Chapter V.

Applications

After dealing with a few preliminaries, we shall show in this chapter how what we can apply what we have learnt so far in many ways in geometry and physics .

5.1 Representation of Curves

5.1.1 Parametric representation: As we have seen in 1.2.3, when we represent a curve by means of an equation y = f(x), we must always restrict ourselves to a single-valued branch. Hence it is often more convenient - when dealing, in particular, with a closed curve - to introduce other analytical methods of representation. The most general and at the same time the most useful representation of a curve is parametric representation. Instead of considering one of the rectangular co-ordinates as a function of the other, we think of both the co-ordinates x and y as functions of a third independent variable t, the socalled parameter; the point with the co-ordinates x and y then describes the curve as t traverses a definite interval. Such parametric representations have already been encountered. For example, for the circle x²+y² = a², we obtain a parametric representation in the form

x = acos t, y =asin t.

Here, as we know already, t has the geometrical meaning of an angle at the centre of the circle. For the ellipse x²/a² + y²/b² = 1 we likewise have the parametric representation x = acos t, y = bsin t, where t is the solaced eccentric angle, that is, the angle at the centre corresponding to the point of the circumscribed circle lying vertically above or below the point P (acos t, bsin t) of the ellipse (Fig. 1). In both these cases, the point with the co-ordinates x, y describes the complete circle or ellipse as the parameter t traverses the interval from 0 to 2p.

In general, we can seek to represent a curve parametrically by taking

that is, by considering two functions of a parameter t; the shorter notation x(t) and y(t) will henceforth be used whenever there is no danger of confusion. For a given curve, these two functions f (t) and y (t) must be determined in such a way that the totality of pairs of functional values x(t) and y(t) corresponding to a given interval of values of t yields all the points on the curve and no points which are not on the curve. If a curve, in the first instance, is given in the form y=f(x), we can arrive at a representation of this kind by first writing x=f(t), where f (t) is any continuous monotonic function which in a definite interval passes exactly once through each of the values of x in question; it then follows that y = f{f(t)}, that is, the second function y(t) is determined by compounding f and f. We thus see that, owing to the arbitrariness in the choice of the function f, we have a great deal of freedom in representing a given curve parametrically; in particular, we may actually take t=x and thus think of the original representation y = f(x) as a parametric representation with the parameter t = x.

The advantage of the parametric representation is that this arbitrariness may be utilized for purposes of simplification. For example, we represent the curve by taking x = t³, y = t², so that f (t) = t³, y (t) = t². The point with the co-ordinates x,y will then describe the entire curve (semi-cubical parabola) as t varies from -¥ to +¥.

On the other hand, if a curve is originally given in parametric representation x=f (t), y = y (t) and we wish to obtain the equation of the curve in non-parametric form, that is, in the form y = f(x), we have only to eliminate the parameter t from the two equations. In the case of the above parametric representations of the circle and ellipse, we can do this at once by squaring and using the equation sin²t + cos²t = 1. (Another example is given below.) In general, we should have to find an expression for t from the equation x = f(t) by means of the inverse function t = F (x) and substitute this into y=y(t), in order to obtain the representation y = y {F(x)} = f(x). Naturally, during such an elimination, we must ordinarily restrict ourselves to a portion of the curve; in fact, to a portion which is not intersected twice by any line parallel to the y-axis.

However, it may happen that the equation y =f(x) obtained in this way represents more than the original parametric representation. For example, the equations x = a sin t, y = b sin t represent only the finite portion of the line y = bx/a between the points x = -a, y = -b and x = a, y = b, whereas the equation y = bx/a represents the entire line.

The parametric representation has associated with it a definite sense in which a curve is described, corresponding to the direction in which the values of the parameter increase; we shall call this direction the positive sense. For example, if the point x = x(t), y = y(t) describes a curve C as t traverses an interval t₀ £ t £ t₁ and the end-points P₀ and P₁ of the curve correspond to t₀ and t₁, respectively, then the curve is traversed positively in the direction from P₀ to P₁. If we introduce t=-t as a new parameter, the curve C will correspond to the values -t₁ £ t £ -t₀ of the variable t, and the points P₀ and P₁ will correspond to t = -t₀ and t = -t₁, respectively. If we now traverse the curve from P₀ to P₁, we proceed in the direction in which the values of the parameter t decrease, that is, in the negative sense. In general, a change of parameter t=t(t) preserves the sense in which a curve is described, if the function t(t) is monotonic increasing, but reverses it, if the function t(t) is monotonic decreasing.

