A New Construction of Holditch Theorem for Homothetic Motions in Cp

Hindawi Advances in Mathematical Physics Volume 2021, Article ID 5583457, 9 pages https://doi.org/10.1155/2021/5583457

Research Article A New Construction of Holditch Theorem for Homothetic Motions in Cp

Tülay Erişir

Erzincan Binali Yıldırım University, Department of Mathematics, 24050 Erzincan, Turkey

Correspondence should be addressed to Tülay Erişir; [email protected]

Received 2 March 2021; Accepted 24 June 2021; Published 13 July 2021

Academic Editor: Antonio Scarfone

Copyright © 2021 Tülay Erişir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this study, the planar kinematics has been studied in a generalized complex plane which is a geometric representation of the generalized complex number system. Firstly, the planar kinematic formulas with one parameter for homothetic motions in the generalized complex plane have been mentioned brieﬂy. Then, the Steiner area formula given areas of the trajectories drawn by the points taken in a generalized complex plane have been obtained during the one-parameter planar homothetic motion. Finally, the Holditch theorem, which gives the relationship between these areas of trajectories, has been expressed for homothetic motions in a generalized complex plane. So, this theorem obtained in this study is the most general form of all Holditch theorems obtained so far.

1. Introduction ered these studies given by Klein and Cayley and divided these geometries into three (parabolic, elliptic, and hyper- The first scientists which introduced the complex numbers, bolic) according to the length measure between two points which are expressed as x + iy where the imaginer unit is i on a line and the angle measure between two lines [9]. This ði2 = −1Þ, are thought to be Italian mathematicians Cardan distinction has showed nine paths by measures of angle and (1501-1576) and Bombelli (1526-1572). In 1545, Cardan length. These nine plane geometries are given in Table 1. published a study called “The Great Art” and defined an alge- The generalized complex number system was introduced braic formula for solving cubic and quartic equations in that by Yaglom [4]. The generalized complex numbers also play a study. But Cardan did not consider complex numbers in role in Cayley-Klein geometry as ordinary complex numbers detail. On the other hand, Bombelli introduced (complex) play a role in Euclidean geometry [4, 9]. Then, Harkin and numbers with Cardan’s formula 30 years after Cardan’s Harkin studied the generalized complex number system tak- study. Then, it has been observed that alternative number ing these studies into account [10]. systems can be created with the help of complex numbers. As a subbranch of physics, mechanics examines the The English geometrician Clifford (1845-1879) introduced motion of systems, their effects that cause motion, and the hyperbolic numbers using i2 =+1 [1–4]. The application to equilibrium states of systems. Mechanics divide into three mechanics of hyperbolic numbers given by Clifford has been parts as statics, kinematics, and dynamics. Statics, kinemat- supported by applications to non-Euclidean geometries. ics, and dynamics examine, respectively, the equilibrium Moreover, the German geometrician study introduced states of systems, the motion of systems without adding force, another number system called “dual numbers” by adding and the factors that change motion. The main elements con- another unit to complex numbers [4–6]. sidered in kinematics are also length and time. In dynamics, Cayley-Klein geometry, which contains Euclidean, Gali- there are three basic elements that are important: length, lean, and Minkowskian geometries, was introduced for the time, and mass. Thus, kinematics can be called a science first time by Klein and Cayley [7, 8]. Then, Yaglom consid- between geometry and dynamics. Briefly, kinematics 2 Advances in Mathematical Physics

Table 1: Nine Cayley-Klein geometries in the plane.

