CMES CMES CMES Computer Modeling in Engineering & Sciences 1526-1506 1526-1492 Tech Science Press USA 12383 10.32604/cmes.2021.012383 Article Lemniscate of Leaf Function Lemniscate of Leaf Function Lemniscate of Leaf Function Shinohara Kazunori shinohara@06.alumni.u-tokyo.ac.jp 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 20 11 2020 126 1 275 292 28 06 2020 14 09 2020 © 2021 Shinohara 2021 Shinohara This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A lemniscate is a curve defined by two foci, F1 and F2. If the distance between the focal points of F1F2 is 2a (a: constant), then any point P on the lemniscate curve satisfy the equation PF1PF2=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 θ 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.

Geometry lemniscate of Bernoulli leaf functions lemniscate functions Pythagorean theorem triangle similarity
Introduction Motivation

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.

d2r(l)dl2=-nr(l)2n-1

r(0)=1

dr(0)dl=0

Another ODE and its initial conditions are given below:

d2r¯(l¯)dl¯2=-nr¯(l¯)2n-1

r¯(0)=0

dr¯(0)dl¯=1

The ODE comprises a function r(l)(or r¯(l¯)) of one independent variable l(or l¯) and the derivatives of this function. The variable n represents a natural number (n=1,2,3,). 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.

Theory of Leaf Functions

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.

d2rdl2drdl=-nr2n-1drdl

The following equation is obtained by integrating both sides of Eq. (7).

12(drdl)2=-12r2n+C

Using the initial conditions in Eqs. (2) and (3), the constant C=12 is determined. The following equation is obtained by solving the derivative dr/dl in Eq. (8).

drdl=±1-r2n

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, 0l12πn (See Appendix A for the constant πn), the above gradient dr/dl becomes negative.

dl=-dr1-r2n

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

l=-1rdt1-t2n=r1dt1-t2n=arccleafn(r)(=1rdl=[l]1r=l(r)-l(1)=l(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) . Therefore, the above equation is described as

r=cleafn(l)

Similarly, the function that satisfies Eqs. (4)(6) is defined as sleafn(l¯). In the domain, 0l¯12πn (See Appendix A for constant πn), the gradient dr¯/dl¯ becomes positive.

dl¯=dr¯1-r¯2n

The following equation is obtained by integrating the above equation from 0 to r¯.

l¯=0r¯dt1-t2n=arcsleafn(r¯)(=0r¯dl¯=[l¯]0r¯=l¯(r¯)-l¯(0)=l¯(r¯))

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: sleafn(l¯) . Therefore, the above equation is described as

r¯=sleafn(l¯)

Literature Comparison

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

arcsin(r¯)=0r¯dt1-t2=arcsleaf1(r¯)=l¯

arccos(r)=r1dt1-t2=arccleaf1(r)=l

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

arcsl(r¯)=0r¯dt1-t4=arcsleaf2(r¯)=l¯

arccl(r)=r1dt1-t4=arccleaf2(r)=l

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

arcsn(r,k)=0rdt1-(1+k2)t2+k2t4

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.

0rdt1-t6,0rdt1-t8,0rdt1-t10

Thus, historically, the inverse functions have not been discussed in the case of n=3 or higher .

Originality and Purpose

A lemniscate is a curve defined by two foci F1 and F2. If the distance between the focal points of F1F2 is 2a (a: constant), then any point P on the lemniscate curve satisfies the equation PF1PF2=a2. 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 . 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 . Deng et al.  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.  investigated the geometric properties of functions. Levin  developed analogs of sine and cosine for the curve to prove the formula. Langer et al.  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 . The trajectory generation algorithm was applied by using the shape of the Bernoulli lemmiscate .

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 .