5.1.2 Interpretation of the Parameter. Change of Parameter: In many cases, we can give an immediate physical interpretation to the parameter t, that is time. Any motion of a point in the plane may be expressed mathematically by the fact that the co-ordinates x and y appear as functions of the time. Hence, these two functions determine the motion along a path or trajectory in parametric form.

An example of this are the cycloids which arise when a circle rolls along a straight line on another circle. We limit ourselves here to the simplest case, in which a circle of radius a rolls along the x-axis and we consider a point on its circumference. This point then describes a common cycloid. If we choose the origin of the co-ordinate system and the initial time in such a way that for time t = 0 the corresponding point of the curve coincides with the origin, we obtain (Fig. 2) the parametric representation for the cycloid

where t denotes the angle through which the circle has turned from its original position; in the case when the velocity of rolling is uniform, it is proportional to the time.

By elimination of the parameter t, we can obtain the equation of the curve in non-parametric form, however, at the cost of the neatness of the expression. We have

whence

and we obtain x as a function of y.

In the parametric representation of a given curve, we have a great deal of freedom in the choice of the parameter (5.1.1). For example, we could take instead of the time t the quantity t = t² as parameter or, indeed, any arbitrary quantity t which is related to the original parameter t by an arbitrary equation of the form t = w (t), where we assume that for the entire interval of values of t under consideration this function has a unique inverse t = k(t). If increasing values of t correspond to increasing values of t, the positive sense of description remains the same; otherwise it is reversed.

Naturally, parametric representation is not limited to rectangular co-ordinates; for example, it can just as well be used with the polar co-ordinates r and q, which are linked to the rectangular co-ordinates by the well-known equations

x = rcos q, rsin q or

the equations of the curve would then be r = r(t), q = q(t).

As an example, the straight line may be represented parametrically (Fig. 3) by

(p and a being constants), from which we immediately obtain the equation of the line in polar co-ordinates

by eliminating the parameter t.

5.1.3 The Derivatives for a Parametrically Represented Curve: If, on the one hand, a curve is given by an equation y = f(x) and, on the other hand, parametrically by x = x(t), y = y(t), then we must have y = f{x(t)}. By the chain rule for differentiation, it follows that

or

where we use as an abbreviation for differentiation with respect to the parameter t a dot · over the variable (Newton's notation) instead of the dash ' ; we shall reserve the latter for differentiation with respect to x.

For example, for the cycloid, we have

These formulae show that the cycloid has a cusp with a vertical tangent at the points i = 0, ±2p, ±4p, ··· , at which it meets the x-axis, because, on approaching these points, the derivative becomes infinite. At these points, y is equal to 0, while everywhere else y > 0.

The equation of the tangent to the curve is

where x and h are the current co-ordinates, that is, the variable co-ordinates corresponding to an arbitrary point on the tangent. For the equation of the normal, i.e., the straight line through a point of the curve, perpendicular to the tangent at that point, we likewise obtain

The direction cosines of the tangent, that is, the cosines of the angles a, b which the tangent makes with the x and y axes, respectively, are given by

as we may verify by elementary methods. The corresponding direction cosines of the normal (Fig. 4) are given by

These formulae show us that at every point, at which are continuous and , the direction of the tangent varies continuously with t. This is the most important case for us; however, it is interesting to illustrate by examples the various possibilities which arise when our assumptions are not fulfilled and we cannot state directly that the tangent keeps on turning continuously. At a point, at which , the tangent may or may not turn continuously. As one example, we have the curve x = t³, y = t², discussed in 2.3.5, which has a cusp at the origin even though are continuous everywhere. Consider as another example the curve x = t³, y = t³, which is the straight line y =x. This curve has the same tangent direction everywhere; the latter is therefore continuous, although the derivatives both vanish for t = 0. Moreover, at a point at which are discontinuous, the direction of the tangent may or may not be continuous. In fact, let f (t) be any continuous monotonic increasing function, defined for t₁ £ t £ t₂, which has a sharp corner at t = t₃, t₁ £ t £ t₂. Then the curve x = t, y = f(t), which is the same curve as y == f(x), has a sharp corner at x = t₃; while the curve x = f(t), y = f(t), which is a segment of the straight line y = x, has a constant tangent direction even though the derivatives do not exist at t = t₃. This indicates that, if we wish to investigate the behaviour of the tangent at a point where our theorem does not apply, we should first use the formulae to find cos a or cos b as functions of t and then investigate these direction cosines themselves.

