14. Geometrical meanings of the inversion curvature. of a plane curve. Jusaku MAEDA. (Received January 10, 1940.) 1. Introduction.

Transcription

1 14. Geometrical meanings of the inversion curvature of a plane curve. By Jusaku MAEDA. (Received January 10, 1940.) 1. Introduction Circular point-transformations. It is well known that any circular point-transformation in the Gaussian plane which transforms z into z* is given by or by (1.1.1) (1.1.2) where z is the conjugate complex number of z, and a, b, c and d are constant complex numbers. Of these transformations (1.1.1) and (1.1.2), the former is said to be direct, while the latter is called indirect The inversion length and the inversion curvature of a plane curve. The problem of determining the necessary and sufficient condition in order that two plane curves given by their natural equations should be transformable into each other by a direct or an indirect circular pointtransformation was first solved by Prof. T. Kubota. (1) If we denote by s, ď and ĕ the arc-length, the curvature and the radius of curvature of a plane curve (M), the fundamental invariants of the curve (M) under circular point-transformations, viz, the inversion length p and the inversion curvature K, originally found by Prof. T. K ubota, are expressed in slightly modified forms as follows: (1) T. Kubota, Beitrage zur Inversionsgeometrie und Laguerre-Geometrie, Jap. Journ. of Math., I (1924), T. Kubota, Beitrage zur Inversionsgeometrie, Sci. Rep. Tohoku Imp. Univ. Series 1, 13 (1924),

2 178 Jusaku MAEDA the accents denoting differentiations with respect to the arc-length s. We know that the quantities ċ, dďds and K are unchanged by any direct circular point-transformation, while they change only their signs by any indirect circular point-transformation; hence both dp and K defined as above remain unaltered by any circular point-transformation. And an equation which gives the inversion curvature K as a function of the inversion length p serves as a natural equation of the curve under circular point-transformations. A geometrical meaning of the inversion length and that of the inversion curvature were given by G. Thomsen. (2) In this paper the writer will give other geometrical meanings of the inversion curvature by considering a certain circle, associated with the curve at each point on it, which plays the role of "normal" of the curve in the geometry of inversion, and will establish some consequences. We shall hereafter confine our investigations to such curves or such portions of curves that the quantity dď/ds relating to each of them has a constant sign. Before concluding this introduction we will add the following remark. If a curve (M) in the Gaussian plane is represented by the equation:z =z(t), (1.2.6) t being a parameter, the quantities dp, ċ and K may be written in the forms :(1.2.7) (2) G. Thomsen, Uber Konforme Geometrie II. Uber Kreisscharen und Kurven in der Ebene und uber Kugelscharen und Kurven im Raum, Abhandl. aus dem Hamb. Math. Sem., 4 (1926),

3 Geometrical meanings of the inversion curvature of a plane curve. 179 where R(z) and _??_(z) denote respectively the real part and the imaginary part of z, and {z, t} stands for the Schwarzian derivative of z with respect to t, i.e. Since this function is infinitely many valued having a and b for its branchpoints, it will be convenient to make use of its Riemann-surface(R) in order to consider the mapping by means of it. Let us denote by {C1} oriented circular arcs, described in (R), joining the points a and b, the positive sense of any one of these circular arcs being taken from a towards b; and denote by {C2} oriented coaxal circles, described in (R), cutting the oriented circular arcs {C1} under the angle +Ĕ/2, the angles being measured from {C1} to {C2}. Next consider oriented curves {L}, described in (R), cutting all the oriented circular arcs {C1} under the angle a, the angles being measured from {C1} to {L}. These curves {L} may be called loxodromic curves, or simply loxodromics, belonging to the angle a and having the poles a and b, where the order of a and b must be taken into account. It may be noted that in the case where a=0, these loxodromics coincide with the oriented circular arcs {C1}, and in the case where a=+Ĕ/2, they may be identif ied with the oriented coaxal circles {C2}. The angle a, under which the curves {L} cut {C1}, may be so taken that its absolute value does not exceed Ĕ. If we reverse the order of a and b, then we may say that the same oriented curves {L} are loxodromics belonging to the angle (a-Ĕ sgn a) having the poles b and

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25 Geometrical meanings of the inversion curvature of a plane curve. 201 If Ċ(s) is any one of the points on the envelope of the circle (K), then it is represented by where t(s) is a real function of s such that Differentiating (3.2.2) with respect to s, we get: By substitution of (3.2.2) in (3.2.4), we have: Next, differentiating (3.2.1) with respect to t, we get: Hence so that (3.2.3) reduces to which gives the two values of t corresponding to the points on the envelope. It will be seen that the necessary and sufficient condition that the circle (K), associated with the curve at each point on it, should arrays pass through a fixed point is that the equation (3.2.7) and the following equation (3.2.8) hold for all values of s. If dě denotes the angle between the circle (K) and its consecutive one, we have: where R1 is the radius of the circle (K) and it is given by (2.2.27).