In this paper, the geometrical properties of leaf functions for n = 2, and the geometric relationship between the angle θ and lemniscate arc length l are presented using the lemniscate curve. The relations between leaf functions sleaf2(l) and cleaf2(l) are derived using the geometrical properties of the lemniscate curve, similarity of triangles, and the Pythagorean theorem. In the literature, the relationship equation of sleaf2(l) and cleaf2(l) is analytically derived; however, it is yet to be derived geometrically . The relation between sleaf2(l) and cleaf2(l) can be expressed as

(sleaf2(l))2+(cleaf2(l))2+(sleaf2(l))2(cleaf2(l))2=1.

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 θ 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 θ, phase l, and leaf functions sleaf2(l) and cleaf2(l)(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.

Geometric Relationship with the Leaf Function <inline-formula id="ieqn-28"><alternatives><inline-graphic xlink:href="ieqn-28.png"/><tex-math id="tex-ieqn-28"><![CDATA[$\textbf{cleaf}_{\textbf{2}}\textbf{\textit{(l)}}$]]></tex-math><mml:math id="mml-ieqn-28"><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">cleaf</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">2</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">(l)</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>

Fig. 1 shows the geometric relationship between the lemniscate curve and cleaf2(l). The y and x axes represent the vertical and horizontal axes, respectively. The equation of the curve is

(x2+y2)2=x2-y2

Geometric relationship between angle <inline-formula id="ieqn-33"><alternatives><inline-graphic xlink:href="ieqn-33.png"/><tex-math id="tex-ieqn-33"><![CDATA[$\theta$]]></tex-math><mml:math id="mml-ieqn-33"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula> and phase <italic>l</italic> of the leaf function <inline-formula id="ieqn-34"><alternatives><inline-graphic xlink:href="ieqn-34.png"/><tex-math id="tex-ieqn-34"><![CDATA[$\mathrm{cleaf}_n(l)$]]></tex-math><mml:math id="mml-ieqn-34"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>

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

OP=cleaf2(l)

ArcAP=l(See Appendix B).

AOP=θ

When point P is circled along the contour of one leaf, the contour length corresponds to the half cycle π2 (See Appendix A for the definition of the constant π2). As shown in Fig. 1, with respect to an arbitrary phase l, angle θ must satisfy the following inequality.

π4(4k-1)θπ4(4k+1)

Here, k is an integer.

Geometric Relationship between the Trigonometric Function and Leaf Function <inline-formula id="ieqn-35"><alternatives><inline-graphic xlink:href="ieqn-35.png"/><tex-math id="tex-ieqn-35"><![CDATA[$\textbf{cleaf}_{\textbf{2}}\textbf{\textit{(l)}}$]]></tex-math><mml:math id="mml-ieqn-35"><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">cleaf</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">2</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">(l)</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>

Fig. 2 shows the foci F and F of the lemniscate curve. The length of a straight line connecting an arbitrary point P and one focal point F is denoted by PF. Similarly, PF denotes the length of the line connecting an arbitrary point P and a second focal point F. On the curve, the product of PF and PF is constant. The relationship equation is described as 

PFPF=(12)2

Lemniscate focus

The coordinates of point P are

P(cleaf2(l)cos(θ),cleaf2(l)sin(θ))=P(rcos(θ),rsin(θ))

PF and PF are given by

PF=(cleaf2(l)cos(θ)-12)2+(cleaf2(l)sin(θ))2=12+(cleaf2(l))2-2cos(θ)cleaf2(l)

and

PF=(cleaf2(l)cos(θ)+12)2+(cleaf2(l)sin(θ))2=12+(cleaf2(l))2+2cos(θ)cleaf2(l),

respectively.

By substituting Eqs. (30) and (31) into Eq. (28), the relationship equation between the leaf function cleaf2(l) and trigonometric function cos(θ) can be derived as

(cleaf2(l))2=2(cos(θ))2-1=cos(2θ)

After differentiating Eq. (32) with respect to l,

-2cleaf2(l)1-(cleaf2(l))4=-2sin(2θ)dθdl

The following equation is obtained by combining Eqs. (32) and (33).

dθdl=cleaf2(l)1-(cleaf2(l))4sin(2θ)=cleaf2(l)1-(cleaf2(l))41-(cos(2θ))2=cleaf2(l)1-(cleaf2(l))41-(cleaf2(l))4=cleaf2(l)

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 0tl yields the following equation (See Appendix C for details).