From a well-known formula in trigonometry or analytical geometry, we find that the angle between the two curves represented parametrically by x=x₁(t), y=y₁(t) and x=x₂(t), y=y₂(t), respectively, (that is, the angle between their tangents or normals) is given by the expression

The indeterminacy of the signs of the square roots in the last few formulae suggests that the angles are not completely determined, since we can still specify either sense of direction on the tangent or normal as positive. Taking the square root as positive, as it is usually done, corresponds to choosing for the positive direction on the tangent the direction in which the parameter increases, and for the positive direction on the normal the direction obtained by rotating the tangent through an angle p /2 in the positive, i.e., the counter-clockwise sense.,

The second derivative y" = d²y/dx² is obtained in the following way by means of the chain rule and the rule for differentiating a quotient:

whence

5.1.4 Change of Axes for Parametrically represented Curves: If we rotate the axes through an angle a in the positive direction, the new rectangular co-ordinates x,h and the old ones x,y are interrelated by the equations

Thus, the new co-ordinates x and h are specified along with x and y as functions of the parameter t. We obtain at once by differentiation

Let the curve be given in polar co-ordinates and both polar and rectangular co-ordinates be given as functions of a parameter t. Then, by differentiation with respect to t, we obtain from the equations x = r cos q, y = r sin q the formulae

which are frequently used in passing from rectangular to polar co-ordinates. As an example, consider the polar equation of a curve, r = f(q ) which might arise from a parametric representation r=r(t), q=q (t) by elimination of the parameter t. The angle y between the radius vector to a point on the curve and the tangent to the curve at that point is then given by

We can convince ourselves of this in the following way. If we think of the curve as being given by an equation y = F(x) and use q as a parameter, so that we have

(cf. Fig. 5 and equations (a) above). In addition, y = a - q, whence

This formula can also be established by geometrical methods.

5.1.5 General Remarks:. In discussing given curves, we sometimes consider properties which do not assert anything about the form of a curve itself, but merely something about the position of the curve with respect to the co-ordinate system; for example, the occurrence of a horizontal tangent, expressed by the equation or the occurrence of a vertical tangent, expressed by Such properties do not persist when the axes are rotated.

In contrast to this, a point of inflection will still be a point of inflection after the axes have been rotated. According to 5.1.4 , the condition for a point of inflection is

If we replace on the left hand side the expressions by their values in terms of the new co-ordinates x,h, we readily obtain

Hence it follows from the equation that so that our equation expresses a property of the point of the Curve which is independent of the co-ordinate system.

We shall often see later on that properties which are truly geometrical are expressed by formulae the form of which is not altered by rotation of the axes.

Exercises 5.1:

1. Find in parametric form the equation of the curve

2. A circle c of radius r rolls on the outside of a fixed circle 0 of radius R. The point P on the circumference of c moves with o and describes a curve called the epicycloid. Find the parametric representation of the epicycloid (consider c to rotate with constant velocity and measure time so that at t = 0 the point P is in contact with the circle C)

3. Sketch the epicycloid for the special case r = R and find its parametric equations, (This particular epicycloid is called the cardioid.)

4. If in 2. the radius r is less than R and c rolls inside C, the point P describes a hypercycloid. Find its parametric equations.

5. Sketch the hypercycloid (1) for R = 4r, (2) for R as 3r.

6. Sketch the hypercyloid for R = 4r (the astroid) and find its non-parametric equation.

7. Find the parametric equations for the curve x³ + y³ = 3axy (the folium of Descartes), choosing as parameter t the tangent of the angle between the x-axis and the radius vector from the origin to the point (x, y).

8. Find the formula for the angle a between two curves r = f(q) and r = g(q) in polar co-ordinates.

9. Find the equation of the curves which everywhere intersect the straight lines through the origin at the same angle a.

10. Let c be a fixed curve and P a fixed point with co-ordinates x₀,y₀. The pedal curve of C with respect to P is defined to be the locus of the foot of the perpendicular from P on the tangent to C. Find the parametric representation of the pedal of C, if C is itself given parametrically by x = f(t), y = g(t).

11. Find the pedal curve of the circle C, (a) with respect to its centre M, (b) with respect to a point P on its circumference.

15. Find the pedal curve of the ellipse x = acos q, y = bsin q with respect to the origin.

Answers and Hints

5.2 Applications to the Theory of Plane Curves

We shall consider two different kinds of geometrical properties or quantities associated with curves. The first type consists of properties or quantities which depend only on the behaviour of the curve in the small, i.e. in the immediate neighbourhood of a point, and which can be expressed analytically by means of the local derivative. Properties of the second type depend on the entire course or a portion of the curve and are expressed analytically by means of the concept of integral. We shall begin by considering properties of the second type.

