Chapter 2: Space Curves
Section 2.5: Principal Normal
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Example 2.5.10
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Obtain and graph the evolute of the ellipse .
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Solution
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Mathematical Solution
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Figure 2.5.10(a) contains the graphs of the ellipse (red) and its evolute (green).
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If is the position vector description of the ellipse, then its radius of curvature and principal normal are respectively
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and
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use plots, Student:-VectorCalculus in
module()
local p1,p2,p3,R,r,N,E;
R:=PositionVector([2*cos(p),sin(p)]);
r:=RadiusOfCurvature(R);
N:=PrincipalNormal(R,normalized);
E:=PositionVector(convert(R+r*N,list));
p1:=PlotPositionVector(R,p=0..2*Pi,curveoptions=[color=red]);
p2:=PlotPositionVector(E,p=0..2*Pi,curveoptions=[color=green]);
p3:=display(p1,p2,scaling=constrained,labels=[x,y]);
print(p3);
end module:
end use:
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Figure 2.5.10(a) Ellipse and its evolute
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Hence, the position-vector description for the evolute is
= =
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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 the radius of curvature
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Write R and press the Enter key.
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Context Panel: Student Vector Calculus≻
Frenet Formalism≻Radius of Curvature≻
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Context Panel: Simplify≻Assuming real
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Context Panel: Assign to a Name≻
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Obtain the principal normal N
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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: Assign to a Name≻N
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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: Simplify≻Assuming Real
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Context Panel: Assign to a Name≻N
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Obtain the evolute as the position vector
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Write , the expression for the evolute.
Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Assuming Real
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Context Panel: Assign to a Name≻E
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Obtain the graph of the ellipse and its evolute
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Write the name of the position vector.
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: Set color and select Constrained Scaling
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Copy the graph of the evolute and paste it onto the graph of the ellipse.
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=
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=
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Maple Solution - Coded
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Install the Student VectorCalculus 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 the ellipse as a position vector.
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Obtain the principal normal via the PrincipalNormal command with the normalized option. Apply the simplify command to this result.
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Apply the PositionVector command to obtain the evolute as a position vector.
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Use the PlotPositionVector command to graph the ellipse (in red) and the evolute (in green).
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