Measure of length Elliptic Parabolic Hyperbolic Elliptic Elliptic geometry Euclidean geometry Hyperbolic geometry Measure of angles Parabolic (Euclidean) Co-Euclidean geometry Galilean geometry Co-Minkowskian geometry Hyperbolic Cohyperbolic geometry Minkowskian geometry Doubly hyperbolic geometry examines how the geometric properties of systems change centuries. The basis of the study in [33] is closely linked to over time. The treatment of kinematics as an independent mathematical linguistics. This approach has led to new science dates back to Ampére’s (1775-1836) time, who first results in analytical geometry used in different applications defined this branch of science and named it. According to in information technology. Moreover, in [33], an architec- Ampére, “Kinematics includes in everything that can be said tural calculation tool was proposed and the existence of sym- about the motions of systems, regardless of the forces that metry in natural languages was briefly demonstrated. The move the systems. It examines all the designs of the force Renishaw Ballbar QC20-W is designed for the diagnosis of independent motion, taking into account many calculations CNC machine tools but is also used in conjunction with such as the area formed during the motion, the time required industrial robots. In the standard measurement situation, to create this area and the various connections that can be not all robot joints move when the measurement plane is par- found between this area and this time” [11]. In kinematics, allel to the robot base. In this regard, in [34], the hypothesis the one-parameter planar motion has been studied by many of motion of all robot joints has been valuated when the scientists in different spaces. Müller introduced the one- desired circular path was placed on an inclined plane. There- parameter planar motion in both Euclidean and complex fore, the hypothesis established in the first part of the exper- planes [12]. The one-parameter planar motion in the Lorent- iments was confirmed by spatial analysis on a simulation zian plane was also studied by Ergin [13] and Görmez [14]. model of the robot. Then, practical measurements were made Then, Yüce and Kuruoğlu introduced the one-parameter pla- evaluating the effect of individual robot joints to deform the nar motion in the hyperbolic plane [15]. Moreover, Akar and circular path, which is shown as a pole plot [34]. Yüce studied in the Galilean plane [16]. Holditch theorem The generalized complex number system was expressed first expressed by Holditch [17] is one of the important the- as orems of kinematics. The most important point of that clas- ÈÉ 2 sical Holditch theorem is that the area is independent of the Cp = x + iy : x, y ∈ R, i = p ∈ R , ð1Þ selected curve. Therefore, due to the Holditch theorem being open to technical application and its independence from the by Yaglom and Harkin [4, 9, 10]. This system involves com- curve, it has gained a lot of attention with the choices of the plex ðp = −1Þ, dual ðp =0Þ, and hyperbolic ðp =+1Þ number plane that carries the curve and has been generalized by systems and also different planes for other values of p ∈ R. many scientists with different methods. Steiner gave the area Considering the aforementioned studies, some kinematic formula bounded by the trajectory curve drawn by any point studies have been carried out in the generalized complex under a one-parameter closed planar motion [18, 19]. Since plane obtained from this number system. Erişir et al. the above-mentioned studies received great interest during obtained the Steiner area formula, the polar moment of iner- the one-parameter closed planar motion, many of the scien- tia, and Holditch-type theorem in Cp [35, 36]. In addition to tists working on kinematics have generalized the Holditch that, Erişir and Güngör gave the Holditch-type theorem for theorem with various studies. The most important of these nonlinear points in a generalized complex plane Cp, [37, studies are [12, 20–25]. Many studies about the Holditch the- 38]. Moreover, Gürses et al. gave the one-parameter planar orem have been also made for homothetic motions. Tutar 2 homothetic motion in Cj = fx + Jy : x, y ∈ R, J = p, p ∈ f−1 and Kuruoğlu obtained the Steiner area formula and Hol- ,0,1 ⊂ C ditch theorem for the one-parameter planar homothetic gg p [39]. motions in the Euclidean plane [26]. In addition, Kuruoğlu This study is on kinematics for one parameter planar and Yüce generalized the area formulas for planar motions homothetic motion in a generalized complex plane which is and gave the corresponding formulas for homothetic a geometric representation of the generalized complex num- 2 motions [27–30]. This study is the most general form of all ber system Cp = fx + iy : x, y ∈ R, i = p ∈ Rg. The