26 202 Jusaku MAEDA 3.3. The points of intersection of the circle (K1) or (K2) with the consecutive one. If we put then Moreover if we put then (3.2.7) becomes The two roots of this equation are equal in magnitudes but they are of contrary signs, which shows that the following statement is true: The two points on the envelope of the circle (K1) or (2) divide har monically the base points of the coaxal system (ƒ ) enunciated in Theorem 1. Let M1, N1 be the points on the envelope of the circle (K1), and t1, t2 be the corresponding values of t. If H is the anharmonic ratio (M1N2PQ), then we have: from which follows: ƒã being +1 or -1 according as t12 is greater than or less than ƒé2. Hence, putting ƒá=ƒãƒ, we have: where ƒã is +1 or -1 according as the quantity in the right-hand side is greater than or less than unity The condition that the circle (K1) or (K2) should arrays pass through a fixed point. We shall find the necessary and sufficient condition that the circle (K1) or (K2), associated with the curve (M) at each point on it, should always pass through a fixed point. Under the conditions (3.3.1) and (3.3.3), the equation (3.2.8) becomes:

27 Geometrical meanings of the inversion curvature of a plane curve. 203 We shall now eliminate the quantity t from (3.3.4) and (3.4.1), If we put then Differentiating (3.4.4) with respect to s, we get: Substituting (3.4.3), (3.4.4) and (3.4.5) in (3.4.1), we have From (3.4.4) follows: Hence (3.4.6) becomes i.e. In the case of the circle (K1), we put then (3.4.2) and (3.4.7) become: from (3.4.9) we obtain

28 204 Jusaku MAEDA In the other case, by integration of (3.4.10), we have: or according as the quantity u2 sinć cosć is less than or greater than unity, p0 being an arbitrary constant. Thus we have the following result: The necessary and sufficient condition that the circle (K1), associated with a curve at each point on it, ć being equal to neither zero nor Ĕ/2, should always pass through a fixed point is that the curve be a loxodromic of the angle ć or that or that where p0 is an arbitrary constant. In this case the circle (K1) coincides with circle (N), and (3.4.9) and (3.4.10) reduce to the normal so that we have the following result The necessary and sufficient condition that the normal circle of a curve at any point on it should always pass through a fixed point is where p0 is an arbitrary constant. In the case of the circle (K2), we put then (3.4.2) and (3.4.7) become:

29 Geometrical meanings of the inversion curvature of a plane curve. 205 If ƒá=0, the circle (K2) coincides with the normal circle (N), and we get again the result above mentioned. If ƒá 0, we get from (3.4.16): p0 being an arbitrary constant. Hence we get the following result: The necessary and sufficient condition that the circle (K2), associated with a curve at each point on it, ƒá being equal to neither zero nor ƒî/2, should always pass through a fixed point is KsinƒÁcosƒÁ-2sin2ƒÁ=tan2{(tan1/2ƒÁ)(p+p0)}, (3.4.17) where p0 is an arbitrary constant. Since the points ƒä on the envelope of the circles (K1) and (K2) are given by (3.4.18) where t2sina+ƒè Œ(ƒÃKsina+2cosa)=0, (3.4.19) by elimination of the parameter a, we get: Z(N2+Z2)+ƒÈ Œ(ƒÃKZ-2N)=0, (3.4.20) which may be rewritten: (3.4.21) Thus we see that the points on the envelope of the circles (K1) and (K2) describe, as the angle ƒá varies, in general a bicircular quartic. If at every point on the curve (M), we have: so that Putting /=2b, b=const..

30 206 Jusaku MAEDA we have: hence u=b(ƒè2 }a2), a=const., from which follows: ƒè-1 =b(s+s0) (3.4.22) or ƒè =atan{ab(s+s0)} (3.4.23) or ƒè=-atanh{ab(s+s0)} (3.4.24) or ƒè= -acoth{ab(s+s0)} (3.4.25) a, b and s0 being arbitrary constants. Thus if the locus (3.4.21), associated with the curve (M) at any point on it, is always a circular cubic, we have the relation (3.4.22) or (3.4.23) or (3.4.24) or (3.4.25); and conversely. The equation (3.4.22) is the natural equation of an equiangular spiral From (2.5.17) and (2.5.18) follows: If the circles (K1) of a curve (M) are osculating circles of another. curve, then the inversion curvature of (M) is constant and is equal to -2cotƒÁ and conversely., or in other words, the curve (M) is a loxodromic of the angle ƒá/2; If the circles (K2) of a curve (M) are osculating circles of another curve, then the inversion curvature of (M) is constant and is equal to 2tanƒÁ, or in other words, the curve (M) is a loxodromzc of the angle and conversely The points of intersection of the circle (K01) with its con secutive one. If we put z0=-ƒê, (3.2.7) and (3.2.8) become: t2sinƒó-2ƒó Œt+ƒÈ Œ(2cosƒÓ+ƒÃKsinƒÓ)=0, 3.5.1) t2cosƒó+(ƒè /ƒè Œ)t+ƒÈ Œ(2sinƒÓ-ƒÃKcosƒÓ)-2t Œ=0. (3.5.2) Let us denote by (K01) the circle passing through M and the two poles P0 and Q0 of the osculating loxodromic (L0), and by (K02) the circle

