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Chapter 2: Space Curves
Section 2.5: Principal Normal
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Example 2.5.3
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At on the graph of , the cycloid defined by , , compute N. Graph , along with and . Does N point towards the center of curvature? Hint: The curvature of was obtained in Example 2.4.4.
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Solution
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Mathematical Solution
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Write the position vector as so that and . Then
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Evaluating at gives and .
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Note that N can be obtained from T by interchanging components and negating the second component to place N to the right of T.
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The center of curvature for the point is given by
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Hence, the center of curvature is to the right of the point P.
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In Figure 2.5.3(a), is represented by the black arrow; and , by the green.
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>
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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:
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Figure 2.5.3(a) Graph of ,
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Vector Calculus
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Loading Student:-VectorCalculus
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Execute the BasisFormat command at the right, or use the
task template.
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Define as the position vector R
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Enter the vector notation for as per Table 1.1.1.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Conversions≻To Position Vector
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Context Panel: Assign to a Name≻R
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Obtain and
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Write R and press the Enter key.
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Context Panel: Student Vector Calculus≻
Frenet Formalism≻Tangent Vector≻
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Context Panel: Student Vector Calculus≻
Normalize≻Euclidean
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Context Panel: Evaluate at a Point≻
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Write R and press the Enter key.
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Context Panel: Student Vector Calculus≻
Frenet Formalism≻Principal Normal≻
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Context Panel: Student Vector Calculus≻
Normalize≻Euclidean
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Context Panel: Evaluate at a Point≻
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Construct Figure 2.5.3(a)
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Control drag and
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Context Panel: Plots≻Arrow from point≻
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Write R
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Conversions≻To List
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Context Panel: Plots≻Plot Builder
Set
Options: Constrained Scaling
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Copy and paste the arrows onto the graph of
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Maple Solution - Coded
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Install the Student Vector Calculus package.
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Use the BasisFormat command to set the display of vectors.
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Use the PositionVector command to define as the position vector R.
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Use the PrincipalNormal command with the normalized option to obtain the general principal normal vector.
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Use the eval and simplify commands to obtain the principal normal vector at .
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Use the TangentVector command with the normalized option to obtain the general tangent vector along the curve.
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Use the eval command to obtain the tangent vector at .
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Use the PlotPositionVector command to graph along with the tangent and principal normal vectors at the single point .
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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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<< Previous Example Section 2.5
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