Steiner for- study about Holditch theorems obtained so far. mula and Holditch theorem for these homothetic motions in In [31], a new analytical geometry method that is symme- Cp have been obtained. So, this study is the most general ver- tries in the Euclidean plane was developed to calculate the sion of all the studies about the Holditch theorem done so far. trajectories of mechatronic systems and CAD/CAM. In addition, some examples of the design of kinematic mechanisms 2. Preliminaries were presented [31]. Then, in [32], a new alternative calculation method of high accuracy calculation of robot trajectory The generalized complex number system consists of ordered for the complex curves was proposed. On the other hand, pairs Z = ðx, yÞ or Z = x + iy, and this number system specif- the similarity method has found a place in science for several ically includes ordinary, dual, and double numbers where i2 Advances in Mathematical Physics 3 is also written as i2 = ðq, pÞ, ði2 = iq + pÞ, and x, y, q, p ∈ R. So, Definition 1. Let the circle which has the center Mða, bÞ and in cases where p + q2/4 is negative, zero, and positive, gener- the radius r be considered. Thus, the equation of this circle is alized complex numbers are isomorphic to ordinary, dual, 2 2 2 and double numbers, respectively [4, 9, 10]. In this paper, ðÞx − a − pyðÞ− b = r ð7Þ especially, q =0, −∞ < p < ∞, and i2 = p ∈ R are considered. So, the generalized complex number system is reduced [10]. ÈÉ 2 Cp = x + iy : x, y ∈ R, i = p ∈ R : ð2Þ ! Let a number in Cp be Z = x + iy which symbolize OT and Figure 2 be as follows. Now, the two generalized complex numbers are consid- So, the angle θp formed by inverse tangent functions can ered Z1 = x1 + iy1 and Z2 = x2 + iy2 ∈ Cp. So, it can be written be defined as as 8 pffiffiffiffiffi > 1 −1 Z ± Z = x + iy ± x + iy = x ± x + iy ± y : > pffiffiffiffiffi tan σ jjp , p <0, 1 2 ðÞ1 1 ðÞ2 2 ðÞ1 2 ðÞ1 2 > p <> jj ð3Þ θ = σ, p =0, 8 p > ð Þ > Moreover, the product in this system is > 1 −1 pffiffiffi :> ffiffiffi tan σ p , p > 0 branch I, III , pp ðÞ ðÞ p M ðÞZ1, Z2 = ðÞx1x2 + py1y2 + ixðÞ1y2 + x2y1 ð4Þ where σ ≡ y/x. Let the point N be the intersection point of OT [4, 10, 40]. with unit circle in Cp. Moreover, the orthogonal projection On the other hand, if two generalized complex vectors on the OM of the point N is the point L and the line QM is which are position vectors of the generalized complex num- M Z Z z = x + iy z = x + iy ∈ also the tangent at the point of the unit circle (see bers 1, 2 are considered 1 1 1 and 2 2 2 Figure 3). Thus, p-trigonometric functions (the p-cosine C p, the scalar product of these vectors is (cos p), p-sine (sin p), and p-tangent (tan p)) can be obtained by z , z =Re Mp z , z =Re Mp z , z = x y − px y hi1 2 p ðÞðÞ1 2 ðÞðÞ1 2 1 1 2 2 8 pffiffiffiffiffi 5 > cos θ p , p <0, ð Þ <> p jj cos pθp = 1, p = 0 branch I , p > ðÞ [10]. Moreover, the -magnitude of the generalized complex > ÀÁffiffiffi Z = x + iy ∈ C : p number p is cos h θp p , p > 0ðÞ branch I , 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffi 9 ÀÁ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffi sin θ p , p <0, ð Þ Z = Mp Z, Z = x2 − py2 , 6 > p jj jjp jj ð Þ > p <> jj sin pθp = θp, p = 0ðÞ branch I , where “−” denotes the ordinary complex conjugation [10]. > > 1 ÀÁffiffiffi Moreover, the unit circle in Cp is characterized by the form > p : pffiffiffi sin h θp p , p > 0ðÞ branch I , of jZjp =1. Thus, the unit circle in the plane Cp can be given p as Figure 1 for the special cases of p. If the special case p <0is chosen, the unit ellipses formed and the ratio QM/OM = NL/OL gives by x2 + jpjy2 =1 are obtained. Moreover, the generalized complex number system Cpðp <0Þ equals to the elliptical sin pθ tan pθ = p : 10 complex number system. In particular, if p = −1 then, the p cos pθ ð Þ 2 p unit circle in Cp corresponds to Euclidean unit circle x + y2 =1 [10]. If the situation p =0 is considered, x2 =1 and Thus, the Maclaurin expansions of the p-trigonometric the unit circles are formed as x =±1. Moreover, the system functions on the branch I are C0 is equal to the parabolic number system and the plane in this situation corresponds to the Galilean plane [10]. ∞ pn Finally, considering the special situation p >0, the hyperbolas cos pθ = 〠 θ2n, x2 − py2 =1 y =±x/ p 2n ! p areffiffiffi obtained by j j which have asymptote n=0 ðÞ pp C ð11Þ . So, the number system p is equal to the hyperbolic ∞ pn p =1 sin pθ = 〠 θ2n+1: complex number system. Particularly, when , the gener- p 2n +1 ! p alized complex plane is the Lorentzian plane [10]. n=0 ðÞ Considering the above-mentioned description of circle for cases of p, the circle in Cp can be defined as follows. By the help of the Maclaurin series, the generalized Euler 4 Advances in Mathematical Physics