31 Geometrical meanings of the inversion curvature of a plane curve. 207 passing through M, with respect to which P0 and Q0 are mutually inverse. Now, if we put ƒó=ƒãƒá, ƒá being determined by (3.1.6), then the circle (K) coincides with the circle (K01); in this case, by virtue of the relations 2cosƒÁ+KsinƒÁ=secƒÁ, 2sinƒÁ-KcosƒÁ=cosecƒÁ, (3.5.3) the equations (3.5.1) and (3.5.2) become: sinƒá(t2-ƒé02)-2ƒá Œt=0, (3.5.4) cosƒá(t2-ƒé02)+(ƒè /ƒè Œ)t-2t Œ=0. (3.5.5) The two roots of the equation (3.5.4) determine the points of intersection of the circle (K01) and its consecutive one. Next, if we put ƒá being determined by (3.1.6), then the circle (K) coincides with the circle (K02), and in this case the equations (3.5.1) and (3.5.2) become: cosƒá(t2+ƒé02)-2ƒãƒá Œt=0, (3.5.6) sinƒá(t2+ƒé02)-ƒã(ƒè /ƒè Œ)t+2ƒÃt Œ=0. (3.5.7) The two roots of the equation (3.5.6) determine the points of intersection of the circle (K02) and its consecutive one. Let M0 and N0 be the points on the envelope of the circle (K01), and t01 and t02 be the corresponding values of t. If H0 is the anharmonic ratio (M0N0MN), then we have: from which follows: H0=(t01t02 0)=t02/t01, -(1+H0)2/4H0=cotƒÁ(dƒÁ/dp)2. Let H0 Œ denote the anharmonic ratio (M0N0P0Q0); then Since t01t02=

32 208 Jusaku MAEDA we get: (1-H0 Œ)/(1+H0 Œ)=(t02-t01)/2ƒÉ0, whence follows: (1-H0 Œ)/(1+H0 Œ)2=1+cot, or - 4H0 Œ/(1+H0 Œ)=cot so that we have: -(1+H0)2/4H0=-4/(1+H0 Œ)2=cot (3.5.8) Therefore if one of the anharmonic ratios H0 and H0 Œ is constant along the curve, then the other is also constant. Suppose H0 and H0 Œ to be constant, so that -(1-H0)2/4H0-4H0 Œ/(1+H0 Œ)2=a2/2, (3.5.9) a being a constant. Then, from (3.5.8) we have: p0 being an arbitrary constant, Putting cot1/2ƒá=u, we get: Hence we have the following result: (3.5.10) where tan ƒá is given by (3.1.6), and p0 is an arbitrary constant. If we eliminate the quantity t2-ƒé02 between (3.5.4) and (3.5.5), we have: t Œ/t=ƒÈ /2ƒÈ Œ+ƒÁ ŒcotƒÁ,

33 Geometrical meanings of the inversion curvature of a plane curve. 209 so that t=a ƒè Œ 1/2sinƒÁ, a being an arbitrary constant. Substituting this in (3.5.4), we get: 2ad/ƒÁdp=a2sin2ƒÁ-2cosec2ƒÁ. (3.5.11) Similarly by elimination of t from (3.5.6) and (3.5.7), we obtain: 2ad/ƒÁdp=a2cos2ƒÁ+2cosec2ƒÁ, (3.5.12) where ƒá is given by (3.1.6), and a is an arbitrary constant. Therefore we may state: The necessary and sufficient condition that the circle (K01) of a plane curve (M) at any point on it should always pass through a fixed point is that (3.5.11) hold along the curve (M). The necessary and sufficient condition that the circle (K02) of a plane curve (M) at any point on it should always pass through a fixed point is that (3.5.12) hold along the curve (M) The angle between the circle (K01) or (K02) and the consecu tive one. If we take z0=-ƒê, (3.6.1) then the radius R1 of the circle (K), with a properly chosen sign, is given by 1/R1=ƒÈcosƒÓ+(ƒÈ /2ƒÈ Œ)sinƒÓ. (3.6.2) Substituting this in (3.2.9), we get: (3.6.3) If we put ƒó=ƒãƒá or ƒó=ƒî/2+ƒãƒá, 0 ƒá ƒî/2, where ƒá is a constant, we obtain the formula (2.5.17) or (2.5.18). Putting ƒó=ƒãƒá or ƒó=ƒî/2+ƒãƒá,

34 210 Jusaku MAEDA where ƒá is determined by (3.1.6), we obtain: (3.6.4) (3.6.5) where dƒµ01 or dƒµ02 is the angle between the circle (K01) or (K02) and the consecutive one. We have also the following relation: (3.6.6) If dƒµ02/dp=0 along the curve (M), by integration of tan1/2ƒádƒá= }dp, we get: (3.6.7) where ƒá is given by (3.1.6), and p0 is an arbitrary constant. Hence we may state: If the circles (K02) of a plane curve (M) are osculating circles of another curve, then the relation (3.6.7) holds; and conversely The angle between the circle (K01) or (K02) and the consecu t ive one. Let us denote by (K01) the circle passing through the point M and touching the locus of the point N at N, and by (K02) the circle passing through the point M and orthogonal to the locus of the point N at N, where N is the base point, other than M, of the coaxal system (ƒ ). When the circle (K) coincides with the circle (K01), we have: tanƒó=-2ƒã/k=ƒãtan2ƒá. (3.7.1) And when the circle (K) coincides with the circle (K02), we have: tanƒó=-ƒãk/2=ƒãcot2ƒá. (3.7.2)