θ=0lcleaf2(t)dt=arctan(sleaf2(l))

Therefore, the following equation holds.

tan(θ)=sleaf2(l)

Eq. (32) can only be described by variable l as

(cleaf2(l))2=cos(20lcleaf2(t)dt)=cos(2arctan(sleaf2(l)))

The phase 0lcleaf2(t)dt of the cos function is plotted in Fig. 3 through numerical analysis. The horizontal and vertical axes represent variables l and θ, respectively. As shown in Fig. 3, angle θ satisfies the inequality

-π4θ=0lcleaf2(t)dtπ4

Curves of leaf functions (<inline-formula id="ieqn-53"><alternatives><inline-graphic xlink:href="ieqn-53.png"/><tex-math id="tex-ieqn-53"><![CDATA[$\mathrm{sleaf}_{2}(l)$]]></tex-math><mml:math id="mml-ieqn-53"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>s</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula id="ieqn-54"><alternatives><inline-graphic xlink:href="ieqn-54.png"/><tex-math id="tex-ieqn-54"><![CDATA[$\mathrm{cleaf}_{2}(l)$]]></tex-math><mml:math id="mml-ieqn-54"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>) and the integrated leaf functions (<inline-formula id="ieqn-55"><alternatives><inline-graphic xlink:href="ieqn-55.png"/><tex-math id="tex-ieqn-55"><![CDATA[$\int_{0}^{l}\mathrm{sleaf}_{2}(t) \mathrm{d}t$]]></tex-math><mml:math id="mml-ieqn-55"><mml:mstyle><mml:msubsup><mml:mrow><mml:mo>∫ </mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>s</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:math></alternatives></inline-formula> and <inline-formula id="ieqn-56"><alternatives><inline-graphic xlink:href="ieqn-56.png"/><tex-math id="tex-ieqn-56"><![CDATA[$\int_{0}^{l}\mathrm{cleaf}_{2}(t) \mathrm{d}t$]]></tex-math><mml:math id="mml-ieqn-56"><mml:mstyle><mml:msubsup><mml:mrow><mml:mo>∫ </mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msubsup></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>d</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:math></alternatives></inline-formula>)

Eq. (38) satisfies the inequality of Eq. (27) under the condition k = 0.

Fig. 4 shows the geometric relationship between functions sleaf2(l) and cleaf2(l). The geometric relation in Eq. (36) is illustrated in Fig. 4.

Geometric relationship between leaf functions <inline-formula id="ieqn-73"><alternatives><inline-graphic xlink:href="ieqn-73.png"/><tex-math id="tex-ieqn-73"><![CDATA[$\mathrm{sleaf}_{2}(l)$]]></tex-math><mml:math id="mml-ieqn-73"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>s</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula id="ieqn-74"><alternatives><inline-graphic xlink:href="ieqn-74.png"/><tex-math id="tex-ieqn-74"><![CDATA[$\mathrm{cleaf}_{2}(l)$]]></tex-math><mml:math id="mml-ieqn-74"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>

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 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, x=OC and y=CP. Substituting these into Eq. (23) gives

(OC2+CP2)2=OC2-CP2,

OCP=90,

and

OAB=90.

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 relationship is then expressed as cleaf2(l)=1=OA and sleaf2(l) = 0 = AB. As θ 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 AP. 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 θ 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 θ reaches 45, P moves to origin O and AB=1. The length of arc AP (or phase l) is π2/2. Moreover, cleaf2(l) = 0 = OP and sleaf2(l) = 1 = AB.

The relationship OC:OA=CP:AB is derived by the similarity of triangles OAB~OCP, as shown in Fig. 4. Thus, the following equation holds.

