Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
Essentials
Curvature
The curvature of a plane curve is a measure of how "curved" it is at each of its points. Table 3.5.1 lists formulas for the calculation of curvature of curves given in various formats.
Curve
Format
Equation
Cartesian, explicit
y=yx
κ=|y″|1+y′23/2
Cartesian, parametric
x=xt,y=yt
κ=x. y..−y. x..x.2+y.23/2
Polar
r=rθ
κ=|r2+2 r′2−r r″|r2+r′23/2
Table 3.5.1 Formulae for curvature of a plane curve
For the explicit Cartesian curve y=yx, the primes in the formula for κ represent derivatives with respect to the independent variable x. For the parametric curve given in Cartesian coordinates, the overdots represent derivatives with respect to the parameter t. For the polar curve given in the form r=rθ, the primes represent derivatives with respect to the independent variable θ.
Most modern calculus texts take the curvature as positive; hence, the absolute values in the numerators of the formulas for κ (the Greek letter "kappa"). Some older texts, and some applications in the sciences, use a signed curvature that omits this absolute value.
Curvature is a measure of the rate at which the tangent line turns as the point of contact moves along the curve. See Figure 3.5.1.
Specifically, κ=dθds, where θ is the angle made by the tangent line and the horizontal, and s=sx is the "arc length" or distance along the curve.
Since y′=tanθ, it follows that θ=arctany′.
The differential of the arc length function is obtained from Figure 3.5.2 by approximating the arc length s by the hypotenuse of the dotted right triangle: ds=dx2+dy2=dx1+dydx2=dx 1+y′2.
Hence, dsdx=1+y′2.
p1 := plot([x^2,Student:-Calculus1:-Tangent(x^2,1)],x=0..2, color=[red,blue], view=[0..1.5,0..2.5]): p2 := plots:-textplot([.65,.11,q], font=[SYMBOL,12]): p3 := plot([[1,1]],style=point,symbol=solidcircle,symbolsize=15,color=green): plots:-display([p1,p2,p3], scaling=constrained, tickmarks=[[0,2],[0,3]], labels=[x,y]);
Figure 3.5.1 Angle made by tangent line and horizontal
p1:=plot(sqrt(x),x=0..4): p2:=plot([[1,1],[3,sqrt(3)]],color=black,linestyle=dot): p3:=plot([[1,1],[3,1]],color=black,linestyle=dot): p4:=plot([[3,1],[3,sqrt(3)]],color=black,linestyle=dot): p5:=plots:-textplot({[2,.85,typeset(dx)],[3.15,1.25,typeset(dy)],[2.2,1.3,typeset(ds)]}): p6:=plot([[[1,1],[3,sqrt(3)]]],style=point,symbol=solidcircle,color=green,symbolsize=15): plots:-display(p||(1..6),scaling=constrained,labels=[x,y],tickmarks=[0,0]);
Figure 3.5.2 Element of arc length
The calculation of κ as the derivative of θ with respect to s is then as follows.
dθds
=dds arctany′
=ddxarctany′ dxds
=y″1+y′2 1ds/dx
=y″1+y′2 11+y′2
=y″1+y′23/2
Second-Order Contact
The graphs of two functions f and g make second-order contact at x=a if the values of f and g, and their first two derivatives, agree at x=a. Table 3.5.2 lists these three conditions as equations, and provides amusing interpretations for this degree of contact between two curves.
Analytic Condition
Humorous Interpretation
fa=ga
Curves touch
f′a=g′a
Curves kiss
f″a=g″a
Curves hug
Table 3.5.2 Conditions for second-order contact
Center and Circle of Curvature
The center of curvature for a plane curve that is the graph of y=fx is the center of the circle of curvature, the circle that makes second-order contact with the plane curve. The radius of the circle of curvature is the radius of curvature. Because the curvature of a circle of radius r is κ=1/r, the radius of curvature is R=1/κ.
Table 3.5.3 lists formulas for h,k, the coordinates of the center of curvature, and for the radius of curvature.
h=a−y.a1+y.2ay..a
k=ya+1+y.2ay..a
R=1+y.2a3/2|y..a|
Table 3.5.3 Center and radius of curvature
The overdots represent differentiation with respect to the independent variable; because some of these derivatives are squared, this notation is used in place of the prime.
The trajectory traced by the center of curvature as the circle of curvature traverses the curve C is called the evolute of C. The curve C is called the involute.
Examples
Example 3.5.1
Show that the curvature of the straight line y=m x+b is zero.
Example 3.5.2
Show that the circle x−h2+y−k2=r2 everywhere has constant curvature, that is, show κ=1/r.
Example 3.5.3
Obtain and graph the curvature κx for y=x2.
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.
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.
Example 3.5.6
Use the appropriate formula from Table 3.5.1 to determine the curvature of yx=x3/2,x≥0, then obtain the curvature from first principles, that is, by calculating the rate at which the tangent turns as arc length increases.
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