Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
Example 3.5.4
At x=1, obtain the equation of the circle of curvature for y=x2.
Show that at x=1, the first and second derivatives for the curve and the circle of curvature agree.
Solution
Part (a): The circle of curvature:
Obtain the circle of curvature
Type yx=… Context Panel: Assign Function
yx=x2→assign as functiony
Type the expression for h1 as per Table 3.5.3. Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻h
1−y.1 1+y.21y..1 = −4→assign to a nameh
Type the expression for k1 as per Table 3.5.3. Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻k
y1+1+y.21y..1 = 72→assign to a namek
Type the expression for R1 as per Table 3.5.3. Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻R
1+y.213/2|y..1| = 525→assign to a nameR
Type the equation of the circle of curvature at x=1 Press the Enter key.
x−h2+y−k2=R2
x+42+y−722=1254
Part (b): Show second-order contact:
Obtain y1,y′1,y″1
Write y1,y′1, and y″1. For each,
Context Panel: Evaluate and Display Inline
y1 = 1
y′1 = 2
y″1 = 2
Show 1,1 is on the circle of curvature
Control-drag the equation of the circle of curvature. Context Panel: Evaluate at a Point≻x=1,y=1
x+42+y−722=1254→evaluate at point1254=1254
For the circle of curvature, evaluate the first derivative at x,y=1,1
Control-drag the equation of the circle of curvature Because y is the function x2, edit: x→u and y→v
Context Panel: Differentiate≻Implicitly Set v,u respectively as dependent and independent variables
Context Panel: Evaluate at a Point≻u=1,v=1
u+42+v−722=1254
→implicit differentiation
−2u+42v−7
→evaluate at point
2
For the circle of curvature, evaluate the second derivative at x,y=1,1
Control-drag the modified equation of the circle of curvature. Press the Enter key.
Context Panel: Differentiate≻Implicitly Set v as the dependent and independent variables in resulting dialog box. Set u,u as the independent variable (threby obtaining the second derivative.)
−24u2+32u+113+4v2−28v8v3−84v2+294v−343
Extra Graphical Insight:
Graph the circle of curvature and trace the evolute
Figure 3.5.4(a) contains a graph of the circle of curvature at 1,1. The green dot is the center of curvature for the point of contact, shown as the black dot.
Figure 3.5.4(b) contains an animation of the circle of curvature (in red) as it rolls along the graph of y=x2. The center of curvature traces the evolute, in green.
p1:=plots:-implicitplot([(x+4)^2+(y-7/2)^2=125/4,y=x^2],x=-10..2,y=-3..10,color=[red,black],gridrefine=2): p2:=plot([[1,1]],style=point,symbol=solidcircle,symbolsize=15,color=black): p3:=plot([[-4,7/2]],style=point,symbol=solidcircle,symbolsize=15,color=green): plots:-display(p1,p2,p3,scaling=constrained);
Figure 3.5.4(a) For y=x2, circle of curvature (red), center of curvature (green dot)
Y:=x^2: H:=unapply(x-diff(Y,x)*(1+diff(Y,x)^2)/diff(Y,x,x),x): K:=unapply(Y+(1+diff(Y,x)^2)/diff(Y,x,x),x): R:=unapply((1+diff(Y,x)^2)^(3/2)/abs(diff(Y,x,x)),x): p1:=plot(Y,x=-2..2,color=black): p2:=plots:-animate(plot,[[H(x),K(x),x=-1..t],x=-1..t,color=green,view=[-10..10,-5..10]],t=-1..1,frames=101,background=p1,digits=3): p3:=plots:-animate(plots:-implicitplot,[(u-H(x))^2+(v-K(x))^2=R(x)^2,u=-10..10,v=-5..10,color=red,gridrefine=2],x=-1..1,frames=101,digits=3): plots:-display(p2,p3,scaling=constrained);
Figure 3.5.4(b) Animation of circle of curvature (red) and evolute (green)
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