the ellipsograph of proclus (april 1, 1900) 2369752

8/13/2019 The Ellipsograph of Proclus (April 1, 1900) 2369752

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The Ellipsograph of Proctus.*BY E. M. BLAKE.

If a plane a containing two points E and E1 moves upon a coincident planea, containing two straight lines g and gl so that E remains upon g and E1 upongl, the two planes form a mechanism possessing the following well-known prop-erties: Every point of a traces an ellipse upon a,, and every point of a, traces alimacon upon a t A circle c of radius a in a rolls upon the inner side of a circlecl of radius 2c in or. Every point of c describes a straight line passing throughthe center of c1. Any two of these lines, with the points which generate them,can be taken for g, g1and E, E1in defining the movement.The object of the present paper is to make a brief study of- 10 the curvesgenerated by the points of a and a,; 20 the ruled surfacesgenerated by anystraight line carriedby a or a1and not parallel to them; 30 the curves envelopedby any straight line of a or ,; 40 the developables enveloped by carriedplanes.The point loci have been given by Cayley I and Schell.? The line loci forthe ellipsograph have been determined by Burmester,11although he omits their

* Read before the Chicago Section of the American Mathematical Society, April 9, 1898, at whicheight thread models of surfaces described in the paper were exhibited. This embodies the author'spaper, "Upon a Ruled Surface of the Fourth Order Mechanically Generated," read before the SocietyDecember 31, 1897. Paper revised, Ithaca, January, 1900.t The discovery of the first property is accredited to Proclus; Chasles, " Apergu historique," 2d ed.,p. 49. The well-known chuck for turning figures with elliptical cross-sections, invented by Leonardoda Vinci, is an application of the mechanism. For other historical notes, see Burmester. " Lehrbuchder Kinematik I, " Leipzig, 1888, pp. 36-42; A. v. Braunmfihl, " Studie fiber Curvenerzeugung" in the"Katalog mathematischer Modelle," by Dyck."On the Kinematics of a Plane. " Quarterly Journal, XVI, 1878, pp. 1-8.

Q"Theorie der Bewegung und Krafte I, pp. 227-230.11Kinematische Flachen erzeugung vermittelst cylindrischer Rollung." Zeit. fulrMath. u. Phys.XXXIII, 1888, pp. 337-348. The surface with real and distinct nodal straight lines is also given byMannheim, Comptes Rendus de l'Acad. de Paris, LXXVI, 1873, pp. 635-639; Bulletin de la Soc. Math.de France, I, 1875, pp. 106-114. Menzel, "Ueber die Bewegung einer starren Geraden." Dissertation,Mfinster, 1891.


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BLAKE: The Ellipsograph of Proclus. 147equation, but those for the inverse movement are believed to be given here forthe first time. The envelope of the straight line of a which passes through Eand El when g and g, are at right angles, was found by Chaslesto be the four-cusped hypocycloid.* The only other lines of a, the equations of whose envel-opes have been previously determined,are those forming a square with E1E.t

Point and Straight Line Loci of the Ellipsograph.Denoting by (x, y) the rectangularcoordinates of any point of a, taking fororigin the center 0 of c, and by (x1,yl) the same for any point of a, taking fororigin the center 01 of cl; the equation

(x2- 2ax + a2 + y2) - 4axy1 y + yl (X2+ 2ax + a2 y2)= (X2+ y2- a2)2 (1)representsthe ellipse in a1 traced by the point (x, y) of a or the lima9ontracedby (x1, Yi)upon a.tFrom the equation we infer the following properties of the congruence ofellipses described by the points of a: They all have the point 01 for center,and rotation through half the angle whose tangent is 31L -rings their axes intoxcoincidence with the coordinate axes. The points of a circumference whosecenter is 0 describe congruent ellipses, which degenerate to the diameters of clwhen the circumference s c. The point 0 describes a circle of radiusa. Thepoints of a straight line through 0 describe ellipses whose axes lie upon twofixed lines.?To obtain the equation of the surfaces generated by the straight linescarriedby a and oblique to it, we proceedas in a formerpaper.I1 Take axes ofz and z, passing respectively through 0 and 01 and perpendicular to a and a1.Let I be a representative generator parallel to y Oz, passing through the point

