Lemniscate of Leaf Function

Shinohara, Kazunori

doi:10.32604/cmes.2021.012383

Department of Mechanical Systems Engineering, Daido University, 10-3 Takiharu-cho, Minami-ku, Nagoya, 457-8530, Japan
*Corresponding Author: Kazunori Shinohara. Email: shinohara@06.alumni.u-tokyo.ac.jp
Received: 28 June 2020; Accepted: 14 September 2020

Abstract: A lemniscate is a curve defined by two foci, F1 and F2. If the distance between the focal points of F1 − F2 is 2a (a: constant), then any point P on the lemniscate curve satisfy the equation PF1.PF2 = a2. Jacob Bernoulli first described the lemniscate in 1694. The Fagnano discovered the double angle formula of the lemniscate (1718). The Euler extended the Fagnano’s formula to a more general addition theorem (1751). The lemniscate function was subsequently proposed by Gauss around the year 1800. These insights were summarized by Jacobi as the theory of elliptic functions. A leaf function is an extended lemniscate function. Some formulas of leaf functions have been presented in previous papers; these included the addition theorem of this function and its application to nonlinear equations. In this paper, the geometrical properties of leaf functions at n = 2 and the geometric relation between the angle images and lemniscate arc length l are presented using the lemniscate curve. The relationship between the leaf functions sleaf2 (l) and cleaf2 (l) is derived using the geometrical properties of the lemniscate, similarity of triangles, and the Pythagorean theorem. In the literature, the relation equation for sleaf2 (l) and cleaf2 (l) (or the lemniscate functions, sl (l) and cl (l)) has been derived analytically; however, it is not derived geometrically.

Keywords: Geometry; lemniscate of Bernoulli; leaf functions; lemniscate functions; Pythagorean theorem; triangle similarity

An ordinary differential equation (ODE) comprises a function raised to the 2n −1 power and the second derivative of this function. Further, the initial conditions of the ODE are defined.

The ODE comprises a function r(l)(or images

) of one independent variable l(or images

) and the derivatives of this function. The variable n represents a natural number ( images

). The above equation and the initial conditions constitute a very simple ODE. However, when this differential equation is numerically analyzed, mysterious waves are generated for all natural numbers. These mysterious waves are regular waves with some periodicity and amplitude. If these waves can be explained, they have the potential to solve various problems of nonlinear ODEs.

No elementary functions satisfy Eqs. (1)–(3). Therefore, in this paper, the function that satisfies Eqs. (1)–(3) is defined as cleafn(l). Function r(l) is abbreviated as r. By multiplying the derivative dr/dl with respect to Eq. (1), the following equation is obtained.

Using the initial conditions in Eqs. (2) and (3), the constant images

is determined. The following equation is obtained by solving the derivative dr/dl in Eq. (8).

We can create a graph with the horizontal axis as the variable l and the vertical axis as the function r. Because function r is a wave with a period, the gradient dr/dl has positive and negative values, and it depends on domain l. In the domain, images

(See Appendix A for the constant images

), the above gradient dr/dl becomes negative.

The following equation is obtained by integrating the above equation from 1 to r.

For integrating the left side of the above equation, the initial condition (Eq. (2), (l, r) = (0, 1)) is applied. The above equation represents the inverse function of the leaf function: cleafn(t) [1]. Therefore, the above equation is described as

Similarly, the function that satisfies Eqs. (4)–(6) is defined as sleaf images

. In the domain, images

(See Appendix A for constant images

), the gradient images

becomes positive.

The following equation is obtained by integrating the above equation from 0 to images

For integrating the left side of the above equation, the initial condition (Eq. (5), (l, r) = (0, 0)) is applied. The above equation represents the inverse function of the leaf function: sleaf images

[2]. Therefore, the above equation is described as

Inverse leaf functions based on the basis n = 1 represent inverse trigonometric functions.

In 1796, Carl Friedrich Gauss presented the lemniscate function [3]. The inverse leaf functions based on the basis n = 2 represents inverse functions of the sin and cos lemniscates [4].

In 1827, Jacobi [5] presented the Jacobi elliptic functions. Compared to Eq. (18), the term t2 is added to the root of the integrand denominator.

