Chapter 2: Space Curves
Section 2.5: Principal Normal
Example 2.5.3
At p=2 π/3 on the graph of C, the cycloid defined by x= p−sinp,y= 1−cosp, p∈0,2 π, compute N. Graph C, along with N2 π/3 and T2 π/3. Does N point towards the center of curvature? Hint: The curvature of C was obtained in Example 2.4.4.
Solution
Mathematical Solution
Write the position vector as R=p−sin(p)1−cos(p) so that R′=1−cos(p)sin(p) and ρ=2−2 cosp. Then
T=12−2 cosp1−cos(p)sin(p), dTds=−14sin(p)cos(p)−11 κ=122−2 cosp, N=12−2 cospsin(p)cos(p)−1
Evaluating at p=2 π/3 gives T2 π/3=1231 and N2 π/3=121−3.
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:2π/3− 3/2,3/2 is given by
R+N/κx=a|f(x)p=2 π/3= 2π/3+3/2−3/2
Hence, the center of curvature is to the right of the point P.
In Figure 2.5.3(a), T2 π/3 is represented by the black arrow; and N2 π/3, by the green.
use Student:-VectorCalculus in module() local R,p1; R:=PositionVector([p-sin(p),1-cos(p)]); p1:=PlotPositionVector(R,p=0..2*Pi, points=[2*Pi/3],normal,tangent, curveoptions=[scaling= constrained,labels=[x,y], size=[300,300]], tangentoptions=[width=.1], normaloptions=[width=.1]); print(p1); end module: end use:
Figure 2.5.3(a) Graph of C, T2 π/3,N2 π/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
p−sinp,1−cosp = →to position Vector →assign to a nameR
Obtain T2 π/3 and N2 π/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=2 π/3
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Principal Normal≻x
R
→tangent vector
→Euclidean-normalize
→evaluate at point
→principal normal
→2-normalize
Construct Figure 2.5.3(a)
Control drag T2 π/3 and N2 π/3
Context Panel: Plots≻Arrow from point≻x=2 π/3− 3/2,y=3/2
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 π Options: Constrained Scaling
Copy and paste the arrows onto the graph of C
12312→plot arrow
12−123→plot arrow
R = →to listp−sinp,1−cosp→
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−sinp,1−cosp:
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 π/3.
N≔evalPrincipalNormalR,normalized,p=2 π/3:
Use the TangentVector command with the normalized option to obtain the general tangent vector along the curve.
Use the eval command to obtain the tangent vector at p=2 π/3.
T≔evalTangentVectorR,normalized,p=2 π/3:
Use the PlotPositionVector command to graph C along with the tangent and principal normal vectors at the single point p=2 π/3.
PlotPositionVectorR,p=0..2 π,points=2 π/3,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 second component so that N points to the right of T.
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