*" Apergu historique," p. 69. The author also remarks that the envelope of any other straight lineis the involute of a hypocycloid.t J. B. Pomey, " Enveloppes des c6t6s d'un carr6 invariable dont deux sommets decrivent deuxdroites rectangulaires." Nouvelles Annales (3), V, 1886, pp. 520-530.t Cayley and Schell, loc. cit.QFor greater detail see J. S. et N. Vanecek. " Sur les ellipses decrites par les points invariable-ments lies Aun segment constant et sur une surface circulaire du huitieme ordre." Bulletin de la Soc.Math. de France, XI, 1883, pp. 76-88.11 merican Journal, XXI, 1899,pp. 260-261.


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148 BLAKE: The Ellipsograph of Proclu.s.(p, 0) of a and making with that plane the angle whose cotangent is s; itslocus isXI (p2 - 2ap + a2 + S2 z2) - 4as xl Y Z1+y2(p2 +2ap+ a2+s2zI) = (p2+s2z2-a2)2 (2)obtained from (1) by substituting for x and y respectively p and szl.There is no difficulty now in establishing the following theorem: Anystraight line oblique to a and carried by it generates a qtuartic scroll having a realisolated double generator in the plane ar at infinity and two other nodal straightlines which are: either real and distinct, coincident, or imaginary according as 1'(the orthogonalprojection of I upon a) intersects c in two real, two coincident, or twoimaginary points.

Point and Straight Line Loci of the Inverse of the Ellipsograph.Regarding the plane a as fixed, the points (x1, y1)of a1 describe upon it acongruence of lima?ons, defined by equation (1) and possessing the followingproperties: Any point on cl describes a cardioide, any point within a limagonhaving a node, and any point without one having a conjugate point. All pointsof a circumference with center at 01 trace congruent curves. They may bebrought into coincidence by rotation about 0 through twice the angle whose

tangent is Yi . The point 01 describes the circle c twice during a cycle of themovement. The real double points of all the curves are upon c. The points ofa line through 01 making the angle 0 with the axis of xl generate curves withthe samie ine of symmetry making the angle 20 with Ox-.As in the preceding case, we have for the locus of I the equationp2 (x2 - 2ax + a2+ y2) - 4apsyz + S2z2(x2+ 2ax + a2+ y2) -(2 + y2 a2)2 (3)and the following theorem: Any straight line oblique to or and carried by it gene-rates a quartic scroll having: if 1' passes through 01, a nodal circle in a (or aparallel plane) and an intersectingnodal straight line; and if 1'does not pass through,a nodal cubical ellipse. Upon the latter are two imaginary pinch-points, antd inaddition either two real, two coincident, or two imaginary pinch-points according as1'intersects cl in two real, two coincident, or two imaginary points.


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BLAKE: The Ellipsograph of Proclus. 149Singularities of Point Loci.

The circle of inflexions* for the ellipsograph is c, and hence, any pointupon it is always describing an inflexion upon its trajectory, i. e., the trajectoryis a straight line. The circle of inflexions for the inverse moveinent has thesame radius as c, and passes, during a complete cycle of the movement, oncearound cl, remaining tangelnt to it and to the concentric circle of radius 4a.Hence, the lirna?ongeneratedby a point of a1 between these two circumferenceshas two real points of inflexion. The two inflexions merge into one of the nexthigher order if the tracing-pointbe taken upon the circumferenceof radius 4a.The movement of a plane orupon another a, is either a rotation about a fixedpoint, a movement such that all straight lines of a remain continually parallel totheir initial positions, or the result of rolling a curve C of a upon a curve GIof a,. Movementsof the latter type are the only ones which give rise to anybut trivial problemsin loci and envelopes. It is easy to show that themovementdefinedby the ellipsographi?S he only movement f the last type which causes allpoints of a to describeconics. For any point of C describes a curve with at leastone cusp. A conic having a cusp consists of two coincident straight lines.Hence, at least two points of a describe straight lines, which defines the move-ment of the ellipsograph.The CurvesEnvelopedby Straight Lines carried Parallel to the Plane of Move-ment, and the DevelopablesEnvelopedby carried Planes.