35 Geometrical meanings of the inversion curvature of a plane curve. 211 If we denote by dƒµ01 or dƒµ02 the angle between the circle (K01) or (K02) and the consecutive one, by virtue of (3.6.3) we obtain: (3.7.3) (3.7.4) where ƒá is given by (3.1.6). From (3.7.3) follows: If the circles (K01) of a curve (M) are osculating circles of another curve, then the inversion curvature of the curve (M) is constant, or in other words, the curve (M) is a loxodromic; and conversely The points on the envelope of the circles (N1) and (N2), and the points on the envelope of the circles (N01) and (N02). Let P and Q be two points given respectively by (3.8.1) (3.8.2) where a is a function of the arc-length s of the curve (M), and ƒé=(-2ƒè Œcosec2a)1/2, sgn(tana)=sgn(-ƒè Œ)=ƒÃ. (3.8.3) If we take z0=ƒécosa-ƒè /2ƒÈ Œ, ƒó=ƒî/2, (3.8.4) then the circle (K), in this case, is one passing through the point P and orthogonal to the curve (M) at the point M. This circle (N) is re presented by ƒä-z 2z Œ/ +ƒécos a-ƒè /2ƒÈ Œ=it, F(t)=0. (3.8.5) By calculation we get: ƒé Œ/ƒÉ-ƒÈ /2ƒÈ Œ=-a Œ cot2a, (3.8.6) 2z0 Œ+z02-2iƒÈz0=-ƒÉa Œcoseca-ƒÈ Œ(ƒÃK+cota)-ƒÈ2-2iƒÈ(ƒÉcosa-ƒÈ /2ƒÈ Œ) (3.8.7)

36 212 Jusaku MAEDA so that (3.2.7) and (3.2.8) become: u2+ƒéaa Œcoseca+ƒÈ Œ(ƒÃK+cota)=0, u Œ+u(ƒÉcosaƒÈ /2ƒÈ Œ)-ƒÈ Œ=0, where we have put: u =t+ƒè (3.8.10) The two roots u1, u2 of the equation (3.8.8) determine the two points, say E, F, on the envelope of the circle (N). We shall consider the anharmonic ratio H=(EFMP). Let u0 denote the value of u corre sponding to the point P; then Since H=(u1u2 u0)=(u2-u0)/(u1-u0), u0=-ƒésina we obtain: (1-H)2/(1+H)2=ƒÃKcota+cot2a+(ƒÉa Œ/ƒÈ Œ)cosacosec2a. (3.8.12) Let us put a=ƒãƒá or a=ƒî+ƒãƒá, 0<ƒÁ<ƒÎ/2. (3.8.13) If ƒá is constant, the circle (N) coincides with the circle (N1) or (N2), and the corresponding anharmonic ratio H satisfies the relation (1-H)2/(1+H)2=KcotƒÁ+cot2ƒÁ. (3.8.14) If we take a such that (3.8.13) holds, the value of ƒá being determined by (3.1.6), then the circle (N) becomes the circle (N01) or (N02) passing through the pole P0 or Q0 of the osculating loxodromic (L0) and orthogonal to the given curve (M) at the point M. If H01 and H02 are the corresponding anharmonic ratios, then we get: -6H01/(1+H0102=6H02/(1+H02)2=d/dpcot3/2ƒÁ. (3.8.15) If H01 or H02 is constant along the curve (M), so that if we obtain: -6H01/(1+H01) 2=6H02/(1+H02)2=a=const. cot3/2ƒá=a(p+p0),, where ƒá is given by (3.1.6), and p0 is an arbitrary constant.

37 Geometrical meanings of the inversion curvature of a plane curve. 213 Since the points on the envelope of the circles (N1) and (N2) are given by where u2=-ƒè Œ(ƒÃK+cota), we get: N2=-ƒÈ Œcota, Z2=-ƒÈ Œ(ƒÃK+cota). Therefore the locus of the points, on the envelope of the circles (N1) and (N2), ƒá being the variable, is in general a bicircular quartic, whose equation is N2-Z2=ƒÃƒÈ ŒK which may be rewritten in the form: {ƒè,s}(x2+y2)2+4{(ƒè /2ƒÈ Œ)x-ƒÈy}(x2+y2)-4(x2-y2)=0. (3.8.19) If this locus (3.8.19), associated with the curve (M) at each point on it, is always a circular cubic, then a, b, c and d being constants; and the converse is also true. If we eliminate u from (3.8.8) and (3.8.9), we get: exp (2ƒ ) E{ƒÉa Œcoseca+ƒÈ Œ(ƒÃK+cota)}+( çƒè Œexpƒ ds+const.)2=0, whereƒ = ç (3.8.20) (ƒécosa-ƒè /2ƒÈ Œ)ds=log ƒè Œ -1/2+ çƒécosads. (3.8.21) The equation (3.8.20) gives the necessary and sufficient condition that the circle (N), associated with the curve (M) at its moving point, should always pass through a fixed point. If we take a such that (3.8.13) holds, ƒá being constant, then we have:

