The Calculus Primer (2011)
Part VIII. Curvature
Chapter 31. CIRCLE OF CURVATURE
8—10.Radius of Curvature. The radius of curvature of a curve at a given point may be defined as the reciprocal of the curvature of the curve at that point. Thus, if the radius of curvature is represented by R, then
Then we may at once write:
It should be noted that while the curvature K was defined as the absolute value of the respective fractions equivalent to , K may be positive or negative. Hence R may also be positive or negative, and will have the same sign as K.If R is positive, the curve is concave upwards at the particular point in question; if R is negative, the curve is concave downwards at that point.
The radius of curvature may be thought of as the measure of the flatness or sharpness of a curve at a point; the smaller the radius of curvature, the sharper the curve.
The curvature of a curve at a point is the rate at which the inclination of the curve is changing with respect to the length of arc, that is, curvature = .
The reader will see, from the following proof, why we take R equal to . Let R = CP be the required radius of curvature at point P on the curve AB; TU and MN are tangents to the curve at P and Q, respectively; C′P and C′Q are the respective normals; angle Δω is the angle between the normals, and therefore equals Δω, the angle between the tangents (sides of the angles are respectively perpendicular); and arc PQ = Δs.
In triangle PC′Q, by the law of sines,
As Q approaches P, Δs → 0; in passing to the limiting value of C′P (that is CP, or R, the radius of curvature), we note that:
(1)chord PQ may be replaced by As,
(2)sin Aw may be replaced by Aw, and
(3)limit of sin Q = 1, since angle Q is approaching 90°.
8—11.Circle of Curvature. At a given point on a curve, there is, in general, but one tangent and one normal. An infinite number of circles, all having their centers lying on the normal, can be drawn through this point. Of these circles, the one whose radius equals the radius of curvature for the curve at that point is called the circle of curvature for the point. Obviously, each point on the curve has a different circle of curvature (except in special cases such as when the curve itself is a circle). It can be shown that the circle of curvature at any point “fits” the curve more closely, near that point, than any other circle.
The circle of curvature is also known as the osculating circle. In general, the circle of curvature of a curve at a point crosses the curve at that point. The center of the circle of curvature is known as the center of curvature.
EXAMPLE 1.Find the radius of curvature of the ellipse at the extremity of the minor axis, that is, at (0,2).
Solution.4x2 + 9y2 = 36.
hence, at point (0,2) the value of = 0.
since, at point (0,2) y = 2 and = 0, the value of at this Point is − .
NOTE 1. Since R is negative, the curve at (0,2) is concave downwards.
NOTE 2. It will be observed that the numerical value of R, or 4, is comparatively large, indicating that the curve at this point is fairly “flat.”
NOTE 3. If we wish to determine the value of R at the extremity of the major axis, we should find that the value of at that point is infinite. In this case, we would interchange the axes, transforming the equation to and the extremity in question becomes (0,3). See below, Exercise 8—2, Problem 11.
Find the radius of curvature for each of the following at the point indicated; in each case sketch the circle of curvature:
1. y = x3 − 4x2 + 3x; (0,0)
2. xy = 20; (4,5)
3. a2x2 + b2y2 = a2b2; (0,a)
4. x2 = 2py; (0,0)
5. y = sin x;
6. y = tan x;
Find the radius of curvature of the following curves at any point:
7. y = x3
8. ρ = a cos θ
9. y2 = x3
10. ρ = 1 − cos θ
11. Find the radius of curvature of the ellipse = 1 at the extremity of the major axis (see NOTE 3, Illustrative Example 1, §8—11). Compare the numerical values of the radii of curvature at the extremities of the two axes; what does this show about the comparative curvature at these two points?
12. Find the radius of curvature of the witch, x2y = 8 − 4y, at the point (0,2).
8—12.Radius of Curvature of Curves with Equations in Parametric Form. We recall from §6—7 that when equations are given in parametric form, such as x = f(t), y = ø(t), then
Substituting (1) and (2) in the formula for the radius of curvature, §8—10, equation , and simplifying, we obtain:
EXAMPLE 1.Find the value of the radius of curvature of the curve x = t2, y = 2t, at the point where t = 1.
EXAMPLE 2.Find the radius of curvature of x = 2 sin t, y = cos t, at the point where t =
8—13.The Center of Curvature. Let P(x,y) be any point on a given curve y = f(x); let t be the tangent to the curve at P. Assume that the curve lies entirely on one side of the tangent. Along the normal, toward the concave side of the curve, lay off the distance PQ, equal to the radius of curvature R at P. Thus Q, the center of the circle of curvature of the given curve for the point P, is called the center of curvature with respect to point P.
Without proof, we state the coordinates (α,β) of the center of curvature in terms of the coordinates (x,y) of P:
EXAMPLE. Find the coordinates of the center of curvature of the curve
at the point where x = 0.
At the point where x = 0, y = 1.
Hence, substituting in equations  and ,