Elements in the bundle and on the sphere, distance, angle and absolute polarity are explained.
The spherical conic has an absolute polar conic, they are equidistant to each other; they have the same evolute curve. The well-known focus-properties are absolutely polarized.
The loxodrome and the spherical evolute curve are presented. Polarizing the loxodrome there results a spherical tractrix. Kinematic generation of these curves is shown

H. Dirnböck

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Mathematical Communications 4(1999), 225–240 225Absolute polarity on the sphere; conics;loxodrome, tractrix∗†Hans Dirnbo¨ck‡Abstract. Elements in the bundle and on the sphere, distance, angleand absolute polarity are explained. The spherical conic has an absolutepolar conic, they are equidistant to each other; they have the same evo-lute curve. The well-known focus-properties are absolutely polarized.The loxodrome and the spherical evolute curve are presented. Polar-izing the loxodrome there results a spherical tractrix. Kinematic gener-ation of these curves is shown.Key words: bundle, sphere, absolute polarity; spherical conic, lox-odrome, spherical tractrix; spherical evolute curve, equidistant curves;kinematic generationAMS subject classifications: 51N05, 51N20, 53A04, 53A17, 53A35Received July 2, 1999 Accepted September 19, 19991. Geometry in the bundleIn the EUKLIDean R3 we fix one point O(0/0/0) and so we get the nonEUKLIDeanelliptic geometry in the bundle with the elements:straight lines 3 O, named “points”planes 3 O, named “straight lines”.Further we have:cones of rotation, spit O, named “circles”cones of 2nd deg., spit O, named “curves of 2nd degree”cones, spit O, named “curves”angles, formed by two straight lines, named “distance between points”angles, formed by two planes, named “angles between straight lines”∗The lecture presented at the Mathematical Colloquium in Osijek organized by CroatianMathematical Society - Division Osijek, April 23, 1999.†The author wants to express his best thanks to the referee and to the editorial staff.‡Institute of Mathematics, Statistics and Didactics, University of Klagenfurt, Univer-sita¨tsstrasse 65, A-9 020 Klagenfurt, Austria, e-mail: hans.dirnboeck@uni-klu.ac.at226 H.Dirnbo¨cka straight line 3 Oand a plane 3 Owhich are orthogonal (⊥)are absolutely polarThis situationis named“a pointand a straight linewhich are orthogonal (⊥)are absolutely polar”Often we calculate with CARTESian coordinates and vectors. All elements andshapes are extended in both directions from O. We write ⊥= 90o = 100g = pi/2.Figure 1.1. The bundle. All elements and shapes are extended in both directionsfrom O!2. Geometry on the sphereWe introduce the unit sphere K with center Ox2 + y2 + z2 = 1; (2.1)all elements and shapes of 1. are intersected with this sphere. Now we get:pairs of points, named “points”great circles, named “straight lines”Absolute polarity on the sphere 227pairs of small circles, named ”circles”“curves of 2nd degree”“curves”“distance between points”“angles between straight lines”If_point1point2 =pi2 , we are allowed to say “orthogonal points”.If ̂straight line1straight line2 = pi2 , we say “orthogonal straight lines”.If a point and a straight line which have the shortest distance = pi2 , are absolutelypolar, we are allowed to say that “they are orthogonal”.This well-known transformation P which transforms points into straight lines,so that distance = pi2 , is the absolute polarity P, in this paper short polar-ity P. We remark that P transforms distances into angles. P is an involutorictransformation, so that P ◦P = I = identity.The geometries in the bundle 1. and on the sphere 2. are isomorphic; thereremain only questions of convenience, visualization, didactics, calculation, comput-ing, constructive and descriptive geometry to choose the method of work. Often weuse spherical trigonometry. It is allowed to use the vocabulary of geography.A line element is a point with direction. P transforms such a line element intoa line element.Let “*” be the symbol for absolutely polarized elements.We write P ∗ = P ◦ P or P ∗ ⊥ P , where P is a point, but P ∗ is a straight line,we speak: P ∗ is the polar line of P .We write g∗ = P ◦ g or g∗ ⊥ g, where g is a