We will commenceby finding the envelope of the line y = d carriedby a.Assume for convenience a=, then the equation of one of the positions ofy = d in a, is Xl +Y1 dI+ db' b =1+ b ~b' +b and b'being respectively the intercepts of y 0 upon the axes of X1 and YiEliminating b',we have

b4-2y b3+ (X2 + y2 1)b2 + 2 (y1 dxl) b + 2y ?=.* Koenigs, " Legons de oinematique," I, pp. 144-154; Sohoenflies, " La geometrie du mouvement."

Paris, 1893, Chapter I.20


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150 BLAKE: The Ellipsograph of Proclus.The discriminantof this with respect to b equated to zero gives(x$ +2)3_(3 + d2)y- 18dx1y3+(21 - 2d2)xPy2 18dx1-(3+d2)x4

+(3 + 20d2) -(36d- 1 6d3) , yi + (3 + 20d2)x1- 16d4+ 8d2-1 -0 , (4)the envelope required, and (x1yi - d)2 = 0. The latter may be regarded as thelocus of nodes,which shouldoccur twice. It is due to the enveloping line havingtwo positions for any value of b.The equation (4) represents, except for a rotation about the origin, thecurve enveloped by any straight line of a. It also represents with the samereservation the cylinder enveloped by any plane carried by, and perpendicularto, cr. The surface enveloped by any plane making an acute angle with a isevidently congruent to one enveloped by a plane passing through y = 0 andmaking the same angle with or. They may be brought into coincidence by rota-tion about and translation along the axis of z1. We proceed immediately to thedetermination and discussionof the equation of the latter surface, in the courseof which we shall be in a position to determine the character of its plane sectionsparallel to a,, which are the same as the curves (4).The equation of the surface enveloped by the plane which passes throughy = 0 and makes with a the angle whose cotangent is s, is obtained from (4) bysubstituting sz, for d. The result is(X2+ y2)3-(3 + s2z2)y4-18sx1 y3 Z1+ (21 - 2s2zl)X2y2-18sxy1z1 -- 3 +s2z1)x4+(3 + 20sz24)y2- (36sz1 - 16s3zl3)xIyl+ (3 + 20s5 2)A - 16sZ + 882z- 1= (5)Rotating the coordinate axes through 45? about the axis of zl, gives this equa-tion the more convenient form(xZ + yl)3 (3 + 82zi)(x2 + Iy2)2 _(x2 _ y2)2 - 9szl x_-+ (3 + 20s2z )(X2+ y2) - (18sz1 - 88 Z3)(X2_ y2) -1684z4+ 8s2z-1=0, (6)to which the discussionwhich follows will apply.The surface is of the sixth order and symmetrical with respect to xl = 0and Y,= 0. Its intersection with y, = 0 consists of the two straight linesZ = i i- - and the parabola x- 4sz1+ 2=0 taken twice. The lines are8 thtangent to the parabolaat 4 2, 3~-) The section by x, 0 consists of the


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BLAKE: The Ellipsograph of Proclu. 1511~~~~~~~~~~lines z1 YL + - and the parabola yi + 4z, + 2 = 0 taken twice. The

lines are tangent to the parabola at (4 2,- 2). The two parabolas arenodal lines of the surface. Each has a segment lying upon the surface and anisolated segment. The former contains the vertex of the parabola, and isincluded between its points of tangency with the straight lines mentioned above.The edge of regression is readily verified to be the intersection of the twocylinders 2 = 44 (SZ1 + 3712X= 4 'SZ1- 3 )

whose right sections are semi-cubic parabolas. Its projecting cylinder upon a1 isx1+ 2- . (8)

By equations (7) the curve is upon the hyperboloid of revolution*x1Y+y -4s 2= 1.

Any two successive positions of a plane fl, subject to a plane movement, intersectin a straight line an element of its envelope making the same (constant) anglewith the plane of movement a and ,) as /3does. Hence, the tangentsof theedge of regression of the envelope of d make a constant angle with the plane ofmovement, i. e., the edge is a curve of constant slope.t The slope of the curve(7) is the reciprocal of s.

