SEELEY G. 1 1UDD LIBRARY LAWRENCE UNIVERSITY Appleton, Wisconsin «__ CURVES AND THEIR PROPERTIES A HANDBOOK ON CURVES AND THEIR PROPERTIES ROBERT C. YATES United States Military Academy J. W. EDWARDS — ANN ARBOR — 1947 97226 NOTATION octangular C olar Coordin =r.t^r ini Tangent and the Rad ,- ^lem a Copyright 1947 by R m Origin to Tangent. i i = /I. f(s ) = C well Intrinsic Egua 9 ;(f «r,p) = C 1- Lithoprinted by E rlll CONTENTS PREFACE nephroid and teacher lume proposes to supply to student curves. Rather ,n properties of plane Pedal Curves e Pedal Equations U vhi C h might be found ^ Yc 31 r , 'f Lr!-ormation and in engine useful in the classroom 3 aid in the s Radial Curves alphabetical arrangement is Roulettes Semi-Cubic Parabola Sketching Evolutes, Curve Sketching, and Spirals Strophoid If 1 :s readily understandable. Trigonometric Functions .... Trochoids Witch of Agnesi bfi Stropho: i including the Astroid, HISTORY: The Cycloidal curves, discovered by Roemer (1674) In his search for the Space Is provided occasionally for the reader to ir ;,„ r e for gear teeth. Double generation was first sert notes, proofs, and references of his own and thus be st form noticed by Daniel Bernoulli in 1725- It is with pleasure that the author acknowledges a hypo loid o f f ur valuable assistance in the composition of this work. 1. DESCRIPTION: The d is y Mr. H. T. Guard criticized the manuscript and offered Le roll helpful suggestions; Mr. Charles Roth and Mr. William radius four Lmes as la ge- -oiling upon the ins fixed circle (See Epicycloids) ASTROID EQUATIONS: = cos 1 1 1 (f)(3 x + - a y sin [::::::: = (f)(3 :ion: (Fig. l) Through P d . METRICAL PROPERTIES: radius i the circle of ^ L = 6a BIBLIOGRAPHY Macmillan (1892) 337- Edwards J.- Calculus , 'g • Dublin (1879) 278. Salmon Higher Flare Curves , Kurven Leipzig (1908). Wleleitner, H. : Spezielle ebene , nans, Green i i , _. i _ dl in , _ (1895) 339- Section on Epicycloids , herein. CARDIOID t-v,c cuicurvere may«uj beuc describeduc=.-i uc as an Epicycloid in Thus the 1 ,. , circle of radius a, or by one of radius £ ays: by a ,.., as shown upon a fixed circle of radius a. I 2. EQUATIONS: Cardioid Is a member of the lamiiy c HISTORY: The 2 2 2 ir 2 2 2 = (x + (Origin at cusp). cloidal Curves, first studied by Roemer (1674) (x + y + 2ax) 4a y ) of gear teeth. vestigation for the best form = (Origin at ci r = 2a(l + cos B), r 2a(l + sin 0) 2 2 2 of circ is an Epicycloid of _ a ) = 8p . (Origin at center fixed 1. DESCRIPTION: The Cardioid 9 ( r point P of a circle rolling cusp: the locus of a t - cos 2t) fx = a(2 cos outside of another of equal size. (Fig. 3a) 2 the =*(-»- e " ) y= a (2slnt-sln2t)> * 3 =4apS r . s=8a-cos(^). 9R = 64a' 3. METRICAL PROPERTIES: - )(.a 2 2 X (^ ) ler cardioid. scial limacon : Let Double Generation: (Pig. Jh) . erated by the point P on the rolling ci Draw ET-, OT'F, and PT' to T. Draw PP t through T, P, D. Since angle DPT = |, t has DT as diameter. Now, PD is parallel 3 parallel. arc T'P = arc T'X. Accordingly, arc TT'X = 2aB = arc TP CARDIOID CARDIOID he oardioid be pivo ted at the cus P and = OD = b; AO = BD = CP = a; BP = DC = c rotated' with constant angular ve locity, a pin, fixed straight lin harmonic motio n. Thu ingle COX. Any point t all times, an;;le F a(l + cos 0), BIBLIOGRAPHY McGraw Hill (1931)- KeoTO and Paires: Mechanism , 2 «) = -k (r - a), L Sketch and -Model 182. v. Press, (1941) 2 fe(r- a) =- k (r- a), il) crossed parallelograms, joined CASSINIAN CURVES ITEMS: If. GENERAL CASSINIAN CURV I plane paral- --'en formed by a of the torus le l to the axis its center, of a Rectanrular [yprrbola. (d) The points P and P' of the linkage shown in 2. EQUATIONS: 2 2 2 [(x - a) + y ]'[U + a) n [Fi = C-a,o) F 2 = (a,o)] 3. METRICAL PROPERTIES: (See Section on Lemniscate) CASSINIAN CURVES CASSINIAN CURVES !