OC=OACPAB=cleaf2(l)sin(θ)sleaf2(l)=cleaf2(l)1(cos(θ))2sleaf2(l)        =cleaf2(l)11+(cleaf2(l))22sleaf2(l)=cleaf2(l)1(cleaf2(l))22sleaf2(l)

Eq. (32) is applied in the transformation process. Similarly, the relationship OP:PC=OB:BA is derived by the similarity of triangles OAB~OCP, as shown in Fig. 4. Therefore, the following equation holds.

PC=OPBAOB=cleaf2(l)sleaf2(l)1+(sleaf2(l))2

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

{(cleaf2(l)1-(cleaf2(l))22sleaf2(l))2+(cleaf2(l)sleaf2(l)1+(sleaf2(l))2)2}2=(cleaf2(l)1-(cleaf2(l))22sleaf2(l))2-(cleaf2(l)sleaf2(l)1+(sleaf2(l))2)2

By rearranging Eq. (44), the following equation is obtained.

(cleaf2(l))2{-1+(sleaf2(l))2+(cleaf2(l))2+(sleaf2(l))2(cleaf2(l))2}{}4(sleaf2(l))2{1+(sleaf2(l))2}2=0{}=2(sleaf2(l))2+6(sleaf2(l))4+4(sleaf2(l))6

{}=2(sleaf2(l))2+6(sleaf2(l))4+4(sleaf2(l))6         (cleaf2(l))2{1+3sleaf2(l))2+4sleaf2(l))4}+(cleaf2(l))4{1+sleaf2(l))2}

For arbitrary l, cleaf2(l)0 and {}0. The relationship between sleaf2(l) and cleaf2(l) can then be obtained as Eq. (22).

Geometric Relationship of the Leaf Function <inline-formula id="ieqn-83"><alternatives><inline-graphic xlink:href="ieqn-83.png"/><tex-math id="tex-ieqn-83"><![CDATA[$\mathbf{sleaf}_{\textbf{2}}(\overline{l})$]]></tex-math><mml:math id="mml-ieqn-83"><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mi>s</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">2</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>

Fig. 5 shows the geometric relationship between length sleaf2(l¯) and lemniscate curve inclined at 45. In Fig. 5, the y and x axes represent the vertical and horizontal axes, respectively. The equation of this curve is given as

(x2+y2)2=2xy

Geometric relationship between angle <inline-formula id="ieqn-89"><alternatives><inline-graphic xlink:href="ieqn-89.png"/><tex-math id="tex-ieqn-89"><![CDATA[$\overline{\theta}$]]></tex-math><mml:math id="mml-ieqn-89"><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula> and phase <inline-formula id="ieqn-90"><alternatives><inline-graphic xlink:href="ieqn-90.png"/><tex-math id="tex-ieqn-90"><![CDATA[$\overline{l}$]]></tex-math><mml:math id="mml-ieqn-90"><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula> of leaf function <inline-formula id="ieqn-91"><alternatives><inline-graphic xlink:href="ieqn-91.png"/><tex-math id="tex-ieqn-91"><![CDATA[$\mathrm{sleaf}_{2}(\overline{l})$]]></tex-math><mml:math id="mml-ieqn-91"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>s</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>f</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>

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

OP¯=sleaf2(l¯)

ArcOP¯=l¯ (See Appendix D)

D¯OP¯=θ¯

In Fig. 5, for an arbitrary variable l¯, the range of angle θ¯ is given by

kπθ¯π2(2k+1).

Here, k is an integer.

Geometric Relationship between Trigonometric Function and Leaf Function <inline-formula id="ieqn-92"><alternatives><inline-graphic xlink:href="ieqn-92.png"/><tex-math id="tex-ieqn-92"><![CDATA[$\textbf{sleaf}_{\textbf{2}}\textbf{(\textit{l})}$]]></tex-math><mml:math id="mml-ieqn-92"><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">sleaf</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">2</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mstyle class="text"><mml:mtext class="textbf" mathvariant="bold">(l)</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>

Fig. 6 shows foci F¯ and F¯ of the lemniscate curve inclined at an angle of 45. This curve has the same relation equation as shown in Fig. 2.

