Chapter 2: Space Curves
Section 2.5: Principal Normal
Example 2.5.10
Obtain and graph the evolute of the ellipse x=2 cosp,y=sinp,p∈0,2 π.
Solution
Mathematical Solution
Figure 2.5.10(a) contains the graphs of the ellipse (red) and its evolute (green).
If R=2 cos(p)sin(p) is the position vector description of the ellipse, then its radius of curvature and principal normal are respectively
r=4−3 cos2p3/2/2
and
N=−14−3 cos2pcos(p)2 sin(p)
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:
Figure 2.5.10(a) Ellipse and its evolute
Hence, the position-vector description for the evolute is
E=R+r N = 2 cos(p)sin(p)−4−3 cos2p3/224−3 cos2pcos(p)2 sin(p) = 32cos3(p)−2 sin3(p)
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
2 cosp,sinp = →to position Vector →assign to a nameR
Obtain the radius of curvature
Write R and press the Enter key.
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Radius of Curvature≻p
Context Panel: Simplify≻Assuming real
Context Panel: Assign to a Name≻r
R
→radius of curvature
−12−3cosp2+43/2csgn13cosp2−4
→assuming real
12−3cosp2+43/2
→assign to a name
r
Obtain the principal normal N
Context Panel: Student Vector Calculus≻Frenet Formalism≻Principal Normal≻p
Context Panel: Assign to a Name≻N
Context Panel: Student Vector Calculus≻ Frenet Formalism≻Principal Normal≻p
Context Panel: Student Vector Calculus≻ Normalize≻Euclidean
Context Panel: Simplify≻Assuming Real
→principal normal
→Euclidean-normalize
N
Obtain the evolute as the position vector R+r N
Write R+r N, the expression for the evolute. Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻E
R+r N = = simplify →assign to a nameE
Obtain the graph of the ellipse and its evolute
Write the name of the position vector. 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: Set color and select Constrained Scaling
Copy the graph of the evolute and paste it onto the graph of the ellipse.
E = →to list32cosp3,−3sinp3→
R = →to list2cosp,sinp→
Maple Solution - Coded
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Use the BasisFormat command to set the display of vectors.
Use the PositionVector command to define the ellipse as a position vector.
R≔PositionVector2 cosp,sinp:
Obtain the principal normal via the PrincipalNormal command with the normalized option. Apply the simplify command to this result.
N≔simplifyPrincipalNormalR,normalized assuming p∷real:
Obtain the radius of curvature via the RadiusOfCurvature command. Apply the simplify command to this result.
r≔simplifyRadiusOfCurvatureR assuming p∷real:
Apply the PositionVector command to obtain the evolute as a position vector.
E≔PositionVectorconvertsimplifyR+r N,list:
Use the PlotPositionVector command to graph the ellipse (in red) and the evolute (in green).
p1≔PlotPositionVectorR,p=0..2 π,curveoptions=color=red:p2≔PlotPositionVectorE,p=0..2 π,curveoptions=color=green:plots:-displayp1,p2,scaling=constrained, labels=x,y
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