5.2.1 Orientation of Area: The idea of area was our starting point for the definition of the integral; but the connection between the definite integral and area is still somewhat incomplete. The areas, with which we are concerned in geometry, are bounded by given closed curves; on the other hand, the area measured by the integral is bounded only partly by the given curve y = f(x), the rest of the boundary consisting of lines which depend on the choice of the co-ordinate system. If we wish to determine the area interior to a closed curve, such as a circle or an ellipse, by means of integrals of this type, we have to use some such device as breaking up the area into several parts, each of which is bounded by a single-valued branch of the curve as well as by the x-axis and the corresponding ordinates.

It is convenient for the discussion of this general case to make first some remarks on the determination of the sign of an area under consideration. For any figure, bounded by an arbitrary closed curve which does not intersect itself, we can relate the sign of its area to the purely geometrical idea of the sense in which the curve is described, following the convention: We say that the boundary of a region is described in the positive sense, if we go around the boundary in such a direction that the interior of the region is on the left; the opposite sense we call negative. If we then consider a region, the boundary of which is traversed in an assigned sense - a solaced oriented region, we consider the area to be positive if this sense is positive, and negative if this sense is negative (Fig. 6).

If we wish to avoid the words right and left in such a context, we say that the triangle, the ordered vertices of which are the origin, the point x = 1, y = 0 and the point x = 0, y = 1, is described in the positive sense, if the vertices are passed in the order mentioned. For every other region, we say that the boundary is positively described. if it is described in the same sense as this triangle, otherwise it is described negatively

In particular, let in the interval a £ x £ b the function f(x) be everywhere positive. We consider the closed curve obtained by starting at the point x = b = x₁, y = 0, traversing the x-axis back to the point x = a = x₀, y = 0, then proceeding along the ordinate to the curve y =f(x), then along the curve to the ordinate x = b, and finally along the ordinate to the x-axis (Fig. 7). The absolute value of the area interior to this curve - the number of square units contained in it - is, as we know, . Hence, denoting by A₀₁ the area with its sign as determined above, the integral yields the value A₀₁ except for its sign. In order to determine the sign, we need only observe that the boundary of the region is traversed in the negative sense, so that A₀₁ is negative; hence

Similarly, if a > b, we find that, according to our convention, A₀₁ is positive, while the integral is negative, whence in either case A₀₁ is given by the above equation.

5.2.2 The General formula for the Area as an Integral:. After these preliminaries, the difficulties mentioned at the beginning can now be avoided in a simple way by representing our curve parametrically. If we introduce formally t into the above integral as a new independent variable, writing x = x(t), y = y(t), we have

where t₀ and t₁ are the values of the parameter corresponding to the abscissae x₀ = a and x₁ = b, respectively. We assume here that the considered branch of the curve y=f(x) is related to an interval t₀ £ t £ t₁ by a (1,1) correspondence, that f(x)* is everywhere positive and that ) never vanishes in this interval. As we have seen, our expression then yields the area of the region bounded by the curve, the lines x = a and x = b, and the x-axis. It is, of course, still subject to the disadvantages mentioned above. We shall now show that, if the curve x=x(t), y=y(t), t₀ £ t £ t₁ is a closed curve bounding a region of area A₀₁, this area is given by an integral which in form is exactly the same as the preceding one.

* i.e., is such that everyone of its points corresponds to a single value of t in the interval t₀ £ t £ t₁ and conversely.

Consider now a closed curve which is represented parametrically by the equations x=x(t), y = y(t), the curve being described just once as t describes the interval t₀ £ t £ t₁. In order that the curve may be closed, it is essential that x(t₀)= x(t₁) and y(t₀)=y(t₁). We shall assume that the derivatives are continuous except at most for a finite number of jump-discontinuities, and that differs from zero except perhaps at a finite number of points which may be corners of the curve.

A continuous curve x = x(t), y = y(t) is said to have a corner at t = t₀, if the positive direction of the tangent approaches a limit as (t - t₀) ® 0 through positive values and approaches a limit as (t - t₀) ® 0 through negative, but the two limits are not the same.

