(11) Epitrochoids and Hypotrochoids. When the generating circle rolls, not on a straight line, but on the circumference of another circle, the curve generated is called an Epitrochoid or a Hypotrochoid, according as the curve rolls on the exterior or interior of the fixed circle. Let O (fig. 22) be the centre of the fixed circle, C that of the generating circle, a, b their radii. Let A and Q be the points originally in contact, P the tracing point. Then if we make CP = h, CN = x, PN = y, AOB = 0, so that QCBO, we find x = OH + HN = (a + b) cos 0 – h cos (*) 0, If we suppose the generating circle to roll in the inside of the fixed circle as in fig. 23, we should find When h = b these become the equations to the Epicycloid and Hypocycloid respectively. When a and b are commensurable the curve will re-enter after a number of revolutions of the generating circle equal to the least common multiple of a and b: in such cases the curve is expressible by an algebraical equation between x and y. When a and b are incommensurable the curve will never re-enter, and is expressible only by some transcendental equation between and y. If h= b and b = a the equations to the epicycloid are x = a (2 cos 0 - cos 20), Whence, squaring and adding, x + yo = a^ {1 + 4 (1 – cos 0)}. But we have also (x – a)? + y* = 4a* (1 – cos 6). Therefore (1x2 + yo – a*)? = 4ao {(x – a)” + yo} is the equation to the curve expressed in rectangular coordinates. If we put x = a + r cos , y = r sin , we find p = 2a (1 – cos ), as the polar equation. From its shape this curve is called the Cardioid : in common with the circle it possesses the property that all lines drawn through its pole and bounded both ways by the curve are of equal length. In the equations to the hypotrochoid, if we make h = b and b= *, we obtain by the elimination of 0 the equation (a’ – 25 – y?)3 = 27 a?r?y" ; which may be put under the form 2} + yi = aš. This hypocycloid occurs in the solution of many problems. ~= (+ n) cos e, y = (y – 5) sin 0. Whence (– 1)*+ (6 + n) ;* = - 1)", which is the equation to an ellipse the axes of which are +h and - h. ce If he the hypocycloid becomes a straight line, which is one of the diameters of the fixed circle. Professor Wallace* has made a very elegant application of the preceding property of the ellipse to generate that curve * Wallace's Conic Sections, p. 182. by continuous motion. A and B (fig. 24) are two wheels the axes of which turn in holes O, C near the ends of the connecting bar OC. The diameter of the wheel B is one half of that of A, and a band EF goes round them. An arm CP is attached to the wheel B, and bears at its extremity P a tracing pencil. If now the wheel A be fixed while the bar OC is turned round 0, the wheel B will, by the action of the band, be made to revolve twice round its centre, while the bar revolves once round 0: the point P will then trace out an ellipse. All Epicycloids and Hypocycloids are rectifiable, as was first shown by Newton*. The length of the arc of the epicycloid comprised between two contiguous cusps—that is, the length of the arc produced by one revolution of the generating circle-is *(a + b), and the corresponding arc of the hypocycloid is 40 (a – 6). The corresponding area of the epicycloid is *** (3a+26) and of the hypocycloid it is — (3a - 26). a The evolute of the epicycloid is a similar figure, the radii of the fixed and generating circles being enerating circles being as and ab a and a + 2ba +26 respectively. An analogous theorem holds for the hypocycloid. (12) The Spiral of Archimedes. While the straight line OM (fig. 25) revolves uniformly round 0, let the point P move uniformly along OM: the locus of the point P is the spiral of Archimedes. To find its equation let AOP=0, OP = r, and when 0 = 24 let r = a. Then , or raremos 2.7 which is the equation to the curve. • Principia, I. Prop. 49. The following are its principal properties. The area of any sector bounded by a line as OQ = r is one third of the circular sector QOR, and it is one half of the area of the segment of a parabola (whose latus rectum is included between the vertex and an ordinate = r. The length of the arc of the sector of the spiral is equal to that of the segment of the parabola. If a tangent be drawn at the extremity of the arc formed by one revolution of the radius, the subtangent will be equal to the circumference of the circle whose radius is a. If at the extremity of the arc formed by two revolutions, it will be double of the circumference, and so on. This curve was invented by Conon, but its principal properties were discovered by the geometer whose name it bears. (13) The Logarithmic Spiral. The definition of this spiral is, that the radius increases in a geometric while the angle increases in an arithmetic ratio. Hence its equation will be of the form p = cea, or, as it is usually written, p = ao. This curve was imagined by Descartes, who also noticed two of its properties; that at every point it makes equal angles with the tangent, and that the length of the curve measured from the origin is proportional to the radius of its extremity. Since r= () when A= - 0, it appears that the curve makes an infinite number of revolutions before it reaches the pole ; a property which was at first disputed by Descartes. From the form of the equation it is easy to see that radii including equal angles are proportional ; for if r = ao and ri = ao+a, The length of an arc of the curve measured from the pole is equal to the portion of the tangent at its extremity cut off by the subtangent, and the area is one half of the triangle contained by the bounding radius, the tangent at its extremity, and the subtangent. But the most remarkable properties of this curve were discovered by James Bernoulli, who showed* that this spiral can be made to reproduce itself in many ways. The evolute and involute of this curve are both spirals equal to the original one, and differing from it in position only ; its caustics both by reflexion and refraction (the pole being the origin of light) are also spirals equal to the primary one ; and if another equal spiral be made to roll on the first, the pole of the rolling spiral will trace out another spiral equal to the original. This property of the logarithmic spiral of constantly reproducing itself appeared so remarkable to Bernoulli that he called it spira mirabilis, and he was pleased to see in it a type of constancy amid changes and in adversity, and a symbol of the resurrection. As a specimen of the fanciful light in which he viewed the properties of this curve, I add the concluding paragraph of his paper. “ Cum autem ob proprietatem tam singularem tamque admirabilem mire mihi placeat spira hæc mirabilis, sic ut ejus contemplatione satiari vix queam; cogitavi illam ad res varias symbolice repræsentandas non inconcinne adhiberi posse. Quoniam enim semper sibi similem et eandem spiram gignit, utcunque volvatur, evolvatur, radiet ; hinc poterit esse vel sobolis parentibus per omnia similis Emblema : Simillima filia matri. ... Aut, si mavis, quia curva nostra mirabilis in ipsa mutatione semper sibi constantissime manet similis et numero eadem, poterit esse vel fortitudinis et constantiæ in adversitatibus; vel etiam carnis nostræ, post varias alterationes et tandem ipsam quoque mortem, ejusdem numero resurrecturæ symbolum; adeo quidem ut si Archimedem imitandi hodienum consuetudo obtineret libenter spiram hanc tumulo meo juberem incidi cum epigraphe: Eadem mutata resurget.” * Opera, p. 497. |