arrow from point≻x= −2,y=3,z=−4
Write R Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder Set p∈−1.3,0 Options: Constrained Scaling
Copy and paste the arrows onto the graph of C
0−122122→plot arrow
100→plot arrow
R = →to list−p3+3p,3p2,p3+3p→
Unfortunately, there is as yet no way to increase the "heft" of the arrows drawn interactively and dropped onto a graph that is drawn to a different scale.
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≔PositionVector3 p−p3,3 p2,3 p+p3:
Use the PrincipalNormal command with the normalized option to obtain the general principal normal vector.
Use the eval command to obtain the principal normal vector at p= −1.
N≔evalPrincipalNormalR,normalized,p= −1:
Use the TangentVector command with the normalized option to obtain the general tangent vector along C.
Use the eval and simplify commands to obtain the tangent vector at p=−1.
T≔simplifyevalTangentVectorR,normalized,p= −1:
Use the PlotPositionVector command to graph C, along with the tangent and principal normal vectors at the single point p= −1.
PlotPositionVectorR,p=−1.3..0,points= −1,normal,tangent,curveoptions=scaling=constrained,labels=x,y,z,tickmarks=2,2,6,axes=frame,orientation=130,80,0,tangentoptions=width=.1,normaloptions=width=.1
The principal normal indeed points towards the center of curvature.
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