We shall first consider a closed curve which has no corners and is convex and of such a type that no straight line intersects it at more than two points. We denote by P₁ and P₂ the points at which the curve has a vertical tangent; these tangents are said to be lines of support at P₁ and P₂, respectively, because the points of the curve in the neighbourhood of P₁ and P₂ lie entirely on one side of the line. We can then (Fig. 8 above) regard the area to be bounded by the curve as the sum of the area A₁₂, bounded by the closed curve P₁MP₂ABP₁, formed as in the preceding section, and the area A₂₁, bounded by the closed curve P₂NP₁BAP₂. We assume here that the curve is described in the positive sense, as in the figure; by our sign convention, A₁₂ is then positive and A₂₁ negative. Let the point x(t), y(t) describe the upper part of the curve from P₁ to P₂ as t moves from t₀ to t, and the lower part from P₂ to P₁ as t moves from t to t₁. We then obtain immediately

whence the total area bounded by the convex curve is

If we denote by the absolute area of a region the number of square units contained in it, - which is, of course, never negative - then the above expression always yields the absolute area bounded by the curve except perhaps for the sign. In order to see what happens when we reverse the sense in which the curve is described, we simply take the same integral from t₁ to t₀ instead of from t₀ to t₁; then our integral becomes

which is equal to -A. We thus recognize the truth of the statement:

The area represented by our formula is positive or negative according to the sense in which the boundary is described.

In drawing the figure, we have assumed that y > 0 for all points of the curve. This really does not restrict the generality of the result. In fact, if we displace the curve by a distance a parallel to the y-axis, without rotating it, in other words, replace y by y+a, the area is unchanged; the value of the integral is likewise unaltered, for the above integral is replaced by

and, since the curve is closed,

Two simple observations enable us to extend our results. Firstly, our formula remains valid for closed curves which do not intersect themselves, even if they are not convex, but have a more general form as is shown in Fig. 9. Secondly, the derivatives may have jump discontinuities or may both vanish at a finite number of points, which may represent corners; according to 4.8, the function remains integrable. (The ordinate to a corner-point is considered to be a line of support, if the curve lies in the neighbourhood of the point entirely to one side of the ordinate). We assume that the curve has only a finite number of lines of support, corresponding to the points P₁, P₂, ··· , P_n and subdivide the

curve into the single-valued branches P₁P₂ ··· P_n-1P_nP₁. Then, as in Fig. 9, we obtain the area bounded by the curve in the form A = A₁₂+A₂₃+···+A_n-1,n+A_n1. (cf. Fig. 9, which illustrates this for the case n = 6.) If we express each of these portions of area parametrically and combine the expressions into a single integral, we find that the area bounded by the curve is given by

which, as before, has the same sign as the sense in which the boundary curve is traversed.

In a certain sense, our formula even gives us the area in the case where the curve intersects itself. But we shall not enter into such a discussion here; readers who wish to do so may turn to A5.2,

We can express our formula for the area in a more elegant symmetrical form if we first apply to the integral integration by parts:

Since the curve is closed,

whence

If we form the arithmetic mean of the two expressions, we obtain the symmetrical form

Instead of finding for the area the second expression above by integration by parts, we could have derived it by using the fact that, as regards the definition of area, the x-axis and the y-axis are interchangeable, except that the sense of rotation which brings the x-axis into the y-axis along the shortest way is opposite to the sense which brings the y-axis into the x-axis along the shortest way.

5.2.3 Remarks and an Example: In connection with these expressions, we must make a remark of a fundamental nature. Both the proof and the statement of the formulae depend on a particular system of rectangular co-ordinates. But the value of the area - a purely geometrical quantity - cannot depend on the chosen co-ordinate system. It is therefore important to show that a change of co-ordinates does not affect our integrals.

If the axes are merely displaced without rotation, obviously the integrals are unaltered (cf. above). Now let us assume that the axes are rotated through an angle a ; instead of x and y, we now have new variables x and h, defined by the equations

the new variables being also functions of the parameter t. If we recall that

a short calculation yields

whence

This equation expresses the fact that the area does not depend on the co-ordinate system.

Our integral expression for the area also does not depend on the choice of parameter. In fact, let us introduce a new parameter t by the equation t = t (t); we have

so that

where t₀ and t₁ are the initial and final values of the new parameter, corresponding to the parametric values t₀ and t₁, respectively.

In this section, we have based the definition of area on the concept of the integral and have shown that this analytical definition has a truly geometrical character, since it yields a quantity independent of the co-ordinate system. However, it is easy to give a direct geometrical definition of the area bounded by a closed curve which does not intersect itself, as follows: The area is the upper bound of the areas of all polygons lying interior to the curve. The proof that the two definitions are equivalent is quite simple, but will not be given here.

As an example of the application of our formulae for the area, we consider the ellipse In order to find its area, we take the upper and lower halves of the ellipse separately and in this way express its area by the integral

However, if we use the parametric representation x = a cos t, y = b sin t, we find immediately that its area is given by

This can be integrated as in 4.2.3; its value is abp.