Eq. (20) represents the inverse Jacobi elliptic function sn, where k is a constant; there are 12 Jacobi elliptic functions, including cn and dn, etc. In Eq. (20), variable t is raised to the fourth power in the denominator. Jacobi did not discuss variable t raised to higher powers as indicated below.

Thus, historically, the inverse functions have not been discussed in the case of images

or higher [6–14].

A lemniscate is a curve defined by two foci F1 and F2. If the distance between the focal points of F1 − F2 is 2a (a: constant), then any point P on the lemniscate curve satisfies the equation images

. Jacob Bernoulli first described the lemniscate in 1694 [15,16]. Based on the lemniscate curve, its arc length can be bisected and trisected using a classical ruler and compass [17]. Based on this lemniscate, a lemniscate function was proposed by Gauss around the year 1800 [3,18]. Nishimura proposed a relationship between the product formula for the lemniscate function and Carson’s algorithm; it is known as the variant of the arithmetic–geometric mean of Gauss [19,20]. The Wilker and Huygens-type inequalities have been obtained for Gauss lemniscate functions [21]. Deng et al. [22] established some Shafer–Fink type inequalities for the Gauss lemniscate function. The geometrical characteristics of the lemniscate have been described [23,24]. Mendiratta et al. [25] investigated the geometric properties of functions. Levin [26] developed analogs of sine and cosine for the curve to prove the formula. Langer et al. [27] presented the lemniscate octahedral groups of projective symmetries. As a kinematic control problem, a five body choreography on an algebraic lemniscate was shown as the potential problem for two values of elliptic moduli [28]. The trajectory generation algorithm was applied by using the shape of the Bernoulli lemmiscate [29].

Leaf functions are extended lemniscate functions. Various formulas for leaf functions such as the addition theorem of the leaf functions and its application to nonlinear equations have been presented [30–32].

In this paper, the geometrical properties of leaf functions for n = 2, and the geometric relationship between the angle images

and lemniscate arc length l are presented using the lemniscate curve. The relations between leaf functions images

and

are derived using the geometrical properties of the lemniscate curve, similarity of triangles, and the Pythagorean theorem. In the literature, the relationship equation of images

and

is analytically derived; however, it is yet to be derived geometrically [33]. The relation between images

and

can be expressed as

The Eq. (22) was analytically derived. However, it cannot be geometrically derived using the lemniscate curve because it is not possible to show the geometric relationship of the lemniscate functions sl (l) and cl (l) on a single lemniscate curve. In contrast, phase l of the lemniscate function and angle images

can be visualized geometrically on a single lemniscate curve. Therefore, in the literature, Eq. (22) is derived using an analytical method without requiring the geometric relationship.

In this paper, the angle images

, phase l, and leaf functions images

and

(or lemniscate functions sl(l) and cl(l)) are visualized geometrically on a single lemniscate curve. Eq. (22) is derived based on the geometrical interpretation, similarity of triangles, and Pythagorean theorem.

Fig. 1 shows the geometric relationship between the lemniscate curve and images

. The y and x axes represent the vertical and horizontal axes, respectively. The equation of the curve is

If P is an arbitrary point on the lemniscate curve, then the following geometric relation exists.

When point P is circled along the contour of one leaf, the contour length corresponds to the half cycle images

(See Appendix A for the definition of the constant images

). As shown in Fig. 1, with respect to an arbitrary phase l, angle images

must satisfy the following inequality.

Fig. 2 shows the foci F and F images

of the lemniscate curve. The length of a straight line connecting an arbitrary point images

and one focal point images

is denoted by images

. Similarly, images

denotes the length of the line connecting an arbitrary point images

and a second focal point images

. On the curve, the product of images

and

is constant. The relationship equation is described as [34]

By substituting Eqs. (30) and (31) into Eq. (28), the relationship equation between the leaf function images

and trigonometric function images

can be derived as

The differential equation can be integrated using variable l. Parameter t in the integrand is introduced to distinguish it from l. The integration of Eq. (34) in the region images

yields the following equation (See Appendix C for details).

The phase

of the cos function is plotted in Fig. 3 through numerical analysis. The horizontal and vertical axes represent variables l and images

, respectively. As shown in Fig. 3, angle images

satisfies the inequality

Figure 3: Curves of leaf functions ( images

and

) and the integrated leaf functions ( images

and

)

Fig. 4 shows the geometric relationship between functions images

and

. The geometric relation in Eq. (36) is illustrated in Fig. 4.