38 214 Jusaku MAEDA so that (3.8.20) reduces to K+cotƒÁ={tan1/2ƒÁ+aexp(-ƒÃpcot1/2ƒÁ)}2, ƒã= }1, (3.8.22) where a is an arbitrary constant; thus we may state: The necessary and sufficient condition that the circle (N1) or (N2), associated with a curve at each point on it, should always pass through a fixed point is given by (3.8.22), where ƒã=+1 in the case of the circle (N1) and ƒã=-1 in the case of the circle (N2). Next, if we take a such that (3.8.13) holds, the value of ƒá being determined by (3.1.6), then (3.8.20) becomes: (3.8.23) where ƒã is +1 or -1 according as the circle (N01) or (N02) is concerned. Thus we may state: The necessary and sufficient condition that the circle (N01) or (N02) of a curve (M) at any point on it should always pass through a fixed point is given by (3.8.23), ƒá being determined by (3.1.6) The angle between the circle (N01) or (N02) and the consecu tive one. Since the reciprocal of the radius of the circle (N), with a properly chosen sign, is equal to ƒè /2ƒÈ Œ-ƒÉcosa, by virtue of (3.2.9) the angle dƒµ between the circle (N) and its consecutive one is given by ( d -ƒéa Œcoseca-ƒÈ Œ(ƒÃK+cota). (3.9.1) When the circle (N) coincides with the circle (N1) or (N2), this formula reduces to (2.5.10) or (2.5.11). From (2.5.10) and (2.5.11) we obtain: If the circles (N1) or (N2) of a curve (M) are osculating circles of another curve, then the inversion curvature of (M) is constant and is equal to -cotƒá, or in other words, the curve (M) is a loxodromic of the angle 1/2cot-1(1/2cotƒÁ;) and conversely.

39 Geometrical meanings of the inversion curvature of a plane curve. 215 In particular, considering the case whereƒá=ƒî/2 we may state as follows: If the normal circles of a curve are osculating circles of another curve, Then the curve is a loxodromic of the angle ƒî/4; and conversely. When the circle (N) coincides with the circle (N01) or (N02), (3.9.1) becomes: (3.9.2) or (3.9.3) where dƒ 01 or dƒ 02 denotes the angle between the circle (N01) or (N02) and the consecutive one. Consequently we get: (3.9.4) From (3.9.2) and (3.9.3) follows: If the circles (N01) or (N02) of a curve (M) are osculating circles of another curve, then 4cot3ƒÁ=9(p+p0)2, (3.9.5) p0 being an arbitrary constant; and conversely The circle (K). We shall proceed to determine the points on the envelope of the circle (K) represented by (3.10.1) ƒó being an arbitrary real number such that -ƒî ƒó ƒî. If ƒä(s) is any one of the points on the envelope of the circle (K), then it is represented by (3.10.2) where ƒó(s) is a real function of the arc-length s of the curve (M) such that (3.10.3)

40 216 Jusaku MAEDA Since -2it(ƒÄ Œ/ ÝƒÄ ÝƒÓ=t2eiƒÓ+2t(iƒÈ-iƒÓ Œ-z0)+e-iƒÓ (2z Œ0+z02-2iƒÈz0)-2t Œ, the condition (3.10.3) is equivalent to t2 cosƒó-2r(z0)t+r{e-iƒó(2z0 Œ+z02-2iƒÈz0)}-2t Œ=0. (3.10.4) The necessary and sufficient condition that the circle (K), associ ated with the curve (K) at its moving point M, should always pass through a fixed point is that the equation (3.10.4) and the following equation (3.10.5) hold along the given curve (M). t2sinƒó+2{ƒè-ƒó Œ-F(z0)}t+F{e-iƒÓ(2z0 Œ+z02-2iƒÈz0)}=0. (3.10.5) The points on the envelope of the circle (K3) or (K03). If we put z0=-ƒê=-ƒè /2ƒÈ Œ+iƒÈ, ƒó=ƒãƒµ, (3.11.1) t=-ƒé=-(2 ƒè Œ cosec2ƒá)1/2, 0<ƒÁ<ƒÎ/2, (3.11.2) ƒá being a function of the arc-length s of the curve (M), then (3.10.4) and (3.10.5) become: 2sinƒµ-cosƒµ+(K+2cosec2ƒÁ+2cot2ƒÁ(2cosec2ƒÁ)1/2dƒÁ/dp=0, ) sinƒµ(2cosec2ƒá-k)-2cosƒµ+2(2cosecƒá)1/2dƒµ/dp=0. (3.11.4) Putting the equation where (3.11.3) may be rewritten: lsinx+mcosx+n=0, (3.11.5) l=2cosƒá+secƒá+ksinƒá, m=2sinƒá-cosecƒá-kcosƒá, (3.11.6) n=2cot2ƒá(2cosec2ƒá)1/2dƒá/dp,} and the equation (3.11.4) may be rewritten: (3.11.7)

41 Geometrical meanings of the inversion curvature of a plane curve. 217 where l1=2sinƒá+cosecƒá-kcosƒá, m1=-2cosƒá+secƒá-ksinƒá, (3.11.8) n1= 2(2cosec2ƒÁ)1/2.} If we put u=tanx/2, (3.11.9) the equation (3.11.5) becomes: (n-m)u2+2lu+(n+m)=0, ( ) whose two roots u1 and u2 determine the points, M and N, on the envelope of the circle (K3) in consideration. If we denote by H the anharmonic ratio (MNPQ), P and Q being determined respectively by (3.8.1) and (3.8.2), then we have: H=(MNPQ)=(u1u20 )=u1/u2, ( ) so that 4H/(1+H)2=(n2-m2)/l2. ( ) First suppose ƒá to be constant; then the circle (K3) coincides with the circle (K3), and ( ) becomes: -4H/(1+H)2=(KcosƒÁ+cosecƒÁ-2sinƒÁ)2/(KsinƒÁ+secƒÁ+2cosƒÁ)2 ( ) Next suppose ƒá to be given by (3.1.6); then the circle (K3) coin cides with the circle (K03), and we have in this case m=0, l=2secƒá. Hence, denoting by H0 the anharmonic ratio, from ( ) we get: 4H0/(1+H0)2=cotƒÁcot22ƒÁ(dƒÁ/dp)2. If the anharmonic ratio I~ is constant along the curve (M), so that if 4H0/(1+H0)2=a2, a being a constant, then we have: ç cot1/2ƒácot2ƒádƒá= }a(p+p0), where p0 is an arbitrary constant.

