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PROPERTIES OF PLANE CURVES, AS DEFINED BY EQUATIONS REFERRED TO POLAR COORDINATES.
Section 1.—The mode of interpretation, and tlie equations, of curves referred to polar coordinates.
261.] In the present Chapter we shall investigate, for polar curves, formulas somewhat analogous to those discussed in the last Chapter for curves referred to rectangular coordinates; but previously it is necessary to extend the usual mode of interpreting polar equations, so as to accommodate them in a greater degree to the law of continuity.
Let r = f(6) be the equation to the curve. Then, taking a fixed point s as the origin, which is called the pole, and a fixed line sx passing through it as the line of origination, which is called the prime radius, see fig. 81, it is manifest that the moveable radius, which is symbolized by r, may revolve about 8 in either of two directions; and thus, if the only datum is that r makes an angle 0 with the prime radius, it is undetermined whether r is above or below sx: that is, whether r revolves up from sx from right to left, or down from left to right. Hence arises the necessity of some symbol of the direction in which r turns, so that angles formed in one direction may be differently symbolized to those formed in another. This indefiniteness will be avoided if we call angles positive when measured up from sx, as in fig. 81: that is, when the radius vector revolves round s in the direction indicated by the curved arrow; and negative when they are measured down from sx, and the radius vector revolves in the direction indicated by the curved arrow in fig. 82. In this case then, -f and —, as affecting angles, indicate the two different directions in which r can revolve in the plane of the paper.
Again, suppose that for a given value of 0, r is affected with a negative sign, a question arises, in what direction is the negative r to be measured? No doubt, if r is affected with a positive sign, the length of it, determined by the equation to the curve, is to be measured from the pole along the revolving radius vector which is inclined at the given angle to the prime radius; as e. g. if a polar equation between r and 0 is such that, when
0 = -t, r = a, then a length = a is to be measured from the 4
pole along the revolving radius, which is inclined at 45° to the prime radius. From analogy therefore to what has been said in Art. 189, on the signs + and —, — r must be measured along the radius vector produced backwards; i. e. if, when
6 = -, r — — a, a line equal to a must be measured from the 4
pole along the revolving radius produced backwards: that is, in a direction making an angle of 225° with the prime radius. That we may the better avoid confusion on this subject, conceive the revolving radius to be an arrow of variable length, such as we have drawn in figs. 81 and 82, the pole being a fixed point in it; then, if 0 is the angle between the prime radius and the part of the arrow towards the barbed end, lines measured from s in the direction Sp will be positive, and in the direction Sq negative. If therefore r is affected with a positive sign, it is to be measured towards the barbed end, but if with a negative sign, towards the feathered end of the arrow. In the figures different positions of the arrow are drawn to indicate different positive and negative directions of r.
In the following Chapter we shall omit those particular values of r which are affected with + V—, as no satisfactory interpretation of such symbols in such a relation exists, and we shall consider those only which are affected with + ; being careful however to make r revolve in both the positive and negative directions, otherwise at certain points the curve will appear to be discontinuous.
And for the purpose of illustration in the sequel, we must here insert an account of the mode of description, and the equations of some polar curves, many of which, having been treated of at length by old geometricians, possess no small historical interest.
262.] The Spiral of Archimedes.
Def.—If the length of the radius vector of a spiral is proportional to the angle through which it has moved from its originating position, the locus of its extremity is the Spiral of Archimedes.
Let a = the length of the radius, when the angle described is equal to unity*; and let r be its length after describing the angle 6; therefore the equation is
see fig. 83.
The curve therefore starts from the pole; and the radius vector, which at the beginning is equal to zero, is equal to Sa, that is a, when it has revolved through the unit angle Asx; and at the end of the first complete revolution is equal to 2na; and this is the distance between the points at which any radius vector is cut by two successive convolutions of the curve. The dotted curve is that described by the generating point, as the radius vector revolves in the negative direction.
263.] The reciprocal spiral.
The reciprocal or hyperbolic spiral is so called from the form of its equation, which is
The form of the curve is given in fig. 84. The radius vector = 00 when 6 = 0, and the curve is asymptotic to the straight line B'ab, as will be shewn in the sequel. Also, when 0 = 1 = A'sx, r = a = Sa'; also r = 0, when 6 = 00 ; therefore, after an infinite number of revolutions, the curve falls into the pole. The curve has also the dotted branch arising from the revolution of r in the negative direction.
264.] The lituus.
This spiral is so called from its form as delineated in fig. 85. Its equation is
r = ±. (3)
The prime radius is an asymptote to the curve; which has a point of inflexion when r = Sb = a \/2, as will be shewn hereafter. Also, when 0 = 1 = Asx, r = Sa = a; there is an apparent discontinuity at the pole and at the extremity of the infinite branch, which arises from our not interpreting r when
* The unit angle is that whose subtending arc is equal to the radius, and expressed in degrees = 57.29578. See Ex. 5, Art. 24.
affected with + -J —, as such it will be if the radius vector is made to revolve in a negative direction.
265.] The logarithmic spiral.
Def.—The logarithmic spiral is that whose radius vector increases in a geometric, as its angle increases in an arithmetic ratio.
Hence the equation is r = a*. (4)
Therefore when 0 = 0, r = 1 = Sa; when 0 = 1, r = a; when 0=oc, r = 30; when 0 = — 00 , r = 0; and therefore the spiral runs into its pole after au infinite number of revolutions in the negative direction; the spiral is also called the equiangular spiral from a property which will subsequently be proved; viz. that it cuts all its radii vectores at a constant angle; that is, the angle Stp is constant, at whatever point p the tangent is drawn. Its form is delineated in fig. 86.
266.] The involute of the circle; fig. 87.
Def.—The involute of the circle is the curve formed by the extremity of an inextensible string, as it is wrapped round the circumference of a circle.
If r is the radius vector of a curve, and p is the perpendicular from the pole ou the tangent, it is frequently convenient to express the equation to the curve in terms of r and p. Such an equation is, as will be subsequently seen, a differential one; but expressing as it does an essential property of a curve, it is sufficient to individualize it, and thus to be a mathematical definition. The equation of the involute of the circle can be easily obtained in this form
Let Sa = a, the radius of the circle; Sp = r; Sy = p; and let A be the point at which P, the generating point of the involute, is in contact with the circle.
Then from the geometry it is plain that Qpy is a right angle, and consequently Qp is parallel to Sy; whence we have
267.] To find the equation to the circle in terms of p aud r, any point being the pole; fig. 88.