Chapter 2: Space Curves
Section 2.4: Curvature
Essentials
Review: Curvature of a Plane Curve
The curvature of a plane curve is a measure of how "curved" it is at each of its points. Table 2.4.1 lists formulas for the calculation of curvature of curves given in various formats.
Curve
Curvature
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 2.4.1 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 2.4.1.
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 2.4.2 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 2.4.1 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 2.4.2 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
Curvature of a Space Curve
Table 2.4.2 lists the definition of curvature of a space curve, and provides a more convenient computational expression for it. The first expression is the definition: curvature is the measure of how the unit tangent vector varies with arc length along the curve. Since it is rare to have arc length as the parameter along a curve, the alternate for computing the length of the derivative of the unit tangent vector is afforded by the chain rule.
The second expression on the right in Table 2.4.2 is an alternate method for calculating curvature. Example 2.4.10 shows that the two expressions for curvature are equivalent.
R=xp i+yp j+zp k
κ = T′s = dTdp1ρ
κ=∥R′×R″∥ρ3
Table 2.4.2 Curvature of space curves, with ρ=∥R′∥ = dsdp.
Center and Circle of Curvature
The circle of curvature is a circle of radius 1/κ that is tangent to a curve, and makes second-order contact with the curve. Second-order contact means that the first and second derivatives agree at the point. The radius 1/κ is called the radius of curvature, and the center of the circle of curvature is called the center of curvature.
Examples
Example 2.4.1
Show that the curvature of the straight line y=m x+b is zero.
Example 2.4.2
Show that the circle x−h2+y−k2=r2 everywhere has constant curvature, that is, show κ=1/r.
Example 2.4.3
Use the appropriate formula from Table 2.4.1 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.
Example 2.4.4
Obtain and graph the curvature of the cycloid defined by x= p−sinp,y= 1−cosp, p∈0,2 π.
Example 2.4.5
Obtain and graph the curvature of the catenary defined by y=coshx.
Example 2.4.6
Obtain the curvature of the helix defined by Rp=cosp i+sinp j+p k.
Example 2.4.7
Obtain and graph the curvature of the curve defined by Rp=p i+3 p2 j+p3 k.
Example 2.4.8
Obtain and graph the curvature of the curve defined by Rp=lncosp i+lnsinp j+2p k, p∈0,π/2.
Example 2.4.9
Obtain and graph the curvature of the curve defined by Rp=3 p−p3 i+3 p2 j+3 p+p3 k.
Example 2.4.10
Show the equivalence of the definition κ = T′s, and the formula κ=R′p×R″p/ρ3.
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