The edge of regression has cusps at the four points (2, 0,3(O, 2, - ( 2, ) (O,-2, -1, ,given in orderalongit. Theportions between are convex toward the origin. The projecting cylinders (7) areeach tangent to one of the nodal parabolas. Hence, at each cusp of the edge ofregression there is a tangent nodal parabola.Due to the convexity toward the origin of the edge of regression, it is pos-sible to pass a plane between the origin on one side and three of its cusps on theother, so near the latter that the edge of regression pierces the plane in six real

* For this result I am indebted to the " Referee," who has shown that the edge of regression of theenvelope of a plane, when the centrodes of the movement are circles, is upon a quadric of revolution.t The " Referee."


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152 BLAKE: The Ellipsograph of Proclus.points, two in the vicinity of each cusp. The edge of regression is thus of thesixth order, and each piercing point is a cusp on the curve of intersection of theplane and developable. In addition to these cusps, the plane section has fourdouble points, the piercing points of the two nodal parabolas. For the ordersix, with four double points and six cusps, Plucker's equations give: class four,no inflexions aindthree double tangents. The multiple points accounted formust be all, since an additional node or cusp would reduce the class to two orone. The two parabolas are the only nodal lilles. The value four for the classof the surface,that being the same as the class of its plane sections, is verified bya theorem of Darboux* which states the class of a surfaceenveloped by a planehaving one degree of freedom to be equal to the order of the trajectory of apoint subjected to the inverse movement. The inverse gives in this case thelimapon.We are now preparedto study the sections of the surface (6) by planes par-allel to ,. As remarked, they are the same as the envelopes of the straightlines of orwhich are parallel to y = 0. The section by z1= 0 is a four-cuspedhypocycloid, the cuspidal tangents bisecting the angles between the coordinateaxes. Its cusps are at the middle points of the arcs of the hypocycloid

+ ya2of double its size, which is the projection of the edge of regression. Sincez1= 0 intersects each of the nodal parabolas in. two imaginary points, thesection has four imaginary double-points. Besides the four reail cusps, ithas two which are imaginary, the remaining points of intersection of theplane with the edge of regression. As the value of c increases from zero,the four real cusps of the section by z1= c move symmetrically away fromthe axis of y1 going along the curve x + y2 = 2a towards its cusps (ih 2, 0).The curve of section has no real nodes until it becomes tangent to the nodalparabola xZ 48Z1+ 2 = 0, Yi = 0 fore = 1 when the section has a tac-node~~~~~~~~2sat the origin. By further increasing c, the tac-node resolves itself into two realnlodes which move along the axis of xl away from the origin in opposite direc-tions. For c = 23 , two cusps and a node unite at each of the points (?t 2, 0)2s3

* Koenigs, '; Legons de cinematique, p. 353.


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BLAKE: The Ellipsograph of Proclus. 153and the curve which has no visible singularities resembles an ellipse whose majoraxis is on the axis of x1. The singular points of this curve are the cusps of theedge of regression. For values of c greater than 23 the sections resemble ellipses,2stheir only real singularities being two conjugate points within and on the axisof x1. The curves become more and more nearlv circular as c increases invalue. The curve of section by z=-- c, whenl rotated in its plane through ariglht angle, is the same as the section by z, = c.In the case of any plane movemnent whose centrodes are circles, the charac-teristic of the envelope of a carried plane through a diameter AB of the movingcentrode, is a straight line passing through the center of the centrode and per-pendicular to AB. Hence, the projection of the edge of regression upon theplane of movement is the evolute of the cycloid enveloped by AB, i. e., a similarcycloid; and all plane sections of the envelope parallel to the plane of movenmentare involutes of this projection. For the special movement under consideration,AB envelopes

x 2+ y- = 1and its evolute is xl + y 2which agrees with equation (8). The involutes of this are the sections of (6) byzi = con st.*Turning, in conclusion, to the inverse of the ellipsograph, a straight line ofa, envelopes a circle in a whose center is upon c, for it is parallel to a diameterof cl, and every diameter of cl passes through a fixed point of c. The envelopeof a carried plane is a cone or a cylinder of revolution.

* My attention was called to this by the " Referee."

the ellipsograph of proclus (april 1, 1900) 2369752

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