Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
|
Example 3.5.4
|
|
a)
|
At , obtain the equation of the circle of curvature for .
|
b)
|
Show that at , 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
Context Panel: Assign Function
|
|
|
•
|
Type the expression for as per Table 3.5.3.
Context Panel: Evaluate and Display Inline
|
•
|
Context Panel: Assign to a Name≻
|
|
=
|
•
|
Type the expression for as per Table 3.5.3.
Context Panel: Evaluate and Display Inline
|
•
|
Context Panel: Assign to a Name≻
|
|
=
|
•
|
Type the expression for as per Table 3.5.3.
Context Panel: Evaluate and Display Inline
|
•
|
Context Panel: Assign to a Name≻
|
|
=
|
•
|
Type the equation of the circle of curvature at
Press the Enter key.
|
|
|
|
|
Part (b): Show second-order contact:
Obtain
|
•
|
Write , and . For each,
|
•
|
Context Panel: Evaluate and Display Inline
|
|
=
=
=
|
Show is on the circle of curvature
|
•
|
Control-drag the equation of the circle of curvature.
Context Panel: Evaluate at a Point≻
|
|
|
For the circle of curvature, evaluate the first derivative at
|
•
|
Control-drag the equation of the circle of curvature
Because is the function , edit: and
|
•
|
Context Panel: Differentiate≻Implicitly
Set respectively as dependent and independent variables
|
•
|
Context Panel: Evaluate at a Point≻
|
|
|
For the circle of curvature, evaluate the second derivative at
|
•
|
Control-drag the modified equation of the circle of curvature.
Press the Enter key.
|
•
|
Context Panel: Differentiate≻Implicitly
Set as the dependent and independent variables in resulting dialog box.
Set as the independent variable (threby obtaining the second derivative.)
|
•
|
Context Panel: Evaluate at a Point≻
|
|
|
|
|
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 . 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 . 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 , 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)
|
|
|
|
|
|
|
|
<< Previous Example Section 3.5
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
|