II II I III I –1 +1 IV

p < 0 p = 0 p > 0

Figure 1: The unit circle in Cp:

N T(x,y) T(x,y) T(x,y) N N

OO�p M(1,0) �p M(1,0) O�p M(1,0)

p < 0 p = 0 p > 0

Figure 2: Elliptic, parabolic, and hyperbolic angles.

formula in Cp is parameter planar homothetic motion in the generalized complex plane Cp is written by iθp e = cos pθp + i sin pθp, ð12Þ iθ x′ = ðÞhx − u e p , ð16Þ where i2 = p: On the other hand, the exponential forms of Z ′ ′ in Cp are where θp is the p-rotation angle of the motion Kp/K p, u = iθp ÀÁ −ue , and h is the homothetic scale in Cp. So, the relative iθ Z = r cos pθ + i sin pθ = r e p , 13 p p p p ð Þ and absolute velocity vectors of X in Kp ⊂ Cp are

r = Z p ′ iθp iθp 17 where p j jp [10]. Moreover, the -rotation matrix given Vr = Vre = hxe , ð Þ by the help of equation (12) is iθp _ _ iθp _ iθp iθp "# Va ′ = Vae = h + iθph xe − u + iθpu e + hxe , ÀÁ cos pθp p sin pθp A θ = 14 p ð Þ ð18Þ sin pθp cos pθp respectively. Using equations (17) and (18) the guide velocity [10]. vector is The one parameter homothetic motions in the p-complex ′ iθp _ _ iθp ′ plane Vf = Vf e = h + iθph xe + u : ð19Þ ÈÉ C = x + Jy : x, y ∈ R, J2 = p, p ∈−1, 0, 1 , 15 J fg ð Þ ′ Theorem 2. Let Kp/K p be the one parameter planar homothetic motion in Cp. So, the relationship between velocity vec- which is the subset of the generalized complex plane Cp was studied by Gürses et al. [39]. Similar to that study, the one- tors is given by parameter homothetic motions in the generalized complex Va = Vf + Vr: ð20Þ plane Cp have been given as follows briefly. ′ Let Kp, K p be the moving and fixed planes in Cp, respec- There are some points that remain fixed in both the fixed ′ ′ ′ tively, and x = x1 + ix2 and x = x 1 + ix 2 be the position vec- K′ K C ! plane p and the moving plane p in p. These points are tors of a point X, and OO′ = u. So, the equation of the one- called pole points. Thus, let the pole points of the one- Advances in Mathematical Physics 5

Q N N Q N Q � � p p �p O LLM O M OML

p < 0 p = 0 p > 0

Figure 3: θp for the special cases of p.