42 218 Jusaku MAEDA Putting we get: cot1/2ƒá=u, Thus if the anharmonic ratio H0 is constant along the curve (M), then the following relation ( ) holds; and conversely. where ƒá is given by (3.1.6), and p0 is an arbitrary constant. ( ) The points on the envelope of the circle (K3) are given by ( ) where 2tanƒµ=K+2cosec2ƒÁ. ( ) Hence, by elimination of ƒµ and ƒá, we obtain: N(N2+Z2)-ƒÈ(2Z+ƒÃKN)=0, ( ) which may be rewritten in the form: ( ) Thus the locus of the points of intersection of the circle (K3) and its consecutive one, ƒá being the parameter, is in general a bicircular quartic The necessary and sufficient condition that the circle (K3) should always pass through a f ixed point. This condition is that the following two equations hold simul taneously for all values of p. K+2cosec2ƒÁ=2tanƒµ, (3.12.1) =0sinƒµ(2cosec2ƒÁ-K)-2cosƒµ+2(2cosec2ƒÁ)1/2dƒµ/dp. (3.12.2)

43 Geometrical meanings of the inversion curvature of a plane curve. 219 Eliminating K we get: (2sin2ƒÁ)1/2cosƒµdƒµ/dp+sin2ƒµ-sin2ƒÁ=0. (3.12.3) If sin2ƒµ=sin2ƒá, we get: (3.12.4) If sin2ƒµ=-sin2ƒá, we have: Since (3.12.5) we get by integration: (3.12.6) Thus the required condition is (3.12.4), or (3.12.6) with (3.12.1). Next we shall find the necessary and sufficient condition that the circle (K03) should always pass through a fixed point. If ƒá is given by (3.1.6), then we have: l=2secƒá, m=0, l1=2cosecƒá, m1=0, so that (3.11.5) and (3.11.7) become: secƒásinx+cot2ƒá(2cosec2ƒá)1/2dƒá/dp=0, (3.12.7) (3.12.8) from which follows: 2dx/dƒÁ=cot2ƒÁ-3, Hence, by integration we obtain: x= a-2ƒá-1/2cotƒá, a being an arbitrary constant.

44 220 Jusaku MAEDA Therefore the required condition is that the following equation hold for all values of p. = tan2ƒásin(a-2ƒá1/2cotƒá+cosƒá(2cosec2ƒá)1/2dƒá/dp 0, (3.12.9) where ƒá is given by (3.1.6), and a denotes an arbitrary constant The angle between the circle (K03) and its consecutive one. The centre c3 of the circle (K) is given by c3=z+2z Œz0/V, (3.13.1) and its radius R3, with a properly chosen sign, is given by where R3=2t/V, (3.13.2) V=t2-z0z0. (3.13.3) Let dƒµ denote the angle between the circle (K) and its consecutive one; then (3.13.4) Now we have: so that (3.13.5) If we put z0=-ƒê,t=-ƒé=-(2 ƒè Œ cosec2ƒá)1/2

45 Geometrical meanings of the inversion curvature of a plane curve. 221 ƒá being a function of the arc-length s of the curve (M), then where U=ƒÈ /2ƒÈ Œ Hence we get finally: 8cosec2ƒÁ(dƒµ/dp)2=4+(K+2cosec2ƒÁ)2-8cosec2ƒÁcot22ƒÁ(dƒÁ/dp)2. (3.13.6) In the case where ƒá is constant along the curve (M), this formula (3.13.6) reduces to (2.5.22). If ƒá is given by (3.1.6), then the circle (K) coincides with the circle (K03), and the angle dƒµ03 between the circle (K03) and its consecutive one is given by (3.13.7) If dƒµ/dp=0 at every point on the curve (M), then we have: ç cot1/2ƒácot2ƒádƒá= }(p+p0), p0 being an arbitrary constant. Integrating this we get: (3.13.8) where ƒá is given by (3.1.6), and p0 is an arbitrary constant. Thus we may state: If the circles (K03) of a curve are osculating circles of another curve, then the relation (3.13.8) holds; and conversely The locus of the point N. The coordinate c of the point N, which is the base point, other than the point M, of the coaxal system (ƒ ), is given by c=z+2z Œ/ƒÊ (3.14.1)

