Chapter 2: Space Curves
Section 2.7: Frenet-Serret Formalism
Example 2.7.8
With c=a=b=0 in the prescription in Table 2.7.2, obtain and graph the plane curve whose curvature is κs=coss. Hint: The integrals defining R can only be obtained numerically.
Solution
Maple Solution - Interactive
Obtain θu=∫0uκt dt
Calculus palette: Definite integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻theta
∫0ucost ⅆt = sinu→assign to a nameθ
Obtain the integrals that define the functions xs and ys
Context Panel: Assign Function
Xs=∫0scosθ ⅆu→assign as functionX
Ys=∫0ssinθ ⅆu→assign as functionY
Graph Rs
Launch the . (See Figure 2.7.8(b).) Set s∈0,4 π Options: Constrained Scaling
Figure 2.7.8(a) shows the resulting graph.
Figure 2.7.8(a) Graph of Rs
Figure 2.7.8(b) Interactive Plot Builder main panel
Alternatively, write the list Xs,Ys and press the Enter key.
Context Panel: Plots≻Plot Builder 2-D plot (parametric) Global Options scaling → constrained
Xs,Ys
∫0scossinuⅆu,∫0ssinsinuⅆu
→
Maple Solution - Coded
Apply the int command.
θ≔intcost,t=0..u:
Apply the Int command.
X≔s→Intcosθ,u=0..s:
Y≔s→Intsinθ,u=0..s:
Apply the plot command with the appropriate syntax for a parametrically defined curve.
Note that since xs and ys are defined as functions, there is no need to reference the independent variable in the command.
plotX,Y,0..4 π,labels=x,y,scaling=constrained
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