Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
Example 3.5.5
Obtain the evolute for C, the graph of yx=x2, and show that it is the locus of the center of curvature.
Solution
Mathematical Solution
The animation in Figure 3.5.5(a) shows the graph of C, the parabola y=x2, 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.
The (varying) radius of curvature is R=124a2+13/2, where x=a is the x-coordinate of the point of contact of the circle of curvature with the parabola. (See Table 3.5.3.)
Similarly, the coordinates of the center of curvature are
h,k=−4 a3,3a2+1/2
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:
Figure 3.5.5(a) Animation: Curve C (black) and evolute (red)
The equations x=h,y=k then provide a parametric representation of the evolute. Eliminating a from these two equations gives the implicit representation of the evolute as
16y3−24y2+12y=27 x2+2
The equation of the circle of curvature, x−h2+y−k2=R2, can be put into the form
x2+y2+8 x a3−y 6a2+1−3a4=0
The equation for the circle of curvature changes as a,a2, the point of contact of the circle and the parabola, changes.
Maple Solution
Although Table 3.5.3 is given in terms of a curve C defined by y=yx, let the parabola be given by gx=x2 so that the letter y will remain unassigned.
Define the curve C
Write gx=… Context Panel: Assign Function
gx=x2→assign as functiong
Obtain h,k and R as per Table 3.5.3
Write the expression for h. Context Panel: Simplify≻Simplify
Context Panel: Assign to a Name≻h
a−g′a 1+g′a2g″a= simplify −4a3→assign to a nameh
Write the expression for k. Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻k
ga+1+g′a2g″a = 3a2+12→assign to a namek
Write the expression for R. Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻R
1+g′a23/2g″a = 124a2+13/2→assign to a nameR
Obtain the parametric and then the implicit representations of the evolute
Write the parametric equations of the evolute; press the Enter key.
Context Panel: Solve≻Eliminate a Variable≻a
x=h,y=k
x=−4a3,y=3a2+12
→eliminate a
a=−3x4y−2,−16y3+27x2+24y2−12y+2
Obtain the equation of the circle of curvature
Write the equation of the circle in the form shown at the right. Press the Enter key.
Context Panel: Simplify≻Simplify
x−h2+y−k2−R2=0
4a3+x2+y−3a2−122−144a2+13=0
= simplify
−3a4+8a3x−6a2y+x2+y2−y=0
The denominator in the expression for R, the radius of curvature, is y″a so that the radius is positive. For gx=x2, the second derivative is the constant 2, which is clearly positive. Hence, the absolute value in the expression for R 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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