Chapter 2: Space Curves
Section 2.4: Curvature
Example 2.4.4
Obtain and graph the curvature of the cycloid defined by x= p−sinp,y= 1−cosp, p∈0,2 π.
Solution
Define the functions xp and yp
Type xp=… Context Panel: Assign Function
xp=p−sinp→assign as functionx
Type yp=… Context Panel: Assign Function
yp=1−cosp→assign as functiony
Obtain and graph the curvature
Write the formula for curvature.
Context Panel: Simplify≻Assuming Real
x′py″p−x″py′px′p2+y′p23/2 = −1−cospcosp+sinp21−cosp2+sinp23/2→assuming real122−2cosp
Figure 2.4.4(a) contains a graph of κp, the curvature function for x=xp,y=yp.
Figure 2.4.4(a) can be obtained interactively via the Context Panel option Plots≻Plot Builder.
Figure 2.4.4(a) Curvature of x=xp,y=yp
An alternative route to the curvature is via the Curvature command in the Student VectorCalculus package, provided the curve is represented as a position vector. The easiest access to this functionality is then through the Context Panel. The one hurdle to overcome is the default format for the display of vectors within this package.
Initialize
Tools≻Load Package: Student Vector Calculus
Apply the BasisFormat command.
Loading Student:-VectorCalculus
BasisFormatfalse:
Write R=… as per Table 1.1.1.
Context Panel: Assign Name
R=xp,yp→assign
Obtain the curvature
Write R
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Frenet Formalism≻Curvature≻p
R = →curvature−1−cospsinp2−2cosp3/2+sinp2−2cosp2+−sinp22−2cosp3/2+cosp2−2cosp22−2cosp→assuming real122−2cosp
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