Chapter 2: Space Curves
Section 2.5: Principal Normal
Example 2.5.5
At x=1/3 on the graph of C, the upper semicircle defined by x2+y2=1, compute N. Graph C, along with N1/3 and T1/3. Does N point towards the center of curvature? Hint: The curvature of a circle was obtained in Example 2.4.2.
Solution
Mathematical Solution
Write the position vector as R=x1−x2 so that R′=1−x1−x2 and ρ=1/1−x2. Then
T=1−x2−x, dTds=− x−1−x2, κ=1, N=−x−1−x2
Evaluating at x=1/3 gives T1/3=132−1 and N1/3=−1312.
Note that N can be obtained from T by interchanging components and negating the second component to place N to the right of T.
The center of curvature for the point P:1/3,2/3 is given by R1/3+N1/3, that is, by
1312− 1312 = 00
Hence, the center of curvature is to the right of the point P.
In Figure 2.5.5(a), T1/3 is represented by the black arrow; and N1/3, by the green.
use Student:-VectorCalculus in module() local R,p1; R:=PositionVector([x,sqrt(1-x^2)]); p1:=PlotPositionVector(R,x=-1..1, points=[1/sqrt(3)],normal,tangent, curveoptions=[scaling= constrained,labels=[x,y], size=[300,300]], tangentoptions=[width=.05], normaloptions=[width=.05]); print(p1); end module: end use:
Figure 2.5.5(a) Graph of C, T1/3,N1/3
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
x,1−x2 = →to position Vector →assign to a nameR
Obtain T1/3 and N1/3
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: Evaluate at a Point≻x=1/3
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Principal Normal≻x
R
→tangent vector
→2-normalize
→evaluate at point
→principal normal
Construct Figure 2.5.5(a)
Control drag T1/3 and N1/3
Context Panel: Plots≻Arrow from point≻x=1/3,y=2/3
Write R Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder Set x∈−1,1 Options: Constrained Scaling
Copy and paste the arrows onto the graph of C
1332−133→plot arrow
−133−1332→plot arrow
R = →to listx,−x2+1→
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,1−x2:
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 x=1/3.
N≔evalPrincipalNormalR,normalized,x=1/3:
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 x=1/3.
T≔evalTangentVectorR,normalized,x=1/3:
Use the PlotPositionVector command to graph C, along with the tangent and principal normal vectors at the single point x=1/3.
PlotPositionVectorR,x=−1..1,points=1/3,normal,tangent,curveoptions=scaling=constrained,labels=x,y,size=300,300,tangentoptions=width=.05,normaloptions=width=.05
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 second component so that N points to the right of T.
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