Draw a perpendicular line from the point P on the lemniscate to the x-axis. This perpendicular is parallel to the y-axis. Let C be the intersection of this perpendicular and the x-axis. Therefore, the angle images

OCP is 90

. Next, draw a line perpendicular to the x-axis from the intersection A(1,0) of the lemniscate and the x-axis. Let B be the intersection of this perpendicular and the extension of the straight line OP. Here, images

and

. Substituting these into Eq. (23) gives

P and B are moving points, and point A is fixed. When angle images

is zero, both P and B are at A. The geometric relationship is then expressed as images

OA and sleaf2(l) = 0 = AB. As images

increases, P moves away from A, and it moves along the lemniscate curve. Here, phase l of cleaf2(l) and sleaf2(l) corresponds to the length of the arc images

. The length of the straight line OP is equal to the value of cleaf2(l). Point B is the intersection point of the straight lines OP and x = 1. In other words, P is the intersection point of the straight line OB and lemniscate curve. As images

increases, B moves away from A and onto the straight line x = 1. That is, it moves in the direction perpendicular to the x axis. The length of straight line AB is equal to the value of sleaf2(l). When images

reaches

, P moves to origin O and images

. The length of arc images

(or phase l) is images

. Moreover, cleaf2(l) = 0 = OP and sleaf2(l) = 1 = AB.

The relationship images

is derived by the similarity of triangles images

, as shown in Fig. 4. Thus, the following equation holds.

Eq. (32) is applied in the transformation process. Similarly, the relationship images

is derived by the similarity of triangles images

, as shown in Fig. 4. Therefore, the following equation holds.

By substituting Eqs. (42) and (43) into Eq. (39), the following equation is obtained.

For arbitrary l, images

and

. The relationship between images

and

can then be obtained as Eq. (22).

Fig. 5 shows the geometric relationship between length images

and lemniscate curve inclined at images

. In Fig. 5, the y and x axes represent the vertical and horizontal axes, respectively. The equation of this curve is given as

is an arbitrary point on the lemniscate curve, the following geometric relation exists.

Fig. 6 shows foci images

and

of the lemniscate curve inclined at an angle of images

. This curve has the same relation equation as shown in Fig. 2.

The coordinates of point images

on the lemniscate curve inclined at an angle of images

are given as

By substituting Eqs. (54) and (55) into Eq. (52), the relationship equation between the leaf function images

and the trigonometric function images

can be derived as

The following equation is obtained by differentiating Eq. (56) with respect to the variable images

The differential equation is integrated by variable images

. Parameter t is introduced to distinguish the parameter from variable images

in the integration region. The integration of Eq. (58) in region images

(See Appendix C) yields

The curve of phase images

is plotted in Fig. 3 through numerical analysis. The horizontal and vertical axes represent variables images

and

, respectively. As shown in Fig. 3, angle images

satisfies the following range.

Eq. (62) satisfies the range of Eq. (51) under the condition k = 0. Fig. 7 shows the lemniscate curve inclined at images

. The geometric relation of Eq. (60) is added to Fig. 7.

Let

be the coordinates on line images

, as shown in Fig. 7. Points images

and

are moving points, and point images

is fixed. When angle images

is zero,

is at origin O, and images

is on the x-axis at images

. The geometric relationship is described as images

and sleaf

. As

increases, images

moves away from origin O and along the lemniscate curve. The phase images

of both cleaf images

and sleaf

corresponds to the length of arc images

. The length of the straight line images

is equal to the value of sleaf images

. Point

is the intersection point of straight lines images

and

. In other words, images

is the intersection point of the straight line images

and the lemniscate curve. As images

increases, images

moves away from point images

,0) on a straight line images

. Here, the length of the straight line images

is equal to the value of cleaf images

. When

reaches

moves to point images

. The length of arc images

(or phase

) becomes

. Furthermore, cleaf images

and sleaf

. The linear equation images

is given by

By substituting Eq. (65) into Eq. (47) and solving for variable x, four solutions can be obtained as

As Eqs. (66) and (67) include imaginary numbers, the solutions for x using both Eqs. (68) and (69) are determined by the intersection points of line images

and the lemniscate curve, as shown in Fig. 7. The larger x value is given by Eq. (69). That is, the coordinates of point images

can be expressed as

The following equation is obtained from the Pythagorean theorem of the triangle images

The elimination of variable t from Eqs. (73) and (74) yields the relation equation, Eq. (22).

