Chapter 2: Space Curves
Section 2.5: Principal Normal
Example 2.5.6
At p=π/3 on the graph of C, the helix defined by Rp=cosp i+sinp j+p k, compute N. Graph C, along with Nπ/3 and Tπ/3. Does N point towards the center of curvature? Hint: The curvature of C was obtained in Example 2.4.6.
Solution
Mathematical Solution
Write the position vector as R=cos(p)sin(p)p so that R′=−sin(p)cos(p)1 and ρ=2. Then
T=12−sin(p)cos(p)1, dTds=−12cos(p)sin(p)0, κ=12, N=−cos(p)sin(p)0
Evaluating at p=π/3 gives Tπ/3=12−3/21/21 and Nπ/3=−12130.
The center of curvature for the point 1/2,3/2,π/2 is given by
R+N/κx=a|f(x)p=π/3= −1/2−3/2π/3
In Figure 2.5.6(a), Tπ/3 is represented by the black arrow; and Nπ/3, by the green. The center of curvature is shown as the gold dot, a visual clue that Nπ/3 points toward the center of curvature.
use plots, Student:-VectorCalculus in module() local R,p1,p2,p3; R:=PositionVector([cos(p),sin(p),p]); p1:=PlotPositionVector(R,p=0..2*Pi, points=[Pi/3],normal,tangent, curveoptions=[scaling= constrained,labels=[x,y,z],tickmarks=[2,2,6],view=[-1..1,-1.. 1.5,0..2*Pi],axes=frame,orientation=[-40, 70,0]],tangentoptions=[width=.1], normaloptions=[width=.1]); p2:=pointplot3d([-1/2,-sqrt(3)/2,Pi/3],symbol=solidsphere,symbolsize=20,color=gold); p3:=display(p1,p2); print(p3); end module: end use:
Figure 2.5.6(a) Graph of C, Tπ/3,Nπ/3
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
cosp,sinp,p = →to position Vector →assign to a nameR
Obtain Tπ/3 and Nπ/3
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= π/3
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Principal Normal≻x
Context Panel: Evaluate at a Point≻p=π/3
R
→tangent vector
→Euclidean-normalize
→evaluate at point
→principal normal
→2-normalize
Construct Figure 2.5.6(a)
Control drag Tπ/3 and Nπ/3
Context Panel: Plots≻Arrow from point≻x= 1/2,y=3/2,z=π/3
Write R Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder Set p∈0,2 π Options: Constrained Scaling
Copy and paste the arrows onto the graph of C
−1423142122→plot arrow
−12−1230→plot arrow
R = →to listcosp,sinp,p→
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≔PositionVectorcosp,sinp,p:
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= π/3.
N≔evalPrincipalNormalR,normalized,p= π/3:
Use the TangentVector command with the normalized option to obtain the general tangent vector along C.
Use the eval command to obtain the tangent vector at =2 π/3.
T≔evalTangentVectorR,normalized,p= π/3:
Use the PlotPositionVector command to graph C along with the tangent and principal normal vectors at the single point p= π/3.
PlotPositionVectorR,p=0..2 π,points= π/3,normal,tangent,curveoptions=scaling=constrained,labels=x,y,z,tickmarks=2,2,6,view=−1..1,−1..1.5,0..2 π,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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