PF¯PF¯=(12)2

Lemniscate curve inclined at <inline-formula id="ieqn-100"><alternatives><inline-graphic xlink:href="ieqn-100.png"/><tex-math id="tex-ieqn-100"><![CDATA[$45^\circ$]]></tex-math><mml:math id="mml-ieqn-100"><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mo>∘</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>

The coordinates of point P¯ on the lemniscate curve inclined at an angle of 45 are given as

P¯(sleaf2(l¯)cos(θ¯),sleaf2(l¯)sin(θ¯))=P¯(r¯cos(θ¯),r¯sin(θ¯))

Lengths PF¯ and PF¯ are expressed by

PF¯=(sleaf2(l¯)cos(θ¯)-12)2+(sleaf2(l¯)sin(θ¯)-12)2=12+(sleaf2(l¯))2-(sin(θ¯)+cos(θ¯))sleaf2(l¯)

and

PF¯=(sleaf2(l¯)cos(θ¯)+12)2+(sleaf2(l¯)cos(θ¯)+12)2=12+(sleaf2(l¯))2+(sin(θ¯)+cos(θ¯))sleaf2(l¯).

By substituting Eqs. (54) and (55) into Eq. (52), the relationship equation between the leaf function sleaf2(l¯) and the trigonometric function sin(θ¯) can be derived as

(sleaf2(l¯))2=2sin(θ¯)cos(θ¯)=sin(2θ¯)

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

2sleaf2(l¯)1-(sleaf2(l¯))4=2cos(2θ¯)dθ¯dl¯

After applying Eq. (56), the equation is transformed as

dθ¯dl¯=sleaf2(l¯)1-(sleaf2(l¯))4cos(2θ¯)=sleaf2(l¯)1-(sleaf2(l¯))41-(sin(2θ¯))2=sleaf2(l¯)1-(sleaf2(l¯))41-(sleaf2(l¯))4=sleaf2(l¯)

The differential equation is integrated by variable l¯. Parameter t is introduced to distinguish the parameter from variable l¯ in the integration region. The integration of Eq. (58) in region 0tl¯ (See Appendix C) yields

θ¯=0l¯sleaf2(t)dt=-arctan(cleaf2(l¯))+π4,

and the following equation holds.

tan(π4-θ¯)=cleaf2(l¯).

Using Eq. (59), Eq. (56) can be described by variable l¯ as

(sleaf2(l¯))2=sin(20l¯sleaf2(t)dt)=sin(2(π4-arctan(cleaf2(l¯))))=cos(2arctan(cleaf2(l¯)))

The curve of phase θ¯=0l¯sleaf2(t)dt is plotted in Fig. 3 through numerical analysis. The horizontal and vertical axes represent variables l¯ and θ¯, respectively. As shown in Fig. 3, angle θ¯ satisfies the following range.

0θ¯=0l¯sleaf2(t)dtπ2

Eq. (62) satisfies the range of Eq. (51) under the condition k = 0. Fig. 7 shows the lemniscate curve inclined at 45. The geometric relation of Eq. (60) is added to Fig. 7.

OCP¯=90

OAB¯=90

Geometric relationship based on the lemniscate curve inclined at <inline-formula id="ieqn-153"><alternatives><inline-graphic xlink:href="ieqn-153.png"/><tex-math id="tex-ieqn-153"><![CDATA[$45^\circ$]]></tex-math><mml:math id="mml-ieqn-153"><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mo>∘</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>