5.2.4 Areas in Polar Co-ordinates: For many purposes, it is important to be able to calculate areas using polar co-ordinates. Let r = f(q) be the equation of a curve in polar co-ordinates. Let A(q) be the area of the region which is bounded by the x-axis (that is, the line q = 0), the line through the origin forming an angle q with the x-axis, and the portion of the curve between these two Lines. Then

In fact, if we consider the radius vector corresponding to the angle q and that corresponding to the angle q+Dq, and denote the smallest radius vector in this angular interval (Fig.10) by r₀ and the largest by r₁, the sector lying between the radius vector q and the radius vector q + Dq will have an area DA which lies between the bounds ½r₀²Dq and ½r₁²Dq). Consequently,

and on passing to the limit as Dq ® 0, we obtain the above relation. By the fundamental theorem of the integral calculus, the area of the sector between the polar angles a and b is then given by

If b > a, this expression cannot be less than zero. Since we readily see that, as q increases, the point with co-ordinates (r,q) describes the boundary of the region in the positive sense, this is in agreement with our previous sign convention.

As an example, consider the area bounded by one loop of a lemniscate/. Its equation is r² = 2 a²cos 2q (cf. A2.1) and we obtain one loop by letting q vary from -p/4 to + p/4. This gives us the expression

for the area. This can be integrated at once by introducing the new variable u=2q ; we find the value of the integral to be a².

5.2.5 Length of a Curve: Another important geometrical concept - the length of arc - leads to integration.

To start with, we shall explain geometrically how we are led to a definition of the length of an arbitrary curve. The elementary process of measuring a length consists of comparing the length to be measured with rectilinear standards of length. The simplest method is to apply our standard length to the curve, with its ends on the curve, and count the number of times that we have to repeat the process in order to pass from the beginning to the end of the curve; we can refine the method as required by using smaller and smaller standards of length. By analogy with this elementary intuitive idea, we set up the definition of the length of a curve as follows: We assume that our curve is given by the equations x = x(t), y = y(t), a £ t £ b (This includes curves in the form y = f(x), since these can be written as y =f(t), x = t.) In the interval between a and b, we choose points t₀ = a, t₁, t₂, ··· , t_n = b in that order. We join the points on the curve, corresponding to these values of t, in order of the line segments, thus obtaining part of a polygon inscribed in the curve; we now measure the perimeter of this polygon. This length will depend on the way in which the points t_n , or, as we may also say, the vertices of the polygon are chosen. We now let the number of the points t_n increase beyond all bounds in such a way that the length of the longest subinterval in the interval a £ t £ b at the same time tends to 0; this causes the number of sides of our polygon to increase without limit, while the length of the longest side tends to 0. The length of the curve is then defined to be the limit of the perimeters of these inscribed polygons, provided that such a limit exists and is independent of the particular way in which the polygons are chosen. It is only when this assumption that the limit exists (the assumption of rectifiability is fulfilled that we can speak of the length of the curve. We shall soon see that very wide classes of curves can be proved to be rectifiable.

In order to express the length analytically by an integral, we think, in fact, of the curve as being represented in the first instance by a function y = f(x) with a continuous derivative y'. We subdivide by the points a = x₁, x₂, ··· , x_n = b the interval a £ x £ b of the x-axis, above which lies our curve, into (n - 1) intervals of lengths Dx₁, ··· , Dx_n-1. We inscribe in the curve a polygon the vertices of which lie vertically above these points. By Pythagoras' theorem, the total length of this inscribed polygon is given by (Fig. 11)

However, by the mean value theorem of the differential calculus, the difference quotient Dy_n/Dx_n is equal to f '(x_n), where x_n is an intermediate value in the interval Dx_n. If we now let n increase beyond all bounds and at the same time let the length of the longest sub-interval Dx tend to zero, then, by the definition of the integral, our expression will tend to the limit

Since this passage to the limit always leads us to the same result, namely, the integral, no matter how the subdivision of the interval is made, we have established the theorem:

Every curve y = f(x), for which the derivative f'(x) is continuous, is a rectifiable curve and its length between x = a and x = b (b ³ a) is given by

If we denote by s the length of arc measured from an arbitrary fixed point to the point with abcissa x, the above equation yields for the derivative of the length of arc with respect to x:

Our expression for the length of arc is still subject to the special and artificial assumption that the curve consists of one single-valued branch above the x-axis. Parametric representation removes this restriction. If a curve of the kind under consideration is given in parametric form by the equations x=x(t), y=y(t), then we obtain by introduction of the parameter t into the above expression the parametric form of the length of arc

where a and b are the values of t which correspond to the points x=a and x=b of the curve, respectively.