46 222 Jusaku MAEDA Hence, by differentiation, we get:c ŒƒÊ 2=2ƒÈ Œz Œƒ, c /c Œ=ƒÈ /2ƒÈ Œ+ƒ Œ/ƒ +2ƒÈ Œƒ /ƒê, (3.14.2) where we have put: ƒ =-ƒãk/2+i=cot2a+i, (3.14.3) ƒãa or ƒá being the angle of the osculating loxodromic (L0) of the given curve (M) at the point M. Also we have:ƒ Œ /ƒ =2a Œ(i-cot2a), 2a -a ŒƒÈ /ƒè Œ=2p Œ2d2a/dp2, ) ƒ =(ƒ Œ/ƒ ) Œ-(ƒ Œ/ƒ )2/2=2a (i-cot2a)+2a Œ2(3+cot22a+2icot2a), whence follows: Thus we obtain: (3.14.5) (3.14.6) Consequently the inversion length p* of the locus of the point N is given by (3.14.7) If dp*/dp=m=const., we have (3.14.8) from which we can deduce the following equation: 8sin22ƒÁ(dƒÁ/dp)=(1 }m)(4ƒá-sin4ƒá)+const.. (3.14.9)

47 Geometrical meanings of the inversion curvature of a plane curve. 223 The curve, for which the locus of the point N is a circle, is obtained by putting m=0 in (3.4.9). In the case where m }1=0, we have the following result: If the inversion lengths of corresponding arcs of a curve (M) and the locus of the point N are always equal, then the following relation ( ) holds; and conversely. cos2ć=ap+b, ( ) where ć is given by (3.1.6), and a and b are arbitrary constants. 4. Miscellaneous Remarks On the ć-princzpal circles. The reciprocals of the radii of the ć-principal circles, with properly chosen signs, of the curve (M), ć being supposed to be equal to neither zero nor Ĕ/2, are equal to ( ď Πtanć)1/2. If the product of these radii is constant and is equal to }a-2 along the curve (M), then so that we have: a coth{(acotć)(s+s0), or atan{(acotć)(s+s0)}, where s0 is an arbitrary constant. If the sum of the squares of the reciprocals of the radii of the ć-principal circles of the curve (M) is constant and is equal to 2a2 along the curve, then so that we have: being an arbitrary constant. If the sum of the radii of the -principal circles of the curve (M) is constant and is equal to 2a along the curve (M), then

48 224 Jusaku MAEDA Integrating this we get: ĕ =a-(sgna)exp{ }(cotć)(s+s0)/a} (4.1.7) where ĕ is the radius of curvature and s0 is an arbitrary constant. If one of the radii of the ć-principal circles of the curve (M) is constant and is equal to a-1 along the curve, we have: ď Πtanć=(ď-a)2 (4.1.8) hence follows: ď =a }(tanć)/(s+s0) (4.1.9) where s0 is an arbitrary constant. If the ratio of the radii of the ć-principal circles of the curve (M) is constant along the curve, we have: ď Πtanć=a2ď2, (4.1.10) where a denotes a constant. From this we get by integration: ĕ= } a2(cotć)(s+s0), s0 being an arbitrary constant; thus the curve (M) is an equiangular spiral On the circles (N1) and (N2). The reciprocals of the radii of the circles (N1) and (N2), with properly chosen signs, of the curve (M), ć being supposed to be equal to neither zero nor Ĕ/2, are equal to If the product of these radii is constant and is equal to }a-2 along the curve, then (4.2.1) or (4.2.2) First we consider the equation (4.2.1). Putting

49 Geometrical meanings of the inversion curvature of a plane curve. 225 we have: ƒè Œ cotƒá+a2u2, (4.2.3) ƒè ds/2ƒè Œ=adu/(u2-a2), u2 berg greater than a2, so that du/(u2-a2)=ds, whence u=-acotha{a(s+s0)}, s0 being an arbitrary constant. Substituting this in (4.2.1), we get: ƒè Œ cotƒá=a2cosech2{a(s+s0)}, hence }ƒè =b-atanƒácotha{a(s+s0)}, (4.2.4) where b is an arbitrary constant. Next we consider the equation (4.2.2). If ƒè Œ cotƒá=a2, the equation (4.2.2) is satisfied, and in this case we have: }ƒè= a2(tanƒá)(s+s0), (4.2.5) s0 being an arbitrary constant. Hence the curve is a clothoid. In the other case, putting u=ƒè /2ƒÈ Œ we get: ƒè Œ cotƒá-a2=u2, (4.2.6) ƒè ds/2ƒè Œ=udu/(u2+a2), so that du/(u2+a2)=ds, whence u=atan{a(s+s0), s0 being an arbitrary constant. Substituting this in (4.2.6), we have: ƒè Œ cotƒá=a2sec2{a(s+s0)},

50 226 Jusaku MAEDA hence }ƒè =a tan ƒá tan {a(s+s0}+b (4.2.7) where b is an arbitrary constant. Thus the curves with the said property are characterized by the equation (4.2.4) or (4.2.5) or (4.2.7). If the sum of the squares of the recipriocals of the radii of the circle (N1) and (N2) is constant and is equal to 2a2 along the curve, then (ƒè /2ƒÈ Œ)2+ ƒè Œ cotƒá=a2. (4.2.8) Putting u=ƒè /2ƒÈ Œ we have: ƒè Œ cotƒá=a2-u2, (4.2.9) ƒè ds/2ƒè Œ=udu/(u2-a2), so that uds=udu/(u2-a2), whence follows: u=0 or u2 being less than a2. In the first case, we have: }ƒè =a2(tanƒá)(s+s0), (4.2.10) s0 being an arbitrary constant. In the second case, we get: u=-atanh{a(s+s0)}, s0 being an arbitrary constant. Substituting this in (4.2.9), we have: ƒè Œ cotƒá=a2sech2{a(s+s0)}, so that }ƒè =atanƒátanh{a(s+s0)}+b, where b is an arbitrary constant.

