Characterization of the Three-Bar Linkage System Generated Symmetric and Asymmetric Lemniscate-Like Curves

INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 2016 Characterization of the Three-Bar Linkage System Generated Symmetric and Asymmetric Lemniscate-like Curves

Hee Seok Nam, Gaspar Porta, and Hyung Ju Nam

Abstract—In this paper, we investigate the characterization of the family of curves generated by three-bar linkage systems. Depending on the location of the marker on the middle rod, symmetric or asymmetric lemniscate-like curves are constructed. An algebraic representation is presented using the distances from the foci as a generalization of the Lemniscate of Bernoulli. It turns out that the corresponding Cartesian equation is of the form of the Hippopede defined as the intersection of a torus and a plane. A geometric Fig. 1 Cassini Ovals construction shows how we can construct a family of symmetric or asymmetric lemniscate-like curves using a circle and a fixed point. It A linkage system corresponds to a set of bars connected to leads to a polar representation. Finally, a parametric representation is each other and the plane by articulations (called linkages) given for the completion of the characterization. typically allowing a marking point on one of the bars to move describing a curve in the plane. More generally a linkage Keywords— asymmetric lemniscate-like curve, Lemniscate of system can have the marking point describe a region as Bernoulli, symmetric lemniscate-like curve, three-bar linkage system opposed to a curve--sometimes with special geometric effects. The formal terms used to describe the anatomy of a Linkage I. INTRODUCTION System throughout the rest of the paper follows. HIS paper is the result of extended observations Toriginating during an exploration of A. B. Kempe’s work 1. Fixed point/Fixed Pivot: A fixed point of a linkage is a “How to Draw a Straight Line; a Lecture on Linkages” [2] by point which does not change its location during the the authors in [3]. motion of the linkage system. Given two fixed points in the plane, Cassini Ovals can be 2. Rod/Bar: A rod or bar is the line segment that connects characterized as the set of points so that the product of the two distinct points. A rod can take only rigid motions distances from the two fixed points (called the foci) is a during the motion of the linkage system. constant. When the foci are 2 apart, if we collect points with 3. Hinge Point/Pivot Point: A hinge point is the “linkage 2 fixed constant for the product of distances from the foci, point”, and thus serves as the connection between two 𝑎𝑎 we obtain either two disconnected loops, the Lemniscate of rods. 𝑏𝑏 Bernoulli, or a connected simple loop enclosing both foci 4. Mover: A mover is the end point on a rod, so called the depending on the range of : < , = and > . See driver, which rotates around a fixed point. It has one 1 Fig. 1 for an illustration . Among many construction methods degrees of freedom and, in fact, we can use the angle of 𝑏𝑏 𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 𝑎𝑎 of the Lemniscate of Bernoulli, linkage systems are of great rotation to represent the location of the marker which interest in this paper due to their usefulness in generalization draws the desired curve as a function of mover. and characterization in several ways. 5. Marker: A marker, also known as tracer, is a point on a rod which will draw the whole or part of the desired curve as the mover rotates around a fixed point on a rod.

H. S. Nam is with the Department of Mathematics, Kettering University, To draw the families of curves that include the Lemniscate Flint, MI 48504 USA (e-mail: [email protected]). of Bernoulli, one can use a three-bar linkage system. This G. Porta is with the Department of Mathematics and Statistics, Washburn University, Topeka, KS 66621 USA (e-mail: [email protected]). system corresponds to a chain of three bars linked (hinged) to H. J. Nam is with Washburn Rural High School, Topeka, KS 66619 USA each other with the outside two bars also linked to the plane at (e-mail: [email protected]). two fixed points. The relative lengths of these bars, the 1 http://mathworld.wolfram.com/CassiniOvals.html

ISSN: 2313-0571 56 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 2016 distance between the fixed points, and the position of the On the other hand, we apply Pappus’s theorem to those marker (along the middle bar) determine the particular curve triangles and get 2 2 2 2 2 being drawn. + | | = 2(1 + | | ) = 2(1 + 1 ), (3) 2 2 2 2 2 + | | = 2(1 + | | ) = 2(1 + 2 ). (4) Combining𝑟𝑟 (3),𝐴𝐴𝐴𝐴 (4) and (2), we𝐴𝐴𝐴𝐴 have the desired𝑑𝑑 result (1). 𝑟𝑟 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 𝑑𝑑

∎ Characterization (1) generalizes the Lemniscate of Bernoulli. In fact, if = 2, then (1) reduces to dist( , ) 2 𝑟𝑟 =√ 1 = 1 2 2 𝐴𝐴 𝐵𝐵 which exactly coincides with the definition of the Lemniscate Fig. 2 Three-bar linkage system 𝑑𝑑 𝑑𝑑 � � of Bernoulli. Fig. 2 is a description of a three-bar linkage system: and

are fixed points; , , are rods; and are hinge Corollary 2 For the three-bar linkage system in Theorem 1, points; is the driver and is the mover rotating around𝐴𝐴 the if = ( 1,0) and = (1,0), then the Cartesian equation of fixed𝐵𝐵 point ; is the𝐴𝐴𝐴𝐴 marker.𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷 The Lemniscate𝐶𝐶 of 𝐷𝐷Bernoulli is 𝐴𝐴𝐴𝐴 𝐶𝐶 the marker F(x,y) is given as a particular case of the full set of curves that may be drawn in 𝐴𝐴 − 𝐵𝐵 2 2 this way. 𝐴𝐴Symmetric𝐹𝐹 lemniscate-like curves, skewed or ( + 1)2 + 2 + 1 ( 1)2 + 2 + 1 asymmetric lemniscate-like curves, and 'teardrop' curves also 2 2 𝑟𝑟 2 𝑟𝑟 appear. � 𝑥𝑥 𝑦𝑦 − � �2 𝑥𝑥 − 𝑦𝑦 − � = 2 . It turns out that the trace of the marker of three-bar linkages 2 𝑟𝑟 with specific rod lengths produces a family of symmetric � − � lemniscate-like curves and it is natural to ask how we can If the two fixed points are 2 away and the connecting rods characterize such a family of curves. This paper will focus on have lengths , 2 , and , then the trace of the marker can be answering this question in various ways. In the following characterized as 𝑎𝑎 sections, we will investigate an algebraic characterization, a 𝑟𝑟 𝑎𝑎2 𝑟𝑟 2 2 2 2 + 2 2 + 2 = 2 2 . (5) geometric construction, a parametric representation, and a 1 2 2 2 2 𝑟𝑟 𝑟𝑟 𝑟𝑟 generalization to skewed lemniscate-like curves, or A proof of this relation (5) is essentially identical to that of �𝑑𝑑 𝑎𝑎 − � �𝑑𝑑 𝑎𝑎 − � � 𝑎𝑎 − � asymmetric lemniscate-like curves. Theorem 1. If we set the two fixed points as ( , 0) and ( , 0), then the Cartesian equation of the graph can be II. ALGEBRAIC CHARACTERIZATION simplified to −𝑎𝑎 𝑎𝑎 2 2 2 2 2 2 2 2 In this section, we give an algebraic characterization of the ( + ) = + ( 4 ) 2 family of symmetric lemniscate-like curves generated by which is exactly the Hippopede of the form 2 2 2 2 2 three-bar linkage systems. See Fig. 2 for an illustration. We 𝑥𝑥 (𝑦𝑦 + )𝑟𝑟 𝑥𝑥 = 𝑟𝑟 +− 𝑎𝑎 𝑦𝑦 initially pick the distance dist( , ) = 2 and | | = 2 for our where > 0 and > . computations and later generalize to encompass any scale. 𝑥𝑥 𝑦𝑦 𝑐𝑐𝑥𝑥 𝑑𝑑𝑦𝑦 𝐴𝐴 𝐵𝐵 𝐶𝐶𝐶𝐶 III.𝑐𝑐 GEOMETRIC𝑐𝑐 CONSTRUCTIO𝑑𝑑 N AND POLAR EQUATION Theorem 1 If a three-bar linkage system is constructed by two A geometric characterization of symmetric lemniscate-like fixed points , with dist( , ) = 2, and three bars , , curves highlights a connection to three-bar linkage systems. with | | = | | = , and | | = 2, then the family of This part is mainly motivated from the work of Akopyan [1]. symmetric lemniscate𝐴𝐴 𝐵𝐵 -like curves𝐴𝐴 𝐵𝐵 generated by the marker𝐴𝐴𝐴𝐴 𝐵𝐵 𝐵𝐵, the𝐶𝐶𝐶𝐶 midpoint𝐴𝐴𝐴𝐴 of the𝐵𝐵 line𝐵𝐵 segment𝑟𝑟 𝐶𝐶𝐶𝐶, can be characterized as Lemma 3 Given three-bar linkage system as in Fig. 3 where 2 2 2 2 𝐹𝐹 dist( , ) = | | = 2 and | | = | | = , if , and 2 + 1 2 + 1 = 2 (1) 1 2 2 𝐶𝐶𝐶𝐶 2 2 𝑟𝑟 𝑟𝑟 𝑟𝑟 are midpoints of , and , respectively and is a where = | | and = | |. 𝐴𝐴 𝐵𝐵 𝐶𝐶𝐶𝐶 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 𝑟𝑟 𝑀𝑀 𝑁𝑁 𝑂𝑂 1 �𝑑𝑑 − �2 �𝑑𝑑 − � � − � parallelogram, then the followings hold. (a) Points , 𝐴𝐴𝐴𝐴, , 𝐴𝐴and𝐴𝐴 are𝐴𝐴𝐵𝐵 collinear. 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 Proof. 𝑑𝑑The proof𝐴𝐴𝐴𝐴 comes𝑑𝑑 from𝐵𝐵𝐵𝐵 the combination of Pappus’s (b) = theorem and the law of cosines. In fact, the quadrilateral (c) | | =𝑀𝑀 | 𝐹𝐹 |𝑋𝑋 = /2 𝑂𝑂 is an anti-parallelogram and = . For the 𝑋𝑋𝑋𝑋��⃗ �𝑂𝑂��𝑂𝑂�⃗ triangles and , the law of cosines gives Proof. 𝐴𝐴Since𝐴𝐴 𝐴𝐴𝐴𝐴 is𝑟𝑟 an anti-parallelogram, we have 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 | |2+22 2 ∠BCD ∠ADC = cos 𝐴𝐴𝐴𝐴𝐴𝐴 4 | 𝐵𝐵𝐵𝐵𝐵𝐵| and is also a parallelogram. Therefore , , and 𝐵𝐵𝐵𝐵 −𝑟𝑟 2 2 2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 ∥ | | +2 are collinear and = which lead to = . 𝐵𝐵 𝐵𝐵 = cos = . ∠𝐵𝐵𝐵𝐵𝐵𝐵 4 | | 𝐶𝐶𝐶𝐶 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑀𝑀 𝐹𝐹 𝑋𝑋 𝐴𝐴𝐴𝐴 −𝑟𝑟 Finally, | | = | | = | |/2 = /2 = | |. ��⃗ ��⃗ ��⃗ ��⃗ Solving this equation leads to the relation 𝐴𝐴𝐴𝐴 𝑂𝑂 𝐹𝐹𝐹𝐹 𝑂𝑂𝑂𝑂 𝑋𝑋𝑋𝑋 𝑂𝑂𝑂𝑂 ∠𝐴𝐴𝐴𝐴𝐴𝐴 2 | | | | = 4 . (2) 2 𝐴𝐴𝐴𝐴 𝑁𝑁𝑁𝑁 𝐵𝐵𝐵𝐵 𝑟𝑟 𝐴𝐴𝐴𝐴 https://en.wikipedia.org/wiki/Hippopede 𝐵𝐵𝐵𝐵 ⋅ 𝐴𝐴𝐴𝐴 − 𝑟𝑟 ISSN: 2313-0571 57 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 2016

Finally, from this polar representation, we can classify the family of curves as follows: ∎ (a) If 0 < < 2, then we have a figure-8 curve, or symmetric lemniscate-like curve. (b) If = 2, 𝑟𝑟then we have two unit circles centered at foci. (c) If > 2, then we have a connected simple loop enclosing𝑟𝑟 both foci. 𝑟𝑟 IV. PARAMETRIC REPRESENTATION

Fig. 3 Characterizing the location of If we want to construct a graph, it is very helpful to get a parametric representation. To find a parametric representation Lemma 3 holds true for general values of and𝐹𝐹 we can of the marker , we assume that = ( 1,0), = (1,0), obtain another characterization of symmetric lemniscate-like | | = | | = , | | = 2, and is the midpoint of as in curves, or the Hippopede. A direct consequence 𝑟𝑟of this lemma Fig. 5. We measure𝐹𝐹 the angle counterclockwise𝐴𝐴 − from𝐵𝐵 the ray can be stated as follows. 𝐴𝐴𝐴𝐴 to the𝐵𝐵𝐵𝐵 driver𝑟𝑟 𝐶𝐶𝐶𝐶 so that 𝐹𝐹 = ( 1 + cos , 𝐶𝐶 𝐶𝐶sin ). From the key fact that the quadrilateral𝜃𝜃 is an anti- Theorem 4 Given three-bar linkage system as in Fig. 4 where parallelogram,𝐴𝐴𝐴𝐴 we 𝐴𝐴𝐴𝐴have =𝐶𝐶 (2 −cos 𝑟𝑟, 𝜃𝜃sin𝑟𝑟 ) and𝜃𝜃 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 dis( , ) = | | = 2, is the midpoint of , | | = = for some constant . The point = ( 1,0) + ��⃗ | | = , and the circle with radius /2 and center , the set (2 cos , sin ) satisfies𝐶𝐶𝐶𝐶 | − 𝑟𝑟| = 𝜃𝜃 w−𝑟𝑟hich gives𝜃𝜃 a 𝐴𝐴 𝐵𝐵 𝐶𝐶𝐶𝐶 𝑂𝑂 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 ��⃗ ��⃗ of all points where = draws a symmetric quadratic𝐴𝐴𝐴𝐴 𝑡𝑡𝐶𝐶𝐶𝐶 equation in : 𝑡𝑡 𝐷𝐷 − 𝐵𝐵𝐵𝐵 𝑟𝑟 𝑟𝑟 𝐴𝐴 lemniscate-like curve generated by the three-bar linkage 𝑡𝑡 − 𝑟𝑟 { (𝜃𝜃2 −𝑟𝑟 cos𝜃𝜃 ) 2}2 + ( 𝐵𝐵 𝐵𝐵sin 𝑟𝑟)2 = 2. 𝐹𝐹 �𝑂𝑂��𝑂𝑂�⃗ 𝑋𝑋𝑋𝑋��⃗ system. Since = 1 is a trivial𝑡𝑡 solution, it is not surprising to factor the equation𝑡𝑡 and− 𝑟𝑟get the𝜃𝜃 solution− set𝑡𝑡 as ⋅ 𝑟𝑟 𝜃𝜃 𝑟𝑟 𝑡𝑡 4 2 , 1 . 4 4 cos + 2 − 𝑟𝑟 We discard the possibility� = 1 to represent� as (2 −cos𝑟𝑟 )(4𝜃𝜃 2𝑟𝑟) (4 2) sin = 1 + , 4 4 cos𝑡𝑡 + 2 4 4𝐷𝐷cos + 2 − 𝑟𝑟 𝜃𝜃 − 𝑟𝑟 − − 𝑟𝑟 𝑟𝑟 𝜃𝜃 and𝐷𝐷 the marker�− as � − 𝑟𝑟 𝜃𝜃 +𝑟𝑟 − 𝑟𝑟 𝜃𝜃 𝑟𝑟 Fig. 4 Geometric construction = 𝐹𝐹 2 (2 cos )(2 cos 𝐶𝐶 )𝐷𝐷 2 sin ( 2 cos ) 𝐹𝐹 From the triangle , the law of cosine gives = 2 , 2 . 2 4 4 cos + 4 4 cos + 2 𝑟𝑟 − 𝑟𝑟 𝜃𝜃 𝜃𝜃 −𝑟𝑟 𝑟𝑟 𝜃𝜃 𝑟𝑟 − 𝜃𝜃 = 1𝐴𝐴𝐴𝐴 + 𝐴𝐴| | 2 | | cos( ) � � 2 − 𝑟𝑟 𝜃𝜃 𝑟𝑟 − 𝑟𝑟 𝜃𝜃 𝑟𝑟 or 𝑟𝑟 � � 𝑂𝑂𝑂𝑂 − 𝑂𝑂𝑂𝑂 𝜋𝜋 −𝜙𝜙 2 | | = cos sin2 . (6) 2 𝑟𝑟 If we apply intersecting secants theorem to the secant , 𝑂𝑂𝑂𝑂 − 𝜙𝜙 − �� − 𝜙𝜙 we have 𝑂𝑂𝑂𝑂𝑂𝑂 | | | | = | | (| | + | |) = 1 1 + 2 2 or 𝑟𝑟 𝑟𝑟 𝑂𝑂𝑂𝑂 ⋅ 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂 ⋅ 𝑂𝑂𝑂𝑂 𝑀𝑀𝑀𝑀 � − � � � 1 2 | | = | | + 1 . (7) | | 4 𝑟𝑟 Fig. 5 Three-bar linkage system: Midpoint marker Substituting (6) to (7), we obtain the following result: 𝑀𝑀𝑀𝑀 − 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂 � − �

V. GENERALIZATION TO SKEWED OR ASYMMETRIC Theorem 5 For a three-bar linkage system with rod lengths r, LEMNISCATE-LIKE CURVES 2, r and foci at ( 1,0) and (1,0), the trace of the marker, or the midpoint of the middle rod can be represented by a polar In this section, we generalize the location of the marker so equation: − that it can be any fixed point on the middle rod. See Fig. 6 where + = 2. We observe a family of skewed or 2 asymmetric lemniscate-like curves if . = 2 sin2 . 2 𝑚𝑚 𝑛𝑛 𝑟𝑟 𝑚𝑚 ≠ 𝑛𝑛 𝜌𝜌 �� − 𝜙𝜙

ISSN: 2313-0571 58 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 2016

Under the same assumptions of Theorem 7, we can construct skewed or asymmetric lemniscate-like curves geometrically. When the marker is located at such that | | = < = | |, we construct the following points in this order (see Fig. 8): 𝐹𝐹 𝐶𝐶𝐶𝐶 𝑚𝑚 𝑛𝑛 𝐹𝐹𝐹𝐹 • is the intersection of and the circle with center at and radius /2 , • 𝑀𝑀 is on such that | 𝐴𝐴𝐴𝐴| = = | |, 𝑌𝑌 𝐴𝐴 𝑛𝑛𝑛𝑛 Fig. 6 Three-bar linkage system: Generalized marker • is on such that | | = | |/2, 𝑂𝑂′ 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴′ 𝑛𝑛 𝐹𝐹𝐹𝐹 • is on such that | | = | |/2, 𝑁𝑁 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 𝑚𝑚 𝐴𝐴𝐴𝐴 For an algebraic characterization, we need a generalized • is the point such that the quadrilateral is a version of Pappus’s theorem. parallelogram.𝑁𝑁′ 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴′ 𝑛𝑛 𝐴𝐴𝐴𝐴 𝑋𝑋 𝐴𝐴𝐴𝐴′𝑂𝑂′𝑋𝑋 Lemma 6 For Fig. 7, we have the relation: Elementary geometry shows that 2 1 , + 2 = 2 + 2. 1 2 + + | | = | | = | |, (10) 1 2 1 2 2 = = 𝑐𝑐 𝑐𝑐 𝑀𝑀𝑀𝑀 ∥ 𝐴𝐴𝐴𝐴𝑚𝑚 In particular,𝑐𝑐 𝑐𝑐 if 1 𝑑𝑑 2 , then 𝑎𝑎the relation reduces𝑏𝑏 to | | = | | = | | = = | |. (11) 𝑐𝑐 1𝑐𝑐 𝑐𝑐 𝑐𝑐 𝑀𝑀𝑀𝑀 𝐴𝐴2𝐴𝐴 𝐴𝐴2𝐴𝐴 2 + 2 = ( 2 + 2). 𝑛𝑛 𝑛𝑛𝑛𝑛 𝑐𝑐 𝑐𝑐 𝑐𝑐 2 𝐴𝐴𝐴𝐴 𝑁𝑁′𝑂𝑂′ 𝐷𝐷𝐷𝐷 𝐴𝐴𝐴𝐴 𝑐𝑐 𝑑𝑑 𝑎𝑎 𝑏𝑏

Fig. 7 Generalized Pappus's Theorem

Proof. We use vectors and the dot product to prove the theorem. In fact, 2 1 = + 1 + 2 1 + 2 Fig. 8 Geometric construction 𝑐𝑐 𝑐𝑐 which leads to ��⃗ ��⃗ ��⃗ 𝐴𝐴𝐴𝐴2 𝐴𝐴2 𝐴𝐴 2 𝐴𝐴𝐴𝐴 2 𝑐𝑐 2 𝑐𝑐 2𝑐𝑐 𝑐𝑐 Let's consider the case that the angle = [ ( 1 + 2) = 2 + 1 + 2 1 2 . sin 1 , + sin 1 ]. The four points , , ′ and are | | = | | = + 2 2 Since 1 2, we have 𝑟𝑟 𝑟𝑟 𝜙𝜙 ∠𝐵𝐵𝑂𝑂 𝑋𝑋 ∈ 𝜋𝜋 − 𝑐𝑐 𝑐𝑐 �𝐴𝐴��𝐴𝐴�⃗� 𝑐𝑐 �𝐴𝐴��2�𝐴𝐴�⃗� 𝑐𝑐 2�𝐴𝐴𝐴𝐴��⃗� 𝑐𝑐 𝑐𝑐 �𝐴𝐴��𝐴𝐴�⃗ ⋅ 𝐴𝐴𝐴𝐴��⃗ collinear− in this order.− From the relation (2) and (10), we have + ( + )2 � � 𝜋𝜋 � � 𝑂𝑂′ 𝑋𝑋 𝐹𝐹 𝑀𝑀 𝐴𝐴𝐴𝐴��⃗ −�� 𝐴𝐴��𝐴𝐴�⃗ =𝐵𝐵��𝐵𝐵�⃗ 𝑐𝑐 𝑐𝑐 1 2 . a system of equations: 2 ��⃗ ��⃗ We plug this dot product�𝐴𝐴𝐴𝐴 into� (8�𝐴𝐴𝐴𝐴) and� −simplify𝑐𝑐 𝑐𝑐to get the 2 �𝐴𝐴��𝐴𝐴�⃗ ⋅ 𝐴𝐴𝐴𝐴��⃗ 4 desired result. | | = | | | | = | |, 2 | | 2 ′ ′ 𝑛𝑛 4 − 𝑟𝑟2 𝑛𝑛 Now we are ready to give a geometric characterization of the∎ 𝑀𝑀𝑀𝑀 𝑀𝑀𝑂𝑂 − 𝑋𝑋𝑂𝑂 ⋅ − 𝐴𝐴𝐴𝐴 | | = | | | | = 𝐴𝐴𝐴𝐴 | |. family of curves generated by a generalized marker of three- 2 | | 2 ′ ′ 𝑛𝑛 − 𝑟𝑟 𝑚𝑚 bar linkage systems. See Fig. 6. Solving 𝐹𝐹𝑂𝑂for | |𝑀𝑀𝑂𝑂 in terms− 𝑀𝑀of𝑀𝑀 | | le⋅ads to: − 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 | | 1 2 Theorem 7 For a three-bar linkage system with two fixed | | = 𝐹𝐹𝐹𝐹′ + |𝑀𝑀𝑀𝑀 |2 + ( )2 1 . 2 4 points and , and three bars , , such that ′ 𝑀𝑀𝑀𝑀 − 𝑚𝑚𝑚𝑚 𝑟𝑟 | | Since𝐹𝐹𝑂𝑂 and are on� the same𝑀𝑀 circle𝑀𝑀 by𝑚𝑚 (11), −𝑛𝑛 we� can− � dist( , ) = = 2 and | | = | | = , the family of 𝑛𝑛 𝑛𝑛 skewed𝐴𝐴 or asymmetric𝐵𝐵 lemniscate𝐴𝐴𝐴𝐴-like𝐵𝐵𝐵𝐵 curves𝐶𝐶𝐶𝐶 generated by the determine the location of as described in the next theorem. 𝑀𝑀 𝑋𝑋 1 𝑌𝑌 1 marker𝐴𝐴 𝐵𝐵 with𝐶𝐶 |𝐶𝐶 | = and𝐴𝐴𝐴𝐴 | | =𝐵𝐵𝐵𝐵 = 2𝑟𝑟 can be Note that if [ ( sin ), sin ], then , 2 2 characterized as 𝐹𝐹 𝑟𝑟 𝑟𝑟 , , and are collinear in −this order, but the− following 𝐹𝐹 𝐶𝐶𝐶𝐶2 𝑚𝑚 𝐹𝐹𝐹𝐹 2 𝑛𝑛 − 𝑚𝑚 2 2 𝜙𝜙 ∈ − 𝜋𝜋 − � � 𝜋𝜋 − � � 𝑋𝑋 2 + 2 + = 2 (9) geometric construction still remains accurate. 1 2 2 2 2 𝑛𝑛𝑟𝑟 𝑚𝑚𝑟𝑟 𝑟𝑟 𝑀𝑀 𝑂𝑂′ 𝐹𝐹 where = | | and = | |. 1 2 ( , ) = �𝑑𝑑 𝑚𝑚𝑚𝑚 − � �𝑑𝑑 𝑚𝑚𝑚𝑚 − � 𝑚𝑚𝑚𝑚 � − � Theorem 8 Let and be two fixed points with dist 2 ( , ) = The proof𝑑𝑑 is similar𝐴𝐴𝐴𝐴 to𝑑𝑑 that of𝐵𝐵 Theorem𝐵𝐵 1 and left to the reader. , and is on the line segment such that dist , 𝐴𝐴 𝐵𝐵 𝐴𝐴 𝐵𝐵 Note that the relation (2) is still valid while we need a dist( , ) = = 2 . For any point on the circle 𝑂𝑂′ 𝐴𝐴𝐴𝐴 𝐴𝐴 𝑂𝑂′ 𝑛𝑛 generalized version of Pappus’s theorem (Lemma 6) to replace with center at and radius /2, let and be determined 𝑂𝑂′ 𝐵𝐵 𝑚𝑚 − 𝑛𝑛 𝑀𝑀 𝑌𝑌 (3) and (4). as 𝐴𝐴 𝑛𝑛𝑛𝑛 𝑋𝑋 𝐹𝐹

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(a) is the other intersection of the circle and the line passing through and , (b) 𝑋𝑋 = where 𝑌𝑌 𝑂𝑂′ 𝑀𝑀 2 2 ��⃗ ��⃗1 1 ( ) 𝑂𝑂′𝐹𝐹 𝑘𝑘 𝑋𝑋𝑋𝑋= + + 1 . 2 | |2 4 − 𝑚𝑚𝑚𝑚 𝑚𝑚 −𝑛𝑛 𝑟𝑟 Then the locus𝑘𝑘 of is� an asymmetric lemniscate� −-like� curve 𝑛𝑛 𝑛𝑛 𝑋𝑋𝑋𝑋 generated by the three-bar linkage system with two fixed points and , and𝐹𝐹 three bars , , such that 2 2 = and = . Moreover, dist( , ) = | | = 𝐴𝐴 𝐵𝐵 𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶 𝐷𝐷𝐷𝐷 2, | | = | | = , | | = and | | = = 2 . ��⃗ 𝑛𝑛 ��⃗ ��⃗ 𝑚𝑚 ��⃗ 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶 𝐴𝐴 𝐵𝐵 𝐶𝐶𝐶𝐶 From𝐴𝐴𝐴𝐴 the triangle𝐵𝐵𝐵𝐵 𝑟𝑟 𝐶𝐶𝐶𝐶 (see Fig𝑚𝑚 . 8), similar𝐹𝐹𝐹𝐹 t𝑛𝑛o the derivation− 𝑚𝑚 of (6) and (7), we have 𝐴𝐴𝐴𝐴𝐴𝐴′ Fig. 9 Asymmetric lemniscate-like curve: = 1.7, = 0.7, 2 = 1.3 | | = cos sin2 , 4 𝑟𝑟 𝑚𝑚 ′ 𝑟𝑟 𝑛𝑛 𝑂𝑂 𝑋𝑋 −𝑛𝑛 2 𝜙𝜙 −𝑛𝑛�2 − 𝜙𝜙2 | | = | | + 1 = 2 sin2 , | | 4 4 ′ 𝑛𝑛 𝑟𝑟 𝑟𝑟 𝑀𝑀𝑀𝑀 − 𝑂𝑂 𝑋𝑋 ′ � − � 𝑛𝑛� − 𝜙𝜙 𝑂𝑂 𝑋𝑋 2 | | = | | + | | = cos + sin2 . 4 ′ ′ 𝑟𝑟 4 2 � Since𝑂𝑂 |𝑀𝑀 | =𝑂𝑂 𝑋𝑋 𝑋𝑋𝑋𝑋, we have− 𝑛𝑛 𝜙𝜙 𝑛𝑛 − 𝜙𝜙 2 | | ′ 𝑛𝑛 −𝑟𝑟 𝑂𝑂 𝑀𝑀 ⋅ 𝐴𝐴𝐴𝐴 2 | | = 2 cos + sin2 . 4 𝑟𝑟 𝐴𝐴𝐴𝐴 − � 𝜙𝜙 � − 𝜙𝜙� Finally, 2 Fig. 10 Relationship between sets of curves | | = | | | | = ( ) cos + 2 sin2 . 2 4 𝑚𝑚 𝑟𝑟 𝑂𝑂′𝐹𝐹 𝑂𝑂′𝑀𝑀 − 𝐴𝐴𝐴𝐴 𝑚𝑚 −𝑛𝑛 𝜙𝜙 � − 𝜙𝜙 VI. CONCLUSION Theorem 9 For a three-bar linkage system with two fixed Fig. 10 indicates the relationship between the sets made up = ( , 0) = ( , 0) points and , and three bars , , of the families of curves we have connected. The geometric ( , ) = + = | | = 2 | | = such that dist and relationship that characterizes Cassini Ovals leads to an | | =𝐴𝐴 −𝑛𝑛 𝐵𝐵 𝑚𝑚 𝐴𝐴𝐴𝐴 𝐵𝐵𝐵𝐵 , then the family of skewed or asymmetric algebraic expression that, in the case of the Lemniscate of 𝐶𝐶𝐶𝐶 𝐴𝐴 𝐵𝐵 𝑚𝑚 𝑛𝑛 𝐶𝐶𝐶𝐶 𝐴𝐴𝐴𝐴 lemniscate-like curves generated by the marker with Bernoulli, can be connected with a construction using a three- 𝐵𝐵𝐵𝐵 𝑟𝑟 | | = and | | = = 2 can be represented by a bar linkage system. We verify this connection algebraically for 𝐹𝐹 polar equation: a three-bar linkage system including the scale-independent 𝐶𝐶𝐶𝐶 𝑚𝑚 𝐹𝐹𝐹𝐹 𝑛𝑛 − 𝑚𝑚 2 case. From the three-bar linkage construction, we extend the = ( ) cos + 2 sin2 . 4 algebraic relation to symmetric lemniscate-like curves, or the 𝑟𝑟 Hippopede, and represent these parametrically. 𝜌𝜌 𝑚𝑚 −𝑛𝑛 𝜙𝜙 � − 𝜙𝜙 With this we incorporate another geometric characterization See Fig. 9 for an illustration generated using Maple. (the geometric representation described in Fig. 4) for which we construct a polar representation. This allows us to broaden the set of curves for which we have an algebraic characterization connected with a geometric characterization to include all our symmetric lemniscate-like curves. We then further expand our algebraic characterization from the three- bar linkage system to include skewed or asymmetric lemniscate-like curves where the marker is located on any fixed point on the middle rod.

REFERENCES [1] A. Akopyan, “The Lemniscate of Bernoulli, without Formulas,” The Mathematical Intelligencer, vol. 36, no. 4, pp. 47–50, 2014.

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[2] A. B. Kempe, “How to Draw a Straight Line; a Lecture on Linkages,” London, Macmillan, 1877. Albion, MI: Cambridge, 1952. [3] H. J. Nam and K. C. Jalosjos, “Parametric Representations of Polynomial Curves Using Linkages,” Parabola incorporating Function, vol. 52, no. 1, 2016.

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