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Chapter 2: Space Curves
Section 2.4: Curvature
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Essentials
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Review: Curvature of a Plane Curve
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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.
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Curve
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Curvature
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Format
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Equation
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Cartesian, explicit
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Cartesian, parametric
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Polar
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Table 2.4.1 Formulae for curvature of a plane curve
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For the explicit Cartesian curve , the primes in the formula for represent derivatives with respect to the independent variable . For the parametric curve given in Cartesian coordinates, the overdots represent derivatives with respect to the parameter . For the polar curve given in the form , the primes represent derivatives with respect to the independent variable .
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Most modern calculus texts take the curvature as positive; hence, the absolute values in the numerators of the formulas for (the Greek letter "kappa"). Some older texts, and some applications in the sciences, use a signed curvature that omits this absolute value.
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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.
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Specifically, , where is the angle made by the tangent line and the horizontal, and is the "arc length" or distance along the curve.
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Since , it follows that .
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The differential of the arc length function is obtained from Figure 2.4.2 by approximating the arc length by the hypotenuse of the dotted right triangle: .
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Hence, .
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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]);
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Figure 2.4.1 Angle made by tangent line and horizontal
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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]);
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Figure 2.4.2 Element of arc length
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The calculation of as the derivative of with respect to is then as follows.
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Curvature of a Space Curve
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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.
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Curve
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Curvature
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= =
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Table 2.4.2 Curvature of space curves, with = .
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Center and Circle of Curvature
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The circle of curvature is a circle of radius 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 is called the radius of curvature, and the center of the circle of curvature is called the center of curvature.
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Examples
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Example 2.4.1
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Show that the curvature of the straight line is zero.
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Example 2.4.2
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Show that the circle everywhere has constant curvature, that is, show .
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Example 2.4.3
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Use the appropriate formula from Table 2.4.1 to determine the curvature of , 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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Example 2.4.4
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Obtain and graph the curvature of the cycloid defined by , .
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Example 2.4.5
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Obtain and graph the curvature of the catenary defined by .
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Example 2.4.6
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Obtain the curvature of the helix defined by .
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Example 2.4.7
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Obtain and graph the curvature of the curve defined by .
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Example 2.4.8
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Obtain and graph the curvature of the curve defined by , .
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Example 2.4.9
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Obtain and graph the curvature of the curve defined by .
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Example 2.4.10
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Show the equivalence of the definition = , and the formula .
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