51 Geometrical meanings of the inversion curvature of a plane curve. 227 Thus the curves with the said property are characterized by the natural equation (4.2.10) or (4.2.11). If the sum of the radix of the circles (N1) and (N2) is constant and is equal to a-1 along the curve, then Putting ƒè Œ cotƒá+a2=(a-ƒè /2ƒÈ Œ)2. (4.2.12) a-ƒè /2ƒÈ Œ=u, (4.2.13) we get: ƒè Œ cotƒè+a2=u2, (4.2.14) ƒè ds/2ƒè Œ=udu/(u2-a2), so that ds=-udu/(u+a)(u-a)2. (4.2.15) Hence we have: so that s+coast.=1/2(u-a)-1+1/2acot-1u/a (4.2.16) From (4.2.14) and (4.2.14), we have: } cotƒèdƒè/du=1+a/u-a), whence }ƒè cotƒá=u+log(u-a)2+coast.. (4.2.17) Thus the curves with the said property are characterized by the equations (4.2.16) and (4.2.17). If the ratio of the radii of the circles (N1) and (N2) is constant, and consequently if along the curve ƒè /2ƒÈ Œ= a( ƒè Œ cotƒá)1/2, (4.2.18) a being a constant, then putting ƒè /2ƒÈ Œ=u, we get: a2 ƒè Œ cotƒá=u2, (4.2.19) ƒá ds/2ƒá Œ=du/u,

52 228 Jusaku MAEDA so that whence s0 being an arbitrary constant. Substituting this in (4.2.19), we get: where b is an arbitrary constant. Since we have: the condition (4.2.18) may be rewritten: hence so that we have: where p0 and s0 are arbitrary constants. If one of the radii of the circles (N1) and (N2) is constant and is equal to 1/2a along the curve, then If a=0, i.e. if, at every point on the curve, one of the circles (N1) and (N2) becomes a straight line, then we have: where s0 and b are arbitrary constants. If a 0, putting we have: If u=a, then where s0 is an arbitrary constant.

53 Geometrical meanings of the inversion curvature of a plane curve. 229 From (4.2.25) we get: ds/2ƒè Œ=du/(u+a), so that du/(a2-u2)=ds In the case: u2<a2, we have: u =a tanh {a(s+s0)}, (4.2.27) s0 being an arbitrary constant. And in the case: u2>a2, we have: u =acoth{a(s+s0)}, (4.2.28) s0 being an arbitrary constant. Substituting (4.2.27) in (4.2.25), we have whence follows: Next, substituting (4.2.28) in (4.2.25), we have: whence follows: Thus the curves with the said property are characterized by the natural equation (4.2.24) or (4.2.26) or (4.2.29) or (4.2.30). The equation (4.2.23) may be rewritten: hence b being an arbitrary constant. Therefore we obtain:

54 230 Jusaku MAEDA Let us next consider such a curve that the locus of the centre of the circle (N1) or (N2) is an involute of it. Putting we have: so that for such a curve we have: Hence follows: s0 being an arbitrary constant, so that we have: where p0 is an arbitrary constant; consequently we obtain Hence the curves with the said property are characterized by the equa tion: 4.3. On the normal circle. The radius of the normal circle, with a properly chosen sign curve (M) is equal to 2ƒÈ Œ/ƒÈ., of the If the normal circle of the curve, at any point on it straight line, we have:, becomes a where a and b are arbitrary contants. Hence the curve is a clothoid. If the radius of the normal circle is constant and is equal to 2a along the curve, we have:

55 Geometrical meanings of the inversion curvature of a plane curve. 231 whence follows: s0 and b being arbitrary constants. If the locus of the centre of the normal circle is an involute of the given curve, then from which follows a, b and s0 being arbitrary constants. The sum of the squares of the reciprocals of the radii of the circles (K1) and (K2) is independent of the angle ƒá, and it is equal to the sum of the squares of the reciprocals of the radii of the normal circle and the osculating circle; this quantity is equal to If this quantity is constant and is equal to 2a2(a>0) along the curve, we have: Putting we get: from which follows: 4.4. On the point N. From (2.4.2) we see that the angular coefficient, referred to the tangent and the normal to the curve (M) at the point as coordinate axes, of the straight line MN, M and N being the base points of the coaxal system (ƒ ), is equal to 2ƒÈƒÈ Œ/ƒÈ If we put

56 232 Geometrical meanings of the inversion curvature of the plane curve. m being a numerical constant, then G(3) is the angular coefficient of the affine normal of (M) at M, G (3/2) is that of the angular normal (8) of (M) at M, and G(-1) is that of the straight line joining M to the centre of second curvature of (M) at M. If the straight line MN coincides always with the straight line which passes through M and whose angular coefficient is equal to G(m), we get by integration: where a and s0 are arbitrary constants. Therefore, in the case: m=-1, we may state as follows: If for a curve (M), the straight line joining the base points of the coaxal system (ƒ ) enunciated in Theorem 1 coincides array's with the straight line joining the point M on the curve to the centre of second curvature of (M) at M, then the curve (M) must be an equiangular spiral; and conversely. In conclusion the present writer wishes to express his hearty thanks to Prof. T. KUBOTA for his kind guidance.