Tab. 1 lists the numerical values for Fig. 4. The angle images

and values of the leaf function (sleaf2(l) and cleaf2(l)) are calculated along the arc length l. The values of leaf functions cleaf2(l) are calculated by Eqs. (1)–(3). The values of leaf functions sleaf2(l) are calculated by Eqs. (4)–(6). Based on these data, the numerical data of functions sleaf2(l) and cleaf2(l) can be confirmed using Eq. (22). The function cleaf2(l) (= r) can also be confirmed by using Eq. (82). Angle images

can be calculated by using Eq. (35).

Table 1: Numerical data of arc length l, angle images

, and leaf functions sleaf2(l) and cleaf2(l) for Fig. 4

Tab. 2 shows the numerical values for Fig. 7. The angle images

and the values of leaf function (sleaf images

and cleaf

) are calculated along the arc length images

. Based on these data, the function sleaf images

can also be confirmed using Eq. (90). The angle images

can be calculated using Eq. (59).

Table 2: Numerical data of arc length images

, angle

, and leaf functions sleaf images

and cleaf

for Fig. 7

Based on the geometric properties of the lemniscate curve, the geometric relationship among angle images

, lemniscate length l, and leaf functions images

and

were shown on the lemniscate curve. Using the similarity of triangles and the Pythagorean theorem, the relationship equation of leaf functions images

and

was derived.

Acknowledgement: The author gratefully acknowledges the helpful comments and suggestions of the reviewers that have improved the presentation of this paper.

Funding Statement: This research is supported by Daido University research Grants (2020).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

References

1. Shinohara, K. (2016). Special function: Leaf function (second report). Bulletin of Daido University, 51, 39–68. [Google Scholar]

2. Shinohara, K. (2016). Special function: Leaf function (first report). Bulletin of Daido University, 51, 23–38. [Google Scholar]

3. Gauss, C. F., Clarke, A. A. (1965). Disquisitiones arithmeticae. United Kingdom: Yale University Press. [Google Scholar]

4. Roy, R. (2017). Elliptic and modular functions from Gauss to Dedekind to Hecke. United Kingdom: Cambridge University Press. [Google Scholar]

5. Jacobi, C. G. J. (2010). Opuscula mathematica. In: Mathematische werke. Berlin: Nabu Press. [Google Scholar]

6. Booth, J. (1852). Researches on the geometrical properties of elliptic integrals. Philosophical Transactions of the Royal Society of London, 142, 311–416. DOI 10.1098/rstl.1852.0019. [Google Scholar] [CrossRef]

7. Byrd, P., Comfort, W., Friedman, M., Friedman, M., Negrepontis, S. (1971). Handbook of elliptic integrals for engineers and scientists. New York: Springer-Verlag.

8. Akhiezer, N., McFaden, H. (1990). Elements of the theory of elliptic functions. In: Translations of mathematical monographs. Providence: American Mathematical Society.

9. Dixon, A. (1894). The elementary properties of the elliptic functions, with examples. New York, United States: Macmillan.

10. Lawden, D. (1989). Elliptic functions and applications. In: Applied mathematical sciences. Berlin, Germany: Springer.

11. McKean, H., Moll, V. (1999). Elliptic curves: Function theory, geometry, arithmetic. Cambridge, United Kingdom: Cambridge University Press.

12. Walker, P. (1996). Elliptic functions: A constructive approach. England: Wiley.

13. Hancock, H. (1958). Lectures on the theory of elliptic functions: Analysis. In: Dover books on advanced mathematics. United States: Dover Publications.