′ x , x F parameter planar homothetic motions Kp/K p be Q = ðq1, q2 1 2Þ. Moreover, X is considered the area of trajectory X Þ ∈ Cp. So, the components of pole points Q = ðq1, q2Þ are drawn by the point . So, this area is given by ðt hi ÀÁÀÁ 1 2 dh du + pu dθ − ph du + u dθ dθ ′ ′ 1 2 p 2 1 p p FX = x , dx , ð25Þ q1 = , 2 2 2 2 t1 dh − ph dθp ÀÁÀÁð21Þ dh du + u dθ − hdu + pu dθ dθ where the determinant product is ½, [41]. Now, considering q = 2 1 p 1 2 p p , 2 2 2 2 equations (16), (22), and (23), equation (25) is equal to dh − ph dθp ðt ðt ðt 1 2 2 1 2 2 1 2 2 where V f =0. In addition to that, the guide velocity vector of FX = xx h dθp − x h qdθp − x h qdθp 2 t 4 t 4 t the fixed point X with respect to Kp can be written in terms of 1 1 1 ðt 1 2 the pole points as + u q − u q − hx q + hx q + x u − x u dh 2 ðÞ1 2 2 1 1 2 2 1 1 2 2 1 ÀÁ t1 ′ iθp ðt dx = dh + ihdθp ðÞx − q e , ð22Þ 1 2 + ðÞu1q1 − pu2q2 − u1x1 + px2u2 hdθp, 2 t where q is the position vector of the pole point Q. 1 26 On the other hand, the following proposition, which can ð Þ be proved easily, for the homothetic motions in Cp can be Q = q , q q given. where the position vector of the pole point ð 1 2Þ is and “−” is the ordinary complex conjugate. We should note X fi K X Proposition 3. Let two arbitrary generalized complex vectors here that is any xed point in p. Particularly, if is con- K X =0 be a = ða1, a2Þ and b = ðb1, b2Þ in Cp. Thus, the following sidered as the origin point of p, then, for the point , equations are satisfied: equation (26) is obtained that hi "#ð ð iθ iθ t t ðÞi ae p , be p = ½a, b , 1 2 2 F0 = ðÞu1q2 − u2q1 dh + ðÞu1q1 − pu2q2 hdθp , 2 t t ÂÃÀÁ 1 1 1 ii a, dh + ihdθ b = a, b dh + ab + ab hdθ , ðÞ p ½ 2 ½p ð27Þ 23 ð Þ _ _ _ where θp ≠ 0 and θp is a continuous function. So, θp can be h _ _ _ where is the homothetic scale and θp <0or θp >0. In here, θp has the same sign everywhere in t , t the interval ½ 1 2. Now, let the mean value theorem of inte- a1 a2 t , t a, b = = a b − a b : 24 gral calculus for the interval ½ 1 2 be considered. So, there ½ 1 2 2 1 ð Þ t ∈ t , t b1 b2 exists at least one point 0 ½ 1 2 so that ðt ðt 2 2 2 2 _ 2 In this paper, the open motions restricted to time interval h dθp = h ðÞt θpðÞt dt = h ðÞt0 δp, ð28Þ t , t branch I C ½ 1 2 on of p are considered. t1 t1

3. Main Theorems and Proofs where δp = θpðt2Þ − θpðt1Þ is the total rotation angle. On the other hand, the Steiner point which is the center ′ Let K p, Kp ⊂ Cp be the fixed and moving generalized com- of gravity of the moving pole curve was first expressed by plex planes, respectively, and any fixed point in Kp be X = ð Steiner [18]. So, let the Steiner point in this study be 6 Advances in Mathematical Physics