>iY are focal radii (measured from F of Q and P be (p,0) and (r,6),^ ; the coordinates ;ely. Since 0, D, and Q 1 .re always at right angles. This 2 2 2 2 2 2 c - 4a sin 8. (O'q) = (DQ) - (DO) point The attached Peaucellier cell inyerts the property P under the BIBLIOGRAPHY Plane Curves Dublin (1879) 44,126. Salmon G. : Higher , Graphics Graphics Press (1909) 74. willson F. N.: , Longmans, Green (1895) 233,533- Williamson, B. : Calculus , Mathematical Sketch and Model Yates, R. C. : Tools , A Book, L. S. U. Press (1941) 186. .: " " ;..,: . < ..! ; ; Let Pig 9, be Fi, it pr ^N. k 2 ? off FiC perpen- rv \ liar to FiF 2 if the circle any radius B h FiX . Dr IV cx ana perpen- die uiar CY. CATENARY 3 i. METRICAL PROPERTIES: A = a-s = 2(area triangle PCB) S x = it(ys + ax) CATENARY stigate tt HISTORY: Galileo w£ s the first to inv noulli some of its in I69I obtained i B true form and ga re 4. GENERAL ITEMS: properties. 1. DESCRIPTION: Th perfectly flexible inextensible chain of unifor n densi al lin hanging from two s rpports not in the = (b) Tangents drawn to the curves y = e , y (c) The path of B, an involute c (e) It is a plane section of the surface of least area (a soap film catenoid) spanning two circular disks, Pig. 11a. (This is the only minimal surface of revolu- 2. EQUATIONS: If = T cos <f ka a a 2 e = 3 sh(^) = (f)(e + ) ; y CATENARY section of a sail bounded by two perpendicular to the plane of t proportional to the sail 'is normal to the element and Routh) square of the velocity, Fig. lib. (See HISTORY: Causti Cayley. ouetelet, Lagrange, and curve 1 s the BIBLIOGRAPHY 1. A ^caustic envelope of Light ra ys, 14th Ed. under "Curves, emitted from a radia nt I fl 458, , 2nd Ed. (I896) Routh, E. J.: An Statics S, afte r re- p. 310. Dublin (1879) 287. refracti on by Trans ." XIV, 625 Vallis! Edinburgh a given curv e f = The caustics by reflect on and refract! on are ailed catacaustic and dia aus- T. Thus 3. The instantaneous ;er of motion of S is the ort ape of normals , TQ, _to - is the evolute of the ortho the reflecting curv 3 locus of P Is the pedal of curve _sln a respect to S. Thus the orthotomic is a to the pedal with double its linear dimensions. ' ; ;.. ,.:,' With usual x,y ax. con whose pole is the radiant point. [radius a, radiant point (c,o)] E E(W2 - a*Kx + lowing forms: With the source S at «, With the source S on the the incident and reflected circle, the incident and rays make angles with reflected rays makes angle the normal at T. Thus the 6/2 with the normal at T. fixed circle 0(a) of Thus the fixed circle and radius a/2 has its arc AB the equal rolling circle equal to the arc AP of the have arcs AB and AP equal. circle through A, P, T of The point P generates a radius a/4. The point P of Cardioid and TPQ is its ta this latter circle gener- gent (AP is perpendicular ates the Nephroid and the to TP). reflected ray TPQ is its tangent (AP is perpendicu- lar to tp). These are the bright curves seen on the surface of cof- (f) f (e) Fig- 15 ee in a cup or upon the table inside of a napkin ring. Dlacausties at a Line L 7- 2512 Caustics by. Refraction ( ) ST Is Incident, QT refracted, and S is the reflectic drawl S in L. Produce TQ_to meet the variable circle of inciden. and S in P . Let the angles through S, Q, PS - PS = SS 8 = and refraction be 6i and 2 and H . The tus of P is tl en an hyper bo La wi th S, S ss/n ty ,"1 PQT a 1 ~mLl "The -ays PQT -bola is (UlUl e, the (Pig 17) THE CIRCLE A circle is a plane continuous curve all 1 DESCRIPTION: points are equidistant from a fixed coplanar of whose d) If the point i -efleoted rays are all noi 2. EQUATIONS: 2 - = A COS 29 + B having 2 E 2 (x - h) + (y - k) = a 2 2 = x + y + Ax + By + C C . METRICAL PROPERTIES: 2 L = 2na 2 = 4na 4. GENERAL ITEMS: r-cle, produc BIBLIOGRAPHY PB circle divides car:.! line : ; constant; i.e., PA = PD-PC (since the arc subtended by / BCD plus that leal Monthly: 28(1921) 182,187- subtended by L BAP Is the entire circumference, tri- "Memoir on Caustics", Philosophical Trans - Dayley A.- angles PAD and PBC are similar). To evaluate this actions ' (1856) constant, p, draw the line through P and the center R. S.: Geometrical Optics (1895) 105- 2 2 Heath, of the circle. Then (P0 - a)(P0 + a) = p = (P0) - a . Curves , Dublin (1879) 98. Salmon G. : Higher Plane The quantity p is called the power of the point P with respect to the circle. If p <, = , > 0, P lies re- The locus of all points P which have equal power I respect to two fixed circles is a line called the Fig.