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¯ is fixed. When angle θ¯ is zero, P¯ is at origin O, and B¯ is on the x-axis at (x,y)=(2,0). The geometric relationship is described as cleaf2(l¯)=1=AB¯ and sleaf2(l¯)=0. As θ¯ increases, P¯ moves away from origin O and along the lemniscate curve. The phase l¯ of both cleaf2(l¯) and sleaf2(l¯) corresponds to the length of arc OP. The length of the straight line OP¯ is equal to the value of sleaf2(l¯)(=r¯). Point B¯ is the intersection point of straight lines OP¯ and y=-x+2. In other words, P¯ is the intersection point of the straight line OB¯ and the lemniscate curve. As θ¯ increases, B¯ moves away from point (x,y)=( 2 ,0) on a straight line y=-x+2. Here, the length of the straight line AB¯ is equal to the value of cleaf2(l¯). When θ¯ reaches 45, P¯ moves to point A¯. The length of arc OA (or phase l¯) becomes π2/2. Furthermore, cleaf2(l¯)=0 and sleaf2(l¯)=1=OA¯. The linear equation CP¯ is given by

y=-x+2t.

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

x=12{2t--1-4t2-1+16t2}

x=12{2t+-1-4t2-1+16t2}

x=12{2t--1-4t2+1+16t2} x=12{2t+-1-4t2+1+16t2}

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 CP¯ and the lemniscate curve, as shown in Fig. 7. The larger x value is given by Eq. (69). That is, the coordinates of point P¯ can be expressed as

P¯(t+12-1-4t2+1+16t2,t-12-1-4t2+1+16t2)

Therefore, the length CP¯ is expressed as

CP¯=12-1-4t2+1+16t2.

The following equation is obtained from the Pythagorean theorem of the triangle OPC¯.

OP¯2=CP¯2+OC¯2

Substitution of Eqs. (48) and (71) into Eq. (72) yields

(sleaf2(t))2=12(-1-4t2+1+16t2)+2t2.

The length ratio is OC¯:OA¯=CP¯:AB¯ owing to the similarity of triangles OAB¯~OCP¯.

2t:1=12-1-4t2+1+16t2:cleaf2(t)

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

Numerical Results

Tab. 1 lists the numerical values for Fig. 4. The angle θ 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 θ can be calculated by using Eq. (35).

Numerical data of arc length <italic>l</italic>, angle <inline-formula id="ieqn-160"><alternatives><inline-graphic xlink:href="ieqn-160.png"/><tex-math id="tex-ieqn-160"><![CDATA[$\theta$]]></tex-math><mml:math id="mml-ieqn-160"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>, and leaf functions sleaf<sub>2</sub>(<italic>l</italic>) and cleaf<sub>2</sub>(<italic>l</italic>) for <xref ref-type="fig" rid="fig-4">Fig. 4</xref>
l θ(radian)(=0lcleaf2(t)dt) θ (degree) sleaf2(l)(= tanθ) cleaf2(l) (= r)
0.0 0.00000 0.00000 0.00000 1.00000
0.1 0.09966 5.7105 0.09999 0.99004
0.2 0.19736 11.3081 0.19996 0.96078
0.3 0.29123 16.6864 0.29975 0.91384
0.4 0.37962 21.7509 0.39897 0.85167
0.5 0.46115 26.4223 0.49689 0.77715
0.6 0.53474 30.6385 0.59230 0.69323
0.7 0.59958 34.3534 0.68352 0.60260
0.8 0.65511 37.5355 0.76831 0.50756
0.9 0.70100 40.1646 0.84400 0.40985
1.0 0.73704 42.2294 0.90768 0.31073
1.1 0.76313 43.7243 0.95643 0.21098
1.2 0.77923 44.6468 0.98774 0.11102
1.3 0.78533 44.9965 0.99987 0.01102

Tab. 2 shows the numerical values for Fig. 7. The angle θ¯ and the values of leaf function (sleaf2(l¯) and cleaf2(l¯)) are calculated along the arc length l¯. Based on these data, the function sleaf2(l¯)(=r¯) can also be confirmed using Eq. (90). The angle θ¯ can be calculated using Eq. (59).