This parametric expression for the length of a curve has a considerable advantage over the previous form in that it is not restricted to single-valued branches of the curves, represented by the equation y=f(x), but instead it holds for any arbitrary arcs of curves, including closed curves, provided that the derivatives are continuous along the arcs.

We recognize this most readily by going back again to the formula for the length of the inscribed polygon. We assume that are continuous along the arc. As in the definition, we subdivide the interval a £ t £ b by points t₀=a, t₁, ···, t_n = b, with the differences Dt_n and use the corresponding points on the curve as vertices of an inscribed polygon; in the passage to the limit n®¥, we assume that the greatest difference Dt_n tends to 0. If we now write the length of the polygon in the form

we see at once that this sum tends to the integral

we need only recall the generalized method of formation of an integral (A2.1). If the curve is composed of several arcs of this type, which may join one another at corners, the expression for the length of the curve is simply the sum of the corresponding integrals. Collecting the results, we have the statement:

If in the interval a £ t £ b the functions x(t) and y(t) are continuous as well as their derivatives , except perhaps for a finite number of jump discontinuities, the arc of x = x(t), y = y(t) has the length

where this integral, if necessary, is to be taken as an improper integral in the sense of 4.8.

By virtue of this formula, in which a must be less than b, there is a meaning in ascribing a negative length, given by the same formula, to an arc of a curve traversed in the direction in which the value of the parameter t decreases. The sign of the length of arc therefore depends on the choice of the parameter. If we introduce a new parametric expression for the same curve which does not reverse the sense of description, that is, if we introduce a new parameter by the equation t = t(t), where dt/dt > 0, we see a priori that our integral formula should give the same value no matter whether t or t is used as parameter, because the two integrals yield the length of the same curve and must therefore be equal. However, this may also be verified directly, because

We now present the expression for the length of arc when the curve is expressed in polar co-ordinates. In the last expression, we need only substitute for their values as given by the formula (a) in 5.1.4 in order to obtain

whence

If we now step over from the parametric expression to the equation in the form r = f(q ) by introducing as parameter t = q itself, so that we find for the length of arc

.

A simple example of the explicit calculation of the length of an arc is given by the parabola y = x²/2; we obtain immediately for its length of arc the integral

which with the substitution x = sinh u becomes

so that the length of arc of the parabola between the abscissae x=a and x=b is given by

For the catenary y = cosh x, we find

Finally, note that it is convenient in many cases to introduce as parameter the length of arc reckoned from some fixed point P₀ on the curve, that is, to take x=x (s) and y = y(s). Points of the curve on opposite sides of P₀ will correspond to values of s with opposite signs. In this case, we have

whence by differentiation

these two relations are applied frequently.

5.2.6 Curvature of a Curve: The axes and the length of arc of a curve depend on its complete course. We now insert a discussion of a concept which has reference only to the behaviour of a curve in the neighbourhood of a point - its curvature.

If we think of a curve as being described uniformly in the positive sense in such a way that equal lengths of arc are passed over in equal periods of time, the direction of the curve will vary at a definite rate, which we take as a measure of its curvature. Hence, denoting the angle between the positive direction of the tangent (5.1.3) and the positive x-axis by a and thinking of a as a function of the length of arc s, we shall define the curvature k at the point corresponding to the length of arc s by the equation k = da /ds. We know that a = artan y', whence, by the chain rule,

(where the positive sign of the square root means that increasing values of x correspond to increasing values of s). Hence, the curvature is given by

Using the parametric formulae for y' and y", we obtain the simple expression for the curvature of a parametrically represented curve:

which, of course, can also be found directly from the equation

In contrast to the previous expression, which depends on the equation y = f(x) and consequently involves a special assumption about the position of the arc with respect to the x-axis, the parametric expression for the curvature holds for all arcs along which are continuous functions of t and In particular, it holds for points where i.e., where dy/dx becomes infinite.

If we introduce the length of arc s as parameter and recall that

we find

We thus obtain a particularly simple expression for the curvature.

The sign of the curvature is changed, if we reverse the sense of description of the curve, that is, if we replace the parameter t or s by the new parameter t = -t or s = -s, because then change their sign, but not as follows from a simple calculation:

(A similar calculation can be made for y.) In the case of the expression found first, this fact is concealed, since it is natural and customary to think of a curve as described from the left to the right hand side, in which case the square root can only be positive.

As an example, consider the curvature of a positively described circle with radius a. If we start from the parametric representation x = acos t, y = asin t, we obtain immediately

Hence the curvature of a positively described circle is the reciprocal of its radius. This result assures us that our definition of curvature is really suitable, because, in the case of a circle, we naturally think of the reciprocal of the radius as a measure of its curvature.