represented by S = ðs1, s2Þ. If this point is adapted to the gen- Let FX, the area of the trajectory drawn by the point X eralized complex plane for homothetic motions, the follow- = ðx1, x2Þ, be constant. So, from equation (31), the equation ing equations are obtained: ! ! ζ ζ 2 F − F x 2 − px 2 − 2 s − 1 x +2 ps − 2 x + ðÞ0 X =0 !1 2 1 h2 t δ 1 2 h2 t δ 2 h2 t δ ðt ðt ðÞ0 p ðÞ0 p ðÞ0 p 1 2 2 S = s + is = 2p h2qdθ − i hdu , 33 1 2 2ph2 t δ p ð Þ ðÞ0 p t1 t1 ðt ðt 2 2 2 2 can be written. So, the following corollaries can be obtained. 2h ðÞt0 δps1 =2 h q1dθp − hdu2, t1 t1 ð ð Corollary 5. The geometric location of the points X with the t2 t2 2 2 same area FX is a circle in Cp with center 2ph ðÞt0 δps2 =2p h q2dθp − hdu1, t t 1 1 ! 29 ζ ζ ð Þ M = s − 1 , s + 2 34 1 2 2 2 ð Þ h ðÞt0 δp ph ðÞt0 δp where h is the homothetic scale in Cp: In here, it is known C that for p = −1 homothetic motion in the complex plane, in for the one-parameter planar homothetic motion in p. the hyperbolic plane for p =+1, and in the dual plane for p =0are mentioned. If p =0, this situation diﬀers from other Corollary 6. If h = 1, the geometric location of the points X cases ðp ≠ 0Þ; instead of a Steiner point, a Steiner line forms with the same area FX is a circle in Cp with center Steiner as s2 = s2ðγðtÞÞ. After all these calculations, equation (26) is point S = ðs1, s2Þ in Cp [35]. obtained that Now, let the Steiner area formula in equation (31) be gen- 1 ÀÁeralized using three linear points for homothetic motions in 2 2 2 C X Y Z FX = F0 + h ðÞt0 δp x1 − px2 − 2x1s1 +2px2s2 p. For this, let three points , , and in the moving plane 2 K X Y ðt ðt p be considered that and are two points and the other 1 2 1 2 ! ! + x1 ðÞ−2hq2 + u2 dh + x2 ðÞ2hq1 − u1 dh: Z XY O′X = x′ O′Y 2 t 2 t point is on . Moreover, let three vectors , 1 1 ! = y′ O′Z = z′ ð30Þ , and be position vectors of these points accord- ′ ing to K p. So, the relationship between these vectors is

Ð t ζ = 1/2 2 −2hq + u dh ζ = So, if the equations 1 t ð 2 2Þ and 2 z′ = αx′ + βy′, α + β =1, α, β ∈ R, 35 Ð 1 ð Þ t2 1/2 ð2hq1 − u1Þdh are considered, the following theorem t1 can be given. where α and β are barycentric coordinates of z′. Thus, considering equation (25), the area of the trajectory drawn by Ð t ′ 2 ′ ′ Theorem 4. For the homothetic motion Kp/K p in Cp, the Stei- Z is written by FZ = 1/2 ½z , dz and the area t1 ner area formula giving the area of the trajectory formed by ﬁ X ðt hi ðt hihi the xed point is 1 2 1 2 F = α2 x′, dx′ + αβ x′, dy′ + y′, dx′ Z 2 2 t1 t1 1 ðt hi 2 1 2 2 FX = F0 + h ðÞt0 δpðÞxx − xs − xs + ζ1x1 + ζ2x2, ð31Þ + β y′, dy′ 2 2 t1 ð36Þ where S = ðs1, s2Þ is the Steiner point of homothetic motion h C : and is the homothetic scale in p is obtained where

h =1 ðt hi Particularly, if is considered, equation (31) is 1 2 F = x′, dx′ , obtained X 2 t1 ðt hi 1 2 1 F = y′, dy′ , 37 F = F + δ xx − xs − xs 32 Y ð Þ X 0 pðÞð Þ 2 t 2 1 ðt hihi 1 2 ′ ′ ′ ′ FXY = x , dy + y , dx : 4 t in [35]. 1 Advances in Mathematical Physics 7

So, equation (35) is written by where h is the homothetic scale, α + β = 1, and FX and FY are the areas formed by the points X and Y, respectively, in Cp: 2 2 FZ = α FX +2αβFXY + β FY , ð38Þ During the one-parameter homothetic motions in Cp,if where FX = FY , equation (44) can be obtained that

1 2 1 2 2 FXY = FO + h ðÞt0 δpðÞxy + xy − ðÞx + y s − ðÞx + y s FX − FZ = αβh ðÞt0 δpd , ð46Þ 4 39 2 1 1 ð Þ + ζ x + y + ζ x + y 2 1ðÞ1 1 2 2ðÞ1 1 where α + β =1. So,