Numerical data of arc length <inline-formula id="ieqn-221"><alternatives><inline-graphic xlink:href="ieqn-221.png"/><tex-math id="tex-ieqn-221"><![CDATA[$\overline{l}$]]></tex-math><mml:math id="mml-ieqn-221"><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>, angle <inline-formula id="ieqn-222"><alternatives><inline-graphic xlink:href="ieqn-222.png"/><tex-math id="tex-ieqn-222"><![CDATA[$\overline{\theta}$]]></tex-math><mml:math id="mml-ieqn-222"><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>, and leaf functions sleaf<inline-formula id="ieqn-223"><alternatives><inline-graphic xlink:href="ieqn-223.png"/><tex-math id="tex-ieqn-223"><![CDATA[$_{2}(\overline{l})$]]></tex-math><mml:math id="mml-ieqn-223"><mml:msub><mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and cleaf<inline-formula id="ieqn-224"><alternatives><inline-graphic xlink:href="ieqn-224.png"/><tex-math id="tex-ieqn-224"><![CDATA[$_{2}(\overline{l})$]]></tex-math><mml:math id="mml-ieqn-224"><mml:msub><mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> for <xref ref-type="fig" rid="fig-7">Fig. 7</xref>
l¯ θ¯(radian) (=0l¯sleaf2(t)dt) θ¯(degree) sleaf2(l¯) (=r¯) cleaf2(l¯) (=tan θ¯)
0.0 0.00000 0.0000 0.00000 1.00000
0.1 0.00499 0.2864 0.09999 0.99004
0.2 0.01999 1.1458 0.19996 0.96078
0.3 0.04498 2.5776 0.29975 0.91384
0.4 0.07993 4.5797 0.39897 0.85167
0.5 0.12474 7.1470 0.49689 0.77715
0.6 0.17922 10.2689 0.59230 0.69323
0.7 0.24306 13.9264 0.68352 0.60260
0.8 0.31571 18.0893 0.76831 0.50756
0.9 0.39642 22.7133 0.84400 0.40985
1.0 0.48411 27.7379 0.90768 0.31073
1.1 0.57746 33.0860 0.95643 0.21098
1.2 0.67482 38.6645 0.98774 0.11102
1.3 0.77436 44.3681 0.99987 0.01102
Conclusion

Based on the geometric properties of the lemniscate curve, the geometric relationship among angle θ, lemniscate length l, and leaf functions sleaf2(l) and cleaf2(l) were shown on the lemniscate curve. Using the similarity of triangles and the Pythagorean theorem, the relationship equation of leaf functions sleaf2(l) and cleaf2(l) was derived.

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.

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Elements of the theory of elliptic functions. In: Translations of mathematical monographs. Providence: American Mathematical Society. Dixon, A. (1894). The elementary properties of the elliptic functions, with examples. New York, United States: Macmillan. Lawden, D. (1989). Elliptic functions and applications. In: Applied mathematical sciences. Berlin, Germany: Springer. McKean, H., Moll, V. (1999). Elliptic curves: Function theory, geometry, arithmetic. Cambridge, United Kingdom: Cambridge University Press. Walker, P. (1996). Elliptic functions: A constructive approach. England: Wiley. Hancock, H. (1958). Lectures on the theory of elliptic functions: Analysis. In: Dover books on advanced mathematics. United States: Dover Publications. Mickens, R. (2019). Generalized trigonometric and hyperbolic functions. United States: CRC Press. Bos, H. J. M. (1974). The lemniscate of Bernoulli. In: Cohen, R. S., Stachel, J. J., Wartofsky, M. W. 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DOI 10.1080/10652469.2012.684054. 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. Smadja, I. (2008). La lemniscate de fagnano et la multiplication complexe. HAL-SHS (Sciences de l’Homme et de la Société), https://halshs.archives-ouvertes.fr/. Schappacher, N. (1997). Some milestones of lemniscatomy. Algebraic geometry (ankara, 1995), Lecture notes in pure and applied mathematics, vol. 193, pp. 257290, New York: Dekker. 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. Levin, A. (2018). A geometric interpretation of an infinite product for the lemniscate constant. American Mathematical Monthly, 113(6), 510520. DOI 10.1080/00029890.2006.11920331. 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), 643682. DOI 10.1007/s00032-010-0124-5. Lopez Vieyra, J. C. (2019). Five-body choreography on the algebraic lemniscate is a potential motion. Physics Letters A, 383(15), 17111715. DOI 10.1016/j.physleta.2019.03.004. 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. 58885893. Shinohara, K. (2018). Exact solutions of the cubic duffing equation by leaf functions under free vibration. Computer Modeling in Engineering & Sciences, 115(2), 149215. 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), 599647. DOI 10.31614/cmes.2019.04472. Shinohara, K. (2020). Addition formulas of leaf functions and hyperbolic leaf functions. Computer Modeling in Engineering & Sciences, 123(2), 441473. DOI 10.32604/cmes.2020.08656. Markushevich, A. (2014). The remarkable sine functions. New York: Elsevier Science. Akopyan, A. (2014). The lemniscate of Bernoulli, without formulas. The Mathematical Intelligencer, 36(4), 4750. DOI 10.1007/s00283-014-9445-5. Cox, D. A. (2011). Chapter 15–-The lemniscate-galois theory. New Jersey, United States: John Wiley & Sons, Ltd., pp. 457508. Appendix A