straight line, but g∗ is a point, wespeak: g∗ is the pole of g.P =pointg = straight linec =circle_PQ =distanceĝh =angleP ⊥ gQ ⊥ h_PQ = ĝhFigure 2.1. The sphere228 H.Dirnbo¨ck3. Spherical conics cA cone Φ of 2nd degree has 3 midpoints, which we put into:U = x− axis ∩KV = y − axis ∩K}are on the “equator” a¨W = z − axis∩K = the “north pole” and the “south pole”.Now Φ can be written asΦ · · · x2tan2 a+y2tan2 b− z2 = 0 (3.1)0 < b ≤ a < pi2, (3.2)where a and b are the half lengths of the axes of Φ in the bundle.Now the spherical conic c isc = K ∩Φ, c = (2.1) ∩ (3.1), let (3.2) (3.3)a and b are the half spherical lengths of the axes of c.Eliminating x or y or z from (3.3) you can get the front view, the side view, thetop view, the formulas for Figure 3.1. Sometimes it is clever to work in the planesz = 1 or z = −1:cz = Φ ∩ (z = ±1) · · · x2tan2 a + y2tan2 b = 1AND z = ±1 (3.4)cz · · · x = cos T tan ay = sinT tan bz = ±1 (3.5)Now we havec · · · R =√(cos T tan a)2 + (sinT tan b)2 + 1x = 1Rcos T tan ay = 1R sinT tan bz = ± 1R (3.6)Absolute polarity on the sphere 229230 H.Dirnbo¨ckFigure 3.3. c and cz, a = pi3 , b =pi6The figure also shows for the next chapter: k = circle of curvature, eva = evolutecurve of c, evaz = [eva O] ∩ (z = −1).Absolute polarity on the sphere 2314. Spherical evolute curve evaLetm = sin2 a−sin2 bsin a cosan = sin2 b−sin2 asin b cos b (4.1)Then we get the evolutic cone of c as[O eva] ...( xm) 23+( yn)23= z23 (4.2)evaz ...(xm) 23 +(yn) 23 = 1AND z = ±1 (4.3)evaz · · · x = m cos3 Ty = n sin3 Tz = ±1 (4.4)eva · · · R =√m2 cos6 T + n2 sin6 T + 1x = mR cos3 Ty = nR sin3 Tz = ± 1R (4.5)(4.1) shows that we have the chance to find a nontrivial equilateral spherical evolutecurve, the condition isa + b =pi2. (4.6)An equilateral evolute curve of an ellipsis cannot appear in EUKLIDean geometry!The case (4.6) and all facts of Section 4 are shown in Figure 3.3.232 H.Dirnbo¨ck5. Absolute polarization of the conic cFigure 5.1. Conic c and the absolute polar conic c∗c is to be polarized. By P the line element P with t is transformed into the lineelement P ∗ with t∗, c is transformed into c∗.c has a and b due to 3.,c∗ has a∗ =pi2− a, b∗ = pi2− b (5.1)as new spherical half lengths of the axes.The distance_Pt∗ = pi2 and n = [P t∗] rolls on eva, while P and t∗ draw thepaths c and c∗, while t and P ∗ envelop c and c∗; c and c∗ are equidistant curves. cand c∗ have the same evolute curve eva.Absolute polarity on the sphere 233Figure 5.2. As in Figure 5.1, we can additionally see the octant P t∗ n∗ and thepath eva∗.We can absolutely polarize eva and we get eva∗. The points P t∗ n∗ form aspherical octant: the three angles have pi2 , the three edges havepi2 .When n is rolling on eva, n∗ draws the path eva∗.In the language of spherical kinematic geometry: eva is the fixed polhode, n is themoving polhode of this spherical motion.234 H.Dirnbo¨ck6. Focus-properties of the spherical conic cA didactical trick: we begin with c∗. The spherical conic c∗ has the well-known fociF ∗1 and F ∗2 . c∗ has the well-known properties_F ∗1 t∗ = r∗1_F ∗2 t∗ = r∗2r∗1 + r∗2 = 2b∗ (6.1)t∗ is a point of c∗.The supplements of these distances arerH∗1 = pi − r∗1rH∗2 = pi − r∗2}(6.2)(6.1) looks like an ellipsis, but if we substitute one of the terms (6.2), it looks like ahyperbola; in fact, there exists only one kind of a spherical conic. The tangent lineP ∗ in t∗ bisects the angle between r∗1 and rH∗2 ;the half angle is denoted as σ (6.3)Now let us use Figure 6.1 and let us polarize these facts step by step!P ◦ c∗ = cP ◦ (point t∗ of c∗) = tangent line t of cP ◦ (tangent line P ∗ of c∗) = point P of cP ◦ F ∗1 = F1 = “focus line” of cP ◦ F ∗2 = F2 = “focus line” of c[F ∗1 t∗] = line v∗1[F ∗2 t∗] = line v∗2P◦ line v∗1 = point v1P◦ line v∗2 = point v2Absolute polarity on the sphere 235Figure 6.1. The well-known focus-properties of c∗, triangle of constant area for cAdditional in Figure 6.1:P ∈ c, P bisects the distance _v1v2 on t,distance_WF ∗1 = e∗,distance_WF ∗2 = e∗,P◦_WF ∗1 = angle e∗,P◦_WF ∗2 = angle e∗,spherical triangle v1v2U : the angle in U = pi − 2e∗ = constant,P◦_F ∗1 t∗ = angle r∗1,P◦_F ∗2 t∗ = angle r∗2,angle r∗1 + angle r∗2 + angle pi − 2e∗ = constant.236 H.Dirnbo¨ckThe spherical triangle v1v2U has a constant area