Chapter 2: Space Curves
Section 2.5: Principal Normal
Example 2.5.7
At p=1 on the graph of C, the curve defined by Rp=p i+3 p2 j+p3 k, 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 2.4.7.
Solution
Mathematical Solution
Write the position vector as R=p3 p2p3 so that R′=16 p3 p2 and ρ=9p4+36p2+1. Then
T=19p4+36p2+116 p3 p2, dTds=−69p4+36p2+123p p2+29 p4−1−p18p2+1, κ=69p4+p2+19p4+36p2+13/2, N=−19p4+36p2+19p4+p2+13pp2+29p4−1−p18p2+1
Evaluating at p=1 gives T1=146163 and N1=1506−9−819.
The center of curvature for the point 1,3,1 is given by
R+N/κx=a|f(x)p=1= −58/11−85/33470/33
In Figure 2.5.7(a), T1 is represented by the black arrow; and N1, by the green. The center of curvature is shown as the gold dot, a visual clue that N1 points toward the center of curvature.
use plots, Student:-VectorCalculus in module() local R,p1,p2,p3; R:=PositionVector([p,3*p^2,p^3]); p1:=PlotPositionVector(R,p=0..1.5, points=[1],normal,tangent, curveoptions=[scaling= constrained,labels=[x,y,z],tickmarks=[2,2,6],axes=frame,orientation=[-40, 70,0]],tangentoptions=[width=.1], normaloptions=[width=.1]); p2:=pointplot3d([-58/11, -85/33, 470/33],symbol=solidsphere,symbolsize=20,color=gold); p3:=display(p1,p2); print(p3); end module: end use:
Figure 2.5.7(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
p,3 p2,p3 = →to position Vector →assign to a nameR
Obtain T2 and N2
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: Student Vector Calculus≻ Frenet Formalism≻Principal Normal≻x
Context Panel: Evaluate at a Point≻p=1
Context Panel: Simplify≻Simplify
R
→tangent vector
→Euclidean-normalize
→evaluate at point
→principal normal
→2-normalize
= simplify
Construct Figure 2.5.7(a)
Control drag T1 and N1
Context Panel: Plots≻Arrow from point≻x= 1,y=3,z=1
Write R Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder Set p∈0,1.5 Options: Constrained Scaling
Copy and paste the arrows onto the graph of C
146463234634646→plot arrow
−95061146−42531146195061146→plot arrow
R = →to listp,3p2,p3→
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≔PositionVectorp,3 p2,p3:
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 p= 2.
N≔simplifyevalPrincipalNormalR,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=2.
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=0..1.5,points= 1,normal,tangent,curveoptions=scaling=constrained,labels=x,y,z,tickmarks=2,2,6,axes=frame,orientation=−40,70,0,tangentoptions=width=.1,normaloptions=width=.1
The principal normal indeed points towards the center of curvature.
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