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Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
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Example 3.5.5
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Obtain the evolute for , the graph of , and show that it is the locus of the center of curvature.
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Solution
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Mathematical Solution
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The animation in Figure 3.5.5(a) shows the graph of , the parabola , in black. Its evolute is shown in red. The slider in the animation toolbar changes the point of contact of the circle of curvature and the parabola. This point of contact is a blue dot. Another blue dot marks the center of curvature, the center of the circle of curvature.
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The (varying) radius of curvature is , where is the -coordinate of the point of contact of the circle of curvature with the parabola. (See Table 3.5.3.)
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Similarly, the coordinates of the center of curvature are
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>
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use plots in
module()
local p1,p2,p3,p4,p5,p6;
p1:=plot(x^2,x=-2..2,color=black):
p2:=implicitplot(-16*y^3+27*x^2+24*y^2-12*y+2,x=-3..3,y=0..4,color=red,gridrefine=3):
p3:=display(p1,p2,scaling=constrained):
p4:=animate(implicitplot,[-3*a^4+8*a^3*x-6*a^2*y+x^2+y^2-y,x=-5..5,y=-5..5,gridrefine=3,color=green,scaling=constrained],a=-2/3..2/3,background=p3,paraminfo=false):
p5:=animate(plot,[[[[-4*a^3,3*a^2+1/2]],[[a,a^2]]],style=point,symbol=solidcircle,symbolsize=15,color=blue],a=-2/3..2/3,paraminfo=false):
p6:=display(p4,p5);
print(p6);
end module:
end use:
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Figure 3.5.5(a) Animation: Curve (black) and evolute (red)
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The equations then provide a parametric representation of the evolute. Eliminating from these two equations gives the implicit representation of the evolute as
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The equation of the circle of curvature, , can be put into the form
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The equation for the circle of curvature changes as , the point of contact of the circle and the parabola, changes.
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Maple Solution
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Although Table 3.5.3 is given in terms of a curve defined by , let the parabola be given by so that the letter will remain unassigned.
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Define the curve
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Write
Context Panel: Assign Function
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Obtain and as per Table 3.5.3
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Write the expression for .
Context Panel: Simplify≻Simplify
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Context Panel: Assign to a Name≻
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Write the expression for .
Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻
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=
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Write the expression for .
Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻
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=
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Obtain the parametric and then the implicit representations of the evolute
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Write the parametric equations of the evolute; press the Enter key.
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Context Panel: Solve≻Eliminate a Variable≻
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Obtain the equation of the circle of curvature
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Write the equation of the circle in the form shown at the right.
Press the Enter key.
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Context Panel: Simplify≻Simplify
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The denominator in the expression for , the radius of curvature, is so that the radius is positive. For , the second derivative is the constant 2, which is clearly positive. Hence, the absolute value in the expression for was deleted.
Maple's form for the equation of the circle of curvature must be manipulated by hand to obtain the form displayed in the Mathematical Solution.
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