The symbols πn are a constant given by

πn=201dt1-t2n(n=1,2,3).

The numerical data of the symbol πn are summarized in the Tab. 3.

Values of constants <inline-formula id="ieqn-306"><alternatives><inline-graphic xlink:href="ieqn-306.png"/><tex-math id="tex-ieqn-306"><![CDATA[$\pi_{n}$]]></tex-math><mml:math id="mml-ieqn-306"><mml:msub><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>
n πn
1 3.1415
2 2.6220
3 2.4286
Appendix B

The Eq. (23) of Cartesian coordinate system is transformed to the following equation of polar coordinates [1,3].

r2=cos(2θ)

The variable r and θ represents OP and AOP in Fig. 1. The arc length in the cartesian and polar coordinates is given by

dx2+dy2=dr2+r2dθ2

The arc length l of the lemniscate with polar coordinates is given by

l=r11+r2(dθdr)2dr

By differentiating Eq. (76) with respect to variable θ,

2rdrdθ=-2sin(2θ)

Then, by applying Eq. (79),

1+r2(dθdr)2=1+r2(-rsin(2θ))2=1+r41-r4=11-r4

The following equation is applied to Eq. (80).

(sin(2θ))2=1-(cos(2θ))2=1-r4

Hence, the arc length l becomes

l=r111-t4dt

Appendix C

The following function is differentiated.

ddlarctan(cleaf2(l))=-1-(cleaf2(l))41+(cleaf2(l))2=-1-(cleaf2(l))21+(cleaf2(l))2=-sleaf2(l)

Integration of the above mentioned equation with respect to l yields

0lsleaf2(l)dt=[-arctan(cleaf2(l))]0l=-arctan(cleaf2(l))+arctan(cleaf2(0))=-arctan(cleaf2(l))+arctan(1)=-arctan(cleaf2(l))+π4.

Similarly, the following function is differentiated with respect to variable l.

ddlarctan(sleaf2(l))=1-(sleaf2(l))41+(sleaf2(l))2=1-(sleaf2(l))21+(sleaf2(l))2=cleaf2(l)

The following equation is obtained by integrating Eq. (85) with respect to l.

0lcleaf2(l)dt=[arctan(sleaf2(l))]0l=arctan(sleaf2(l))-arctan(sleaf2(0))=arctan(sleaf2(l))-arctan(0)=arctan(sleaf2(l))

Appendix D

Eq. (47) of the Cartesian coordinate system is transformed to the following equation in the polar coordinates [2,35].

r¯2=sin(2θ¯)

By differentiating Eq. (87) with respect to the variable θ¯,

2r¯dr¯dθ¯=2cos(2θ¯)

Then, applying Eq. (88),

1+r¯2(dθ¯dr¯)2=1+r¯2(r¯cos(2θ¯))2=1+r¯41-r¯4=11-r¯4

Hence, the arc length l¯ becomes

l¯=0r¯11-t4dt