Let us set r = 1/k. In general, the quantity |r| = 1/|k| is called the radius of curvature of a curve at the point in question. For a given point on a curve, that circle which touches the curve at the point and has there the same sense of description and the same curvature as the curve and, moreover, has its centre on the positive or negative side of the normal according to whether k is positive or negative, is called the circle of curvature, corresponding to the point. Let us think of the equation of the circle (or an arc of the circle containing the point in question) as being written in the form y=g(s). Then, at the point in question, we do not only have f(x) = g(x) and f'(x) = g'(x), as follows from the fact that the circle and curve touch, but, by virtue of the relation

we also have

The centre of the circle of curvature is called the centre of curvature, corresponding to the given point. Its co-ordinates are expressed parametrically by

In order to prove this, we need only employ the formulae for the direction cosines of the normal, on which the centre of curvature lies at a distance 1/|k|=|r| from the tangent. These formulae yield an expression for the centre of curvature in terms of the parameter t. As t describes its range, the centre of curvature describes a curve, the socalled evolute of the given curve and since, together with x and y, we must regard as known functions of t, the formulae above yield parametric equations for the evolute.

For special examples, refer to 5.3 and to A5.1.

5.2.7 Centre of Mass and Moment of a Curve: We now come to some applications which take us into mechanics. We consider a system of n particles in a plane. Let m₁, m₂, ··· , m_n be the masses of these particles, and y₁, y₂, ··· , y_n their respective ordinates. We then call

the moment of the system of particles with respect to the x-axis. The expression h = T/M, where M denotes the total mass of the system, gives us the height of the centre of mass of the system of particles above the x-axis. We define the moment about the y-axis and the abscissa of the centre of mass in a corresponding way.

We shall now see that this idea is readily extended to yield a definition of the moment of a curve along which a mass is distributed uniformly, and of the co-ordinates x and h of the centre of mass of such a curve. Merely for the sake of brevity, we assume that the density along the curve is constant, say m; any continuous distribution could equally well be discussed in the same manner.

In order to arrive at this extension, we return to the consideration of a system of a finite number of particles and then pass on to the limit. For this purpose, we assume that the length of arc s is introduced as a parameter on the curve and that the curve is subdivided by (n—1) points into arcs of lengths Ds₁,Ds₂,···,Ds_n. We represent as concentrated the mass mDs_i of each arc Ds_i at an arbitrary point s of the arc, say, with the co-ordinate y_i.

By definition, the moment of this system of particles with respect to the x-axis has the value

If now the largest of the quantities Ds_i tends to 0, this sum tends to a definite limit given by

which we shall therefore naturally accept as the definition of the moment of a curve with respect to the x-axis. Since the total mass of the curve equals its length, multiplied by m, is

we are immediately led to the expressions for the co-ordinates of the centre of mass of the curve:

These statements are actually definitions of the moment and centre of mass of a curve, but they are such straightforward extensions of the simpler case of a number of particles, so that we naturally expect that - as is actually the case - any statement in mechanics, which involves the centre of mass or the moment of a system of particles, will also apply to curves. In particular, the position of the centre of mass with respect to a curve does not depend on the co-ordinate system.

5.2.8 Area and Volume of a Surface of Revolution:. If we rotate the curve y =f(x), for which f(x) ³ 0, about the x-axis, it describes a so-called surface of revolution. The area of this surface, the abscissae of which we assume to lie between the bounds x₀ and x₁ > x₀, can be obtained by a discussion analogous to the preceding work. In fact, if we replace the curve by an inscribed polygon, we shall have instead of the curved surface a figure composed of a number of thin truncated cones. Following these intuitive suggestions, we define the area of a surface of revolution as the limit of the areas of these conical surfaces as the length of the longest side of the inscribed polygon tends to zero. We know from elementary geometry that the area of each truncated cone is equal to its slant height multiplied by the circumference of the circular section of mean radius. If we add these expressions and then carry out the passage to the limit, we obtain for the area the expression

Expressed in words, this result states that the area of a surface of revolution is equal to the length of the generating curve multiplied by the distance, traversed by the centre of mass (Guldin's rule).

In the same way, we find that the volume interior to a surface of revolution, which is bounded at the ends by the planes x = x₀ and x = x₁ > x₀, is given by

This formula is obtained by following the intuitive suggestion that the volume in question is the limit of the volumes of the above-mentioned figures consisting of truncated cones. The remainder of the proof is left to the reader.

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