14. Mickens, R. (2019). Generalized trigonometric and hyperbolic functions. United States: CRC Press. [Google Scholar]

15. Bos, H. J. M. (1974). The lemniscate of Bernoulli. In: Cohen, R. S., Stachel, J. J., Wartofsky, M. W. (Eds.For Dirk Struik: Historical and political essays in honor of Dirk J. Struik, pp. 3–14, Dordrecht: Springer Netherlands. [Google Scholar]

16. Ayoub, R. (1984). The lemniscate and fagnano’s contributions to elliptic integrals. Archive for History of Exact Sciences, 29, 131–149. [Google Scholar]

17. Osler, T. (2016). Bisecting and trisecting the arc of the lemniscate. Mathematical Gazette, 100(549), 471–481. DOI 10.1017/mag.2016.112. [Google Scholar] [CrossRef]

18. Rosen, M. (1981). Abel’s theorem on the lemniscate. American Mathematical Monthly, 88(6), 387–395. DOI 10.1080/00029890.1981.11995279. [Google Scholar] [CrossRef]

19. Nishimura, R. (2015). New properties of the lemniscate function and its transformation. Journal of Mathematical Analysis and Applications, 427(1), 460–468. DOI 10.1016/j.jmaa.2015.02.066. [Google Scholar] [CrossRef]

20. Carlson, B. C. (1971). Algorithms involving arithmetic and geometric means. American Mathematical Monthly, 78(5), 496–505. DOI 10.1080/00029890.1971.11992791. [Google Scholar] [CrossRef]

21. Neuman, E. (2013). On lemniscate functions. Integral Transforms and Special Functions, 24(3), 164–171. DOI 10.1080/10652469.2012.684054. [Google Scholar] [CrossRef]

22. Deng, J. E., Chen, C. P. (2014). Sharp shafer-fink type inequalities for Gauss lemniscate functions. Journal of Inequalities and Applications, 2014(1), 496. DOI 10.1186/1029-242X-2014-35. [Google Scholar] [CrossRef]

24. Schappacher, N. (1997). Some milestones of lemniscatomy. Algebraic geometry (ankara, 1995Lecture notes in pure and applied mathematics, vol. 193, pp. 257–290, New York: Dekker. [Google Scholar]

25. Mendiratta, R., Nagpal, S., Ravichandran, V. (2014). A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli. International Journal of Mathematics, 25(9), 1450090. DOI 10.1142/S0129167X14500906. [Google Scholar] [CrossRef]

26. Levin, A. (2018). A geometric interpretation of an infinite product for the lemniscate constant. American Mathematical Monthly, 113(6), 510–520. DOI 10.1080/00029890.2006.11920331. [Google Scholar] [CrossRef]

27. Langer, J., Singer, D. (2010). Reflections on the lemniscate of Bernoulli: The forty-eight faces of a mathematical gem. Milan Journal of Mathematics, 78(2), 643–682. DOI 10.1007/s00032-010-0124-5. [Google Scholar] [CrossRef]

28. Lopez Vieyra, J. C. (2019). Five-body choreography on the algebraic lemniscate is a potential motion. Physics Letters A, 383(15), 1711– 1715. DOI 10.1016/j.physleta.2019.03.004. [Google Scholar] [CrossRef]

29. Saraiva, R., De Lellis Costa de Oliveira, M., Bruhns Bastos, M., Trofino, A. (2019). An algebraic solution for tracking Bernoullis lemniscate flight trajectory in airborne wind energy systems. IEEE 58th Conference on Decision and Control, Nice, France, pp. 5888–5893. [Google Scholar]

30. Shinohara, K. (2018). Exact solutions of the cubic duffing equation by leaf functions under free vibration. Computer Modeling in Engineering & Sciences, 115(2), 149–215. [Google Scholar]

31. Shinohara, K. (2019). Damped and divergence exact solutions for the duffing equation using leaf functions and hyperbolic leaf functions. Computer Modeling in Engineering & Sciences, 118(3), 599–647. DOI 10.31614/cmes.2019.04472.

32. Shinohara, K. (2020). Addition formulas of leaf functions and hyperbolic leaf functions. Computer Modeling in Engineering & Sciences, 123(2), 441–473. DOI 10.32604/cmes.2020.08656. [Google Scholar] [CrossRef]

33. Markushevich, A. (2014). The remarkable sine functions. New York: Elsevier Science. [Google Scholar]

34. Akopyan, A. (2014). The lemniscate of Bernoulli, without formulas. The Mathematical Intelligencer, 36(4), 47–50. DOI 10.1007/s00283-014-9445-5. [Google Scholar] [CrossRef]

35. Cox, D. A. (2011). Chapter 15–-The lemniscate-galois theory. New Jersey, United States: John Wiley & Sons, Ltd., pp. 457–508. [Google Scholar]