1 or F − F = h2 t δ XZ YZ , 47 X Z 2 ðÞ0 pjjjj ð Þ 1 F = F + h2 t δ x y − px y − x + y s + px + y s XY 0 2 ðÞ0 pðÞ1 1 2 2 ðÞ1 1 1 ðÞ2 2 2 where jXZj = βd and jYZj = αd. Thus, the following main 1 1 + ζ x + y + ζ x + y , theorem can be given with the above proof. 2 1ðÞ1 1 2 2ðÞ1 1 Theorem 8. X Y K ð40Þ (Main Theorem). Let two points and in p ⊂ Cp be ﬁxed and the point Z be on the line XY during the Ð t Ð t ζ = 1/2 2 −2hq + u dh ζ = 1/2 2 2hq − u XY where 1 t ð 2 2Þ and 2 t ð 1 1Þ homothetic motions. Moreover, when the endpoints of 1 1 Z XY ﬀ dh. draw the same curve, the point on draws a di erent X = Y curve. So, the relationship between areas formed by these Let be considered in equation (40). So, the equa- p Z X Y p tion curves depends on the -distances of to and , the -rotation angle of the homothetic motion, and homothetic scale h in C 1 p. F = F + h2 t δ xx − xs − xs + ζ x + ζ x 41 X 0 2 ðÞ0 pðÞ1 1 2 2 ð Þ This theorem is called Holditch theorem for the one- C : is obtained. As can be seen from here, equation (41) is the parameter homothetic motions in p So, this theorem is same as the Steiner area formula in equation (31) for the the most general form of all Holditch theorems obtained so homothetic motions in Cp. Thus, equation (40) is a more far. general form of the formula in equation (31). In addition, 4. Conclusion considering some calculations, the equation

1 ÀÁCurves are very important in kinematic mechanisms. The F − 2F + F = h2 t δ x 2 − px 2 − 2x y +2px y + y 2 − py 2 X XY Y 2 ðÞ0 p 1 2 1 1 2 2 1 2 trajectory drawn by a point or set of points (rigid body; such 42 as a robot) along the motion creates a curve. This curve can ð Þ be special curves such as a circle, ellipse, hyperbola, or a ran- dom curve formed by trajectory drawn by any point. It is is obtained. Now, let the distance between the points X and Y important to characterize the motion to make calculations d fi C be . So, using the de nition of distance (6) in p, the dis- such as the area and moment of the trajectory (curve) drawn tance is along the motion. The Holditch theorem is a theorem that expresses the area of the trajectory drawn during the motion. 2 2 2 fi d = ðÞx1 − y1 − pxðÞ2 − y2 ð43Þ To be more speci c, the Holditch theorem in plane geometry emphasizes that if a fixed-length chord is allowed to rotate in a convex closed curve, the position of a point on the chord x for branch I of Cp. So, equation (41) can be written as units from one end and y units from the other end, the curve 1 1 drawn by this point is less than the area of the original curve F = F + F − h2 t δ d2: 44 πxy. This theorem was first given in 1858 by the English XY 2 ðÞX Y 4 ðÞ0 p ð Þ mathematician Hamnet Holditch. Although not emphasized Z by Holditch, the proof of the theorem requires the chord to Thus, for the area of the trajectory drawn by , the fol- be short enough that the position of the point taken is a sim- lowing theorem can be given. ple closed curve. The fact that the area of trajectories expressed in the Holditch theorem is independent of the Theorem 7. C During the homothetic motions in p, the area of curve (circle, ellipse, etc.) makes this theorem very interest- the trajectory drawn by Z is ing. Thus, the Holditch theorem has been included as one of Clifford A. Pickover’s 250 milestones in the history of 1 2 2 mathematics. It should be noted again that the most impor- FZ = αFX + βFY − αβh ðÞt0 δpd , ð45Þ 2 tant feature of the theorem is that the formula that gives 8 Advances in Mathematical Physics the area πxy is independent of both the shape and size of the [12] H. R. Müller, Verallgemeinerung einer formel von Steiner, original curve, and this formula gives the same formula as the vol. 29, Abhandlungen der Braunschweigischen Wissenschaf- area of an ellipse with axes x and y. Until now, Holditch’s the- tlichen Gesellschaft Band, 1978. orem has been generalized to many planes and spaces. But [13] A. A. 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