LL EMNISCATE OF L EAF F UNCTION
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Kazunori Shinohara ∗ Department of Mechanical Systems EngineeringDaido University10-3 Takiharu-cho, Minami-ku, Nagoya 457-8530, Japan [email protected]
June 30, 2020 A BSTRACT
A lemniscate is a curve defined by two foci: F and F . If the distance between the focal pointsof F − F is 2a (a: constant), then any point P on the lemniscate curve satisfies the equation P F · P F = a . Jacob Bernoulli first described the lemniscate in 1694, and the lemniscatefunction was subsequently proposed by Gauss around the year 1800. A leaf function is an extendedlemniscate function. I have previously presented formulas that describe leaf functions, such as theaddition theorem of this function and its application to nonlinear equations. In this paper, I discussthe geometrical properties of leaf functions at n = 2 using the lemniscate curve, and explain thegeometric relationship between the angle θ and lemniscate arc length l . The relationship between theleaf functions sleaf ( l ) and cleaf ( l ) is derived using the geometrical properties of the lemniscate,similarity of triangles, and Pythagorean theorem. In the literature, the relation equation for sleaf ( l ) and cleaf ( l ) has been analytically derived, but not geometrically derived. K eywords Geometry; Lemniscate of Bernoulli; Leaf functions; Lemniscate functions; Pythagoreantheorem; Triangle similarity
A lemniscate is a curve defined by two foci: F and F . If the distance between the focal points of F − F is 2a(a: constant), then any point P on the lemniscate curve satisfies the equation P F · P F = a . Jacob Bernoulli firstdescribed the lemniscate in 1694 [1] [2]. Based on the lemniscate curve, its arc length can be bisected and trisectedusing a classical ruler and compass [3]. Based on this lemniscate, the lemniscate function was proposed by Gaussaround the year 1800 [4]. The relationships among these functions and the leaf functions are as follows [5]: sleaf ( l ) = sl( l ) = sin lemn( l ) (1) cleaf ( l ) = cl( l ) = cos lemn( l ) . (2)Nishimura proposed a relationship between the product formula for the lemniscate function and Carson’s algorithm, andit is known as the variant of the arithmetic–geometric mean of Gauss [6] [7]. The leaf functions are extended lemniscatefunctions. Various formulas for leaf functions, such as the addition theorem of the leaf functions and its application tononlinear equations were presented [8] [9] [10]. In this paper, I discuss the geometrical properties of leaf functionsfor n = 2 using the lemniscate curve, and explain the geometric relationship between the angle θ and lemniscate arclength l . The relations between leaf functions sleaf ( l ) and cleaf ( l ) are derived using the geometrical properties of thelemniscate curve, similarity of triangles, and Pythagorean theorem. In the literature, the relation equation of sleaf ( l ) and cleaf ( l ) is analytically derived, but not geometrically derived [11]. The relation between Eqs. (1) and (2) can beexpressed as (sleaf ( l )) + (cleaf ( l )) + (sleaf ( l )) (cleaf ( l )) = 1 . (3) ∗ a r X i v : . [ m a t h . G M ] J un PREPRINT - J
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30, 2020 ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ lcleaf P y x o A l Figure 1: Geometric relationship between angle θ and phase l of leaf function cleaf n ( l ) The aforementioned formula was analytically derived. However, it cannot be geometrically derived using the lemniscatecurve because it is not possible to show the geometric relation of the lemniscate functions sl ( l ) and cl ( l ) on a singlelemniscate curve. In contrast, phase l of the lemniscate functions and angle θ can be geometrically visualized on asingle lemniscate curve. Therefore, in the literature, Eq. (3) has been derived through an analytical method withoutrequiring the geometric relation.In this paper, the angle θ , phase l , and lemniscate functions sl ( l ) and cl ( l ) are geometrically visualized on a singlelemniscate curve. Eq. (3) is derived based on the geometrical interpretation, similarity of triangles, and Pythagoreantheorem. cleaf ( l ) Fig. 1 shows the geometric relationship between the lemniscate curve and cleaf ( l ) . The y and x axes represent thevertical and horizontal axes, respectively. The equation of the curve is as follows. ( x + y ) = x − y (4)If P is an arbitrary point on the lemniscate curve, then the following geometric relation exists. OP = cleaf ( l ) (5) Arc (cid:95)
AP = l (See refs . [ ][ ][ ]) (6) ∠ AOP = θ (7)When point P is circled along the contour of one leaf, the contour length corresponds to half cycle π of cleaf ( l ) (SeeAppendix A for the definition of the constant π ). As shown in Fig. 1, with respect to an arbitrary phase l , angle θ mustsatisfy the following inequality. π k − (cid:53) θ (cid:53) π k + 1) (8)Here, k is an integer. 2 PREPRINT - J
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30, 2020 cleaf ( l ) Fig. 2 shows foci F and F’ of the lemniscate curve. The length of a straight line connecting an arbitrary point P andone focal point F is denoted by PF . Similarly, PF (cid:48) denotes the length of the line connecting an arbitrary point P anda second focal point F (cid:48) . On the curve, the product of PF and PF (cid:48) is constant. The relation equation is described asfollows. PF · PF (cid:48) = (cid:18) √ (cid:19) (9)The coordinates of point P are as follows. P(cleaf ( l )cos( θ ) , cleaf ( l )sin( θ )) (10) PF and PF (cid:48) are given by PF = (cid:115)(cid:18) cleaf ( l )cos( θ ) − √ (cid:19) + (cleaf ( l )sin( θ )) = (cid:114)
12 + (cleaf ( l )) − √ θ )cleaf ( l ) (11)and PF (cid:48) = (cid:115)(cid:18) cleaf ( l )cos( θ ) + 1 √ (cid:19) + (cleaf ( l )sin( θ )) = (cid:114)
12 + (cleaf ( l )) + √ θ )cleaf ( l ) , (12)respectively. By substituting Eqs. (11) and (12) into Eq. (9), the relation equation between the leaf function cleaf ( l ) and trigonometric function cos( θ ) can be derived. (cleaf ( l )) = 2(cos( θ )) − θ ) (13)After differentiating Eq.(13) with respect to l , the following equation is obtained. − ( l ) (cid:112) − (cleaf ( l )) = − θ ) · d θ d l (14)The following equation is obtained by combining Eqs. (13) and (14). d θ d l = cleaf ( l ) (cid:112) − (cleaf ( l )) sin(2 θ ) = cleaf ( l ) (cid:112) − (cleaf ( l )) (cid:112) − (cos(2 θ )) = cleaf ( l ) (cid:112) − (cleaf ( l )) (cid:112) − (cleaf ( l )) = cleaf ( l ) (15)The differential equation can be integrated by variable l . Parameter t in the integrand that is introduced to distinguish itfrom the variable l . Integration of Eq. (15) in the region (cid:53) t (cid:53) l provides the following equation (See Appendix B fordetails). θ = (cid:90) l cleaf ( t )d t = arctan(sleaf ( l )) (16)Therefore, the following equation holds. tan( θ ) = sleaf ( l ) (17)Eq. (13) can only be described by variable l . (cleaf ( l )) = cos (cid:32) (cid:90) l cleaf ( t )d t (cid:33) = cos(2arctan(sleaf ( l ))) (18)3 PREPRINT - J
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30, 2020The phase (cid:82) l cleaf ( t )d t of the cos function is plotted in Fig. 3 through numerical analysis. The horizontal and verticalaxes represent variables l and θ , respectively. As shown in Fig, 3, angle θ satisfies the following inequality. − π (cid:53) θ = (cid:90) l cleaf ( t )d t (cid:53) π (19)Eq. (19) satisfies the inequality in Eq. (8).Fig. 4 shows the geometric relationship between functions sleaf ( l ) and cleaf ( l ) . The geometric relation in Eq. (17) isillustrated in Fig. 4. Here, x = OC and y = CP . Substitution of these into Eq. (4) gives (OC + CP ) = OC − CP , (20) ∠ OCP = 90 ◦ , (21)and ∠ OAB = 90 ◦ . (22)P and B are moving points, and point A is fixed. When angle θ is zero, both P and B are at A. The geometric relationshipis then expressed as cleaf ( l ) = 1 = OA and sleaf ( l ) = 0 = AB. As θ increases, P moves away from A, moving alongthe lemniscate curve. Here, phase l of cleaf ( l ) and sleaf ( l ) corresponds to the length of the arc (cid:95) AP . The length of thestraight line OP is equal to the value of cleaf ( l ) . Point B is the intersection point of the straight lines OP and x = 1 . Inother words, P is the intersection point of the straight line OB and lemniscate curve. As θ increases, B moves awayfrom A and onto the straight line x = 1 . That is, it moves in the direction perpendicular to the x axis. The length ofstraight line AB is equal to the value of sleaf ( l ) . When θ reaches ◦ , P moves to origin O and AB=1. The length ofarc (cid:95) AP (or phase l ) is π / . Moreover, cleaf ( l ) = 0 = OP and sleaf ( l ) = 1 = AB.The relation
OC : OA = CP : AB is derived by the similarity of triangles (cid:52)
OAB ∼ (cid:52)
OCP , as shown in Fig. 4. Thus,the following equation holds.
OC = OA · CPAB = cleaf ( l )sin( θ )sleaf ( l ) = cleaf ( l ) (cid:112) − (cos( θ )) sleaf ( l )= cleaf ( l ) (cid:113) − ( l )) sleaf ( l ) = cleaf ( l ) (cid:112) − (cleaf ( l )) √ ( l ) (23)Eq. (13) is applied in the transformation process. Similarly, the relation OP : PC = OB : BA is derived by thesimilarity of triangles (cid:52)
OAB ∼ (cid:52)
OCP , as shown in Fig .4. Therefore, the following equation holds.
PC = OP · BAOB = cleaf ( l )sleaf ( l ) (cid:112) ( l )) (24)By substituting Eqs. (23) and (24) into Eq. (20), the following equation is obtained. (cid:32) cleaf ( l ) (cid:112) − (cleaf ( l )) √ ( l ) (cid:33) + (cid:32) cleaf ( l )sleaf ( l ) (cid:112) ( l )) (cid:33) = (cid:32) cleaf ( l ) (cid:112) − (cleaf ( l )) √ ( l ) (cid:33) − (cid:32) cleaf ( l )sleaf ( l ) (cid:112) ( l )) (cid:33) (25)By rearranging Eq. (25), the following equation is obtained. (cleaf ( l )) {− ( l )) + (cleaf ( l )) + (sleaf ( l )) (cleaf ( l )) }{· · · } ( l )) { ( l )) } = 0 (26) {· · · } = 2(sleaf ( l )) + 6(sleaf ( l )) + 4(sleaf ( l )) − (cleaf ( l )) { ( l )) + 4sleaf ( l )) } + (cleaf ( l )) { ( l )) } (27)For arbitrary l , cleaf ( l ) (cid:54) = 0 and {· · · } (cid:54) = 0 . The relation equation between sleaf ( l ) and cleaf ( l ) can then beobtained as Eq. (3). 4 PREPRINT - J
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30, 2020 ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ F F lcleaf sin,cos lcleaflcleafP y x o A l Figure 2: Lemniscate focus sleaf ( l ) Fig. 5 shows the geometric relationship between length sleaf ( l ) and lemniscate curve inclined at ◦ . In Fig. 5, the y and x axes represent the vertical and horizontal axes, respectively. The equation of this curve is as follows. ( x + y ) = 2 xy (28)If P is an arbitrary point on the lemniscate curve, then the following geometric relation exists. OP = sleaf ( l ) (29) Arc OP = l (See refs . [ ][ ][ ]) (30) ∠ AOP = θ (31)In Fig. 5, for an arbitrary variable l , the range of angle θ is given by kπ (cid:53) θ (cid:53) π k + 1) . (32)Here, k is an integer. sleaf ( l ) Fig. 6 shows foci F and F (cid:48) of the lemniscate curve inclined at an angle of ◦ . This curve has the same relation equationas shown in Fig. 2. PF · PF (cid:48) = (cid:18) √ (cid:19) (33)The coordinates of point P on the lemniscate curve inclined at an angle of ◦ are as follows. P(sleaf ( l )cos( θ ) , sleaf ( l )sin( θ )) (34)5 PREPRINT - J
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30, 2020 ‐ ‐ ‐ ‐ ‐ l dttsleaf lcleaf lsleaf l dttcleaf l Figure 3: Curves of leaf functions ( sleaf ( l ) and cleaf ( l ) ) and integrated leaf functions ( (cid:82) l sleaf ( t )d t and (cid:82) l cleaf ( t )d t )Lengths PF and PF (cid:48) are expressed by PF = (cid:115)(cid:18) sleaf ( l )cos( θ ) − (cid:19) + (cid:18) sleaf ( l )sin( θ ) − (cid:19) = (cid:114)
12 + (sleaf ( l )) − (sin( θ ) + cos( θ ))sleaf ( l ) (35)and PF (cid:48) = (cid:115)(cid:18) sleaf ( l )cos( θ ) + 12 (cid:19) + (cid:18) sleaf ( l )cos( θ ) + 12 (cid:19) = (cid:114)
12 + (sleaf ( l )) + (sin( θ ) + cos( θ ))sleaf ( l ) . (36)By substituting Eqs. (35) and (36) into Eq. (33), the relation equation between the leaf function sleaf ( l ) andtrigonometric function sin( θ ) can be derived as follows. (sleaf ( l )) = 2sin( θ )cos( θ ) = sin(2 θ ) (37)The following equation is obtained by differentiating Eq. (37) with respect to variable l . ( l ) (cid:113) − (sleaf ( l )) = 2cos(2 θ ) · d θ d l (38)After applying Eq. (37), the equation is transformed as follows. d θ d l = sleaf ( l ) (cid:113) − (sleaf ( l )) cos(2 θ ) = sleaf ( l ) (cid:113) − (sleaf ( l )) (cid:113) − (sin(2 θ )) = sleaf ( l ) (cid:113) − (sleaf ( l )) (cid:113) − (sleaf ( l )) = sleaf ( l ) (39)6 PREPRINT - J
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30, 2020 ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ lcleaf P y x o A l B C lsleaf lsleaf Figure 4: Geometric relationship between leaf functions sleaf ( l ) and cleaf ( l ) The differential equation is integrated by variable l . Parameter t is introduced to distinguish the parameter from variable l in the integration region. Integration of Eq. (37) in region (cid:53) t (cid:53) l (See Appendix B) yields θ = (cid:90) l sleaf ( t )d t = − arctan(cleaf ( l )) + π , (40)and the following equation holds: tan (cid:16) π − θ (cid:17) = cleaf ( l ) . (41)Using Eq. (40), Eq. (37) can be described by variable l . (sleaf ( l )) = sin (cid:32) (cid:90) l sleaf ( t )d t (cid:33) = sin (cid:16) π − arctan(cleaf ( l ))) (cid:17) = cos (cid:0) ( l ))) (cid:1) (42)The curve of phase θ = (cid:82) l sleaf ( t )d t is plotted in Fig. 3 through numerical analysis. The horizontal and vertical axesrepresent variables l and θ , respectively. As shown in Fig. 3, angle θ satisfies the following range. (cid:53) θ = (cid:90) l sleaf ( t )d t (cid:53) π (43)Eq. (43) satisfies the range of Eq. (32). Fig. 7 shows the lemniscate curve inclined at an angle of ◦ . The geometricrelation of Eq. (41) is plotted in Fig. 5. ∠ OCP = 90 ◦ (44) ∠ OAB = 90 ◦ (45)Let C ( t, t ) be the coordinates on line OA , as shown in Fig. 7. Points P and B are moving points, and point A isfixed. When angle θ is zero, P is at origin O, and B is on the x -axis at ( x, y ) = ( √ ,0). The geometric relationshipis described as cleaf ( l ) = 1 = AB and sleaf ( l ) = 0 . As θ increases, P moves away from origin O and along the7 PREPRINT - J
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30, 2020 ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ lsleaf P o l y x Figure 5: Geometric relationship between angle θ and phase l of leaf function sleaf ( l ) lemniscate curve. Phase l of cleaf ( l ) and sleaf ( l ) corresponds to the length of arc (cid:95) OP . The length of straight line OP is equal to the value of sleaf ( l ) . Point B is the intersection point of straight lines OP and y = − x + √ . In otherwords, P is the intersection point of straight line OB and the lemniscate curve. As θ increases, B moves away frompoint ( x, y ) = ( √ ,0) on straight line y = − x + √ . Here, the length of straight line AB is equal to the value ofcleaf ( l ) . When θ reaches ◦ , P moves to point A . The length of arc (cid:95) OA (or phase l ) becomes π / . Furthermore,cleaf ( l ) = 0 and sleaf ( l ) = 1 = OA . The linear equation CP is given by y = − x + 2 t. (46)By substituting Eq. (46) into Eq. (28) and solving for variable x , four solutions can be obtained as follows. x = 12 (cid:26) t − (cid:113) − − t − (cid:112) t (cid:27) (47) x = 12 (cid:26) t + (cid:113) − − t − (cid:112) t (cid:27) (48) x = 12 (cid:26) t − (cid:113) − − t + (cid:112) t (cid:27) (49) x = 12 (cid:26) t + (cid:113) − − t + (cid:112) t (cid:27) (50)As Eqs. (47) and (48) include imaginary numbers, the solutions for x using both Eqs. (49) and (50) are determined bythe intersection points of line CP and the lemniscate curve, as shown in Fig 7. The larger x value is given by Eq. (50).That is, the coordinates of point P can be expressed as follows. P (cid:18) t + 12 (cid:113) − − t + (cid:112) t , t − (cid:113) − − t + (cid:112) t (cid:19) (51)8 PREPRINT - J
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30, 2020 ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ lsleaf o A F F l sin,cos lsleaflsleafP y x Figure 6: Lemniscate curve inclined at an angle of ◦ Therefore, length CP is expressed as CP = 1 √ (cid:113) − − t + (cid:112) t . (52)The following equation is obtained from the Pythagorean theorem of the triangle (cid:52) OPC . OP = CP + OC (53)Substitution of Eqs. (29) and (52) into Eq. (53) yields (sleaf ( t )) = 12 ( − − t + (cid:112) t ) + 2 t . (54)The length ratio is OC : OA = CP : AB owing to the similarity of triangles (cid:52)
OAB ∼ (cid:52)
OCP . √ t : 1 = 1 √ (cid:113) − − t + (cid:112) t : cleaf ( t ) (55)Elimination of variable t from Eqs. (54) and (55) yields the relation equation, Eq. (3). Based on the geometric properties of the lemniscate curve, the geometric relation among angle θ , lemniscate length l , and leaf functions sleaf ( l ) and cleaf ( l ) is shown on the lemniscate curve. Using the similarity of triangles andPythagorean theorem, I derived the relation equation of leaf functions sleaf ( l ) and cleaf ( l ) . References [1] H. J. M. Bos.
The Lemniscate of Bernoulli , pages 3–14. Springer Netherlands, Dordrecht, 1974.[2] Raymond Ayoub. The lemniscate and fagnano’s contributions to elliptic integrals.
Archive for History of ExactSciences , 29, 06 1984. 9
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30, 2020 ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ ‐ lsleaf o A l sin,cos lsleaflsleafP B lcleaf ttC , y x Figure 7: Geometric relation of leaf functions sleaf ( t ) and cleaf ( t ) based on the lemniscate curve inclined at ◦ [3] Thomas Osler. Bisecting and trisecting the arc of the lemniscate. The Mathematical Gazette , 100:471–481, 112016.[4] Carl Friedrich Gauss and Arthur A. Clarke.
Disquisitiones Arithmeticae . Yale University Press, 1965.[5] R. Roy.
Elliptic and Modular Functions from Gauss to Dedekind to Hecke . Cambridge University Press, 2017.[6] Ryo Nishimura. New properties of the lemniscate function and its transformation.
Journal of MathematicalAnalysis and Applications , 427:460–468, 07 2015.[7] B. C. Carlson. Algorithms involving arithmetic and geometric means.
The American Mathematical Monthly ,78(5):pp. 496–505, 05 1971.[8] Kazunori Shinohara. Exact solutions of the cubic duffing equation by leaf functions under free vibration.
ComputerModeling in Engineering & Sciences , 115(2):149–215, 2018.[9] Kazunori Shinohara. Damped and divergence exact solutions for the duffing equation using leaf functions andhyperbolic leaf functions.
Computer Modeling in Engineering & Sciences , 118(3):599–647, 2019.[10] Kazunori Shinohara. Addition formulas of leaf functions and hyperbolic leaf functions.
Computer Modeling inEngineering & Sciences , 123(2):441–473, 2020.[11] A.I. Markushevich.
The Remarkable Sine Functions . Elsevier Science, 2014.[12] David A Cox.
The Lemniscate , chapter 15, pages 457–508. John Wiley & Sons, Ltd, 2011.[13] Kazunori Shinohara. Special function: Leaf function r = sleaf n ( l ) (first report). Bulletin of Daido University ,51:23–38, 2016.[14] Kazunori Shinohara. Special function: Leaf function r = cleaf n ( l ) (second report). Bulletin of Daido University ,51:39–68, 2016.
Appendix A π is a constant that is given by π = 2 (cid:90) d t √ − t . (56)10 PREPRINT - J
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30, 2020
Appendix B
The following function is differentiated. dd l arctan(cleaf ( l )) = − (cid:112) − (cleaf ( l )) ( l )) = − (cid:115) − (cleaf ( l )) ( l )) = − sleaf ( l ) (57)Integration of the abovementioned equation with respect to l yields (cid:90) l sleaf ( l )d t = [ − arctan(cleaf ( l ))] l = − arctan(cleaf ( l )) + arctan(cleaf (0))= − arctan(cleaf ( l )) + arctan(1) = − arctan(cleaf ( l )) + π . (58)Similarly, the following function is differentiated with respect to variable l . dd l arctan(sleaf ( l )) = (cid:112) − (sleaf ( l )) ( l )) = (cid:115) − (sleaf ( l )) ( l )) = cleaf ( l ) (59)The following equation is obtained by integrating the above equation with respect to l . (cid:90) l cleaf ( l )d t = [arctan(sleaf ( l ))] l = arctan(sleaf ( l )) − arctan(sleaf (0))= arctan(sleaf ( l )) − arctan(0) = arctan(sleaf ( l ))))