Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
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Essentials
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Curvature
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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.
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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 3.5.1 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 3.5.1.
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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 3.5.2 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 3.5.1 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 3.5.2 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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Second-Order Contact
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The graphs of two functions and make second-order contact at if the values of and , and their first two derivatives, agree at . Table 3.5.2 lists these three conditions as equations, and provides amusing interpretations for this degree of contact between two curves.
Analytic Condition
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Humorous Interpretation
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Curves touch
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Curves kiss
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Curves hug
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Table 3.5.2 Conditions for second-order contact
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Center and Circle of Curvature
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The center of curvature for a plane curve that is the graph of 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 is , the radius of curvature is .
Table 3.5.3 lists formulas for , the coordinates of the center of curvature, and for the radius of curvature.
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Table 3.5.3 Center and radius of curvature
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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 is called the evolute of . The curve is called the involute.
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Examples
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Example 3.5.1
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Show that the curvature of the straight line is zero.
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Example 3.5.2
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Show that the circle everywhere has constant curvature, that is, show .
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Example 3.5.3
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Obtain and graph the curvature for .
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Example 3.5.4
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a)
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At , obtain the equation of the circle of curvature for .
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b)
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Show that at , the first and second derivatives for the curve and the circle of curvature agree.
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Example 3.5.5
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Obtain the evolute for , the graph of , and show that it is the locus of the center of curvature.
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Example 3.5.6
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Use the appropriate formula from Table 3.5.1 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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