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Zeros and Multiple Points of Certain
Periodic Polar Curves



Translatory-harmonic motion is represented in Cartesian coordinates by the equation
y = f(x) = a + b cos px/q

while rotary-harmonic motion requires the polar equation
r = f(t) = a + b cos pt/q.

In both case a, b, p and q are constants. This latter cyclic-harmonic curve has been studied kinematically by Moritz [4]. When he varied the ratio p/q he was able to describe four species of cyclo-harmonic curves; curtate, cuspitate, prolate and foliate.

FIGURE 1 shows cases

1/7, 2/7, 2/5,
2/11, 3/11, 4/11

r = 1 + tan pt/q when q is odd. Zeros, double points and intersections of axes can be counted and thus verify the theorems since the ratio p/q which is in the upper right corner. For example: the trajectory of the 2/7 case crosses the pole 4 times, and the x axis 18 times and has 24 double points (4 of which are off the screen).

Now when a tangent function replaces the cosine function in the polar equation quite different properties occur. For instance, the sum and product of the integers p and q determine how often the resulting new curve crosses itself (multiple points), how often the curve crosses the x and y axes, and how often it passes through the pole (zeros of r). These polar curves are of special interest because they represent the cross section of the surfaces (with z constant, z= b) in the class of ruled surfaces of rational order r = a + z tan pt/q which were examined by Emch [1]. Such a cross section has been used to prove that a Moebius band (in this case p/q = 1/2) self intersects in a straight line [3].
This is a study of these polar curves for the case a = b = 1. The matter is presented in theorem form followed by computer generated illustrations. The first theorem concerns the zeros of the curves, the second theorem concerns multiple points (where the trajectory crosses itself) and the last two concern points on the x and y axes. Consider the periodic trajectory of a particle moving along the polar curve
r = f(t) = 1 + tan pt/q

where p and q are relatively prime integers and p < q . The members of this collection of curves form curious variations of the single loop (one complete trajectory) which is characteristic of the polar equation r = tan t/2. This curve passes through the pole only once and self-intersects at the point (1, pi /2). Its domain for one period is (-pi , pi ) . In the general case, however, when p and q are arbitrary and relatively prime positive integers, the domain of one period is (-q pi , q pi ) for q odd, and (-q pi/2, q pi /2) for q even. Outside each of these periods, the particle retraces a path which is composed of p or 2p loops depending on whether q is even or odd. Since the range of the tangent function sweeps through all real numbers, the variable r takes on all real values from -pi to +pi in the course of each loop. These curves self-intersect at the pole and have multiple points apart from the pole. Although p and q vary independently their product and sum can be used to classify properties of these polar curves. These properties are listed in the following four theorems.

FIGURE 2 shows cases

1/2, 1/8, 3/8,
3/4, 3/10, 5/12

r = 1 + tan pt/q when q even. For example: the trajectory of the 3/10 case crosses the pole 3 times, the y axis 13 times and has 15 double points (3 of which are off the screen).




Theorem l:

At the pole the curve r = 1 + tan pt/q has a multiple point of order p or 2p depending on whether q is even or odd.

Proof: The curve (1) passes through the pole (where r = 0) so the multiple points occur when tan pt/q = -1. If t' is a solution of this equation then t' + kq pi /p for k="0", 1, 2, 3, . . . are also solutions because the period of this function is q pi/p and so
tan p( t' + kq/p)/q = tan (pt'/q + k ) = tan pt'/q = -1.
Of these solutions, some have different principal values t* which are the distinct values of 0 in the interval [0, 2 ], that is for 0 < t* < 2 . These values t* occur when proper multiples of 2 (complete revolutions of the ray) are subtracted from t' + kq/p so that t' and t* are related by the equation
t' + k /p - 2N = t* (for integral values of N).
Notice that t' and t* will not be distinct solutions of (1) when ktq/p - 2N is an even integer since this would cause t' and t* to differ by an integral multiple of 2 , so would not belong to the set of principle values. If q is even, then k = p is the first value of k for which kq/p - 2N is even and t* is not essentially different from t'. If q is odd, then the first such integer is k = 2p. Therefore, the equation tan(pt/q) = -1 has p distinct solutions to obtained for k = 0, 1, 2, . . ., p - 1 when q is even or 2p distinct solutions t* obtained for k="0", I, 2, ..., 2p -1 when q is odd, and the theorem follows.

These trajectories self-intersect (have double points) at other points in the plane besides the pole and the frequency of their occurrence again depends on the values of the constants p and q. The rays along which double points occur are seen in the case p/q = 1/2, Fig. 3a. Each ray t = ti cuts the curve in two distinct points (ri, ti) and ( rj, tj), where the arguments differ by that is tj= +ti. This occurs for all values of 0 except along the ray t = ti where non zero values of ri = - rj.
This is the only double point for p/q = 1/2, because of the fact that for proper multiples of pi

r= 1 + tan [(ti +k pi)/2] =

1 + tan [ti/2] = ri for k even
1 + tan [(ti+k pi )/2] = rj for k odd

since the period of the tangent function is pi, and the equation tan [ti/2] = tan[(ti+pi )/2] has only one solution resulting in ri =- rj.

QED

This principle is now used to prove the second theorem concerning the general p/q case.


Theorem 2:

Apart from the pole, the curve r = 1 + tan p t/q has

I. pq/2 double points if q is even
II. 2p(q-1) double points if q is odd.


Proof of Theorem 2 part I: for q even:
A similar process is used now to find the distinct double points for larger values of q, for instance q = 8 in Fig. 3b. We examine one loop of the trajectory and count the double points. These points occur along the rays t = tk + k when k is an odd integer: 1, 3, 5, ...., q-1. These are all distinct values of rj or -rj because the greater values q+1, q+3, ... q+k duplicate either the values tan ptk/q or tan p(tk+ )/q. This is true since
tan p[ti + (q+k) ]/q = tan [p(ti+k )/q + p ] = tan pti/q for p even or tan (pt + pi )/q for p odd
since the period of the tangent function is . So along each loop there are an odd number of double points depending on the size of q. That is k can be 1, 3, 5, . . . , q-1 which amounts to a total of q/2 double points (rk, tk). From the reasoning in Theorem I, we know there are p loops when q is even so there are pq/2 distinct double points when q is even.

QED

Proof of theorem 2 Part II. q is odd:
A similar procedure proves the second part of Theorem 2. Examine one of the loops, Fig. 3c to count the double points (rk, tk). These occur along the rays t = tk where k = 1, 2, 3, 4, . . . q - 1. As above the values of k > q-1 duplicate tan ptk/q or tan p(tk+k )/q already obtained but since q is odd, even values do not duplicate. So in the case q odd, each loop has q-1 double points. Now from Theorem 1, we recall that there are 2p loops for q odd. So altogether we have 2p(q-1) double points apart from the pole.

QED

Theorems 3 and 4 can be proven somewhat like theorem I and 2 by counting the number of times a loop intersects an arbitrary ray from the origin.




Theorem 3:

The curve r = 1 + tan pt/q intersects the x axis in 2(p + q)points if q is odd.


Proof: Since y = r sin t the curve (1) may be represented by the equation y = sin t (1 + tan pt/q)( 2 ) Now sin t = 0 when for k = 0,1, 2, 3,...(2q - 1) as is clear from the proof of Theorem 1. Therefore, sin t accounts for 2q distinct roots. On the other hand, the other factor in (2) (I + tan pt/q) has 2 p zeros when q is odd according to Theorem 1. Altogether, there are 2p + 2q roots for the equation y = sin t ( 1 + tan pt/q) = r sin o and the theorem follows, so the curve r = 1 + tan pt/q intersects the x-axis in 2(p + q) points.

QED

FIGURE 3 shows cases

1/2, 3/8, 1/3

Double points (apart from the pole) when ri = -rj along the ray tk:
3a Special case p/q = 1/2: only one double point (rk, tk ) along tk = /4
3b. Case of q even: pq/2 = 3(8)/2 = 12 double points for q even (3 are off screen)
3c. Case of q odd: 2p(q-1 )= 2(1)(3-1 ) = 4 double points for q odd




Theorem 4:

The curve r = 1 + tan pt/q intersects the y-axis in (p + q) points if q is even.


Proof: A proof is used which is similar to that of theorem 3. Substituting x = r cos t in (1) we write (1) as x = cos t(1 + tan pt/q) ( 3 )
A proof is used which is similar to the proof for theorem 3. Substituting x = r cos t in (1) we write (1) as x = cos t (1 + Tan pt/q). As in the proof for theorem 3, the zeros for the factors of (3) can be added together. Since cos t = O whenever t = (2k-1) /2 for k = 1, 2, 3,. . . q then cos t has q zeros. The factor (1 + tan pt/q ), however, has p zeros according to Theorem I. So the total number of roots for (3) is (p + q) and the theorem is proven: that is r = 1 + tan pt/q intersects the y-axis in (p + q) points.

QED

Figures I and 2 illustrate members of this collection of curves for the varying values of p and q. The ratio p/q for each graph is stated in the upper right hand corner. Because of the fact that the graphs extend only to six units along the x and y axes, some points of intersection cannot be seen, only extrapolated.
This collection of polar curves reveals properties not easily found in Cartesian curves. Although the integers p and q are independent, except for the obvious fact that p and q are relatively prime, this study illustrates these relations.
1. The number of times the curves passes through the pole depends on whether q is odd or even and is a simple function of p.
2. The number of double points depends on the product of p and q
3. The frequency in passing through the axes depends on the sum of p and q . The points of these trajectories in the polar plane can be calculated using a computer for any ratio p/q and then by using computer graphics these four theorems can be demonstrated.

References


[1] A. Emch, "On a certain class of rational ruled surfaces",
American Journal of Mathematics, 42 (1920) 189-210

[2] W. C. Graustein, Introduction to Higher Geometry, Macmillan, New York 1940

[3] J. MacDonnell, S.J., "A ruled Moebius band which self-intersects in a straight line",
American Mathematical Monthly, 91 (1984) 125-127

[4] R. E. Moritz, "On the construction of certain curves given in polar coordinates", American Mathematical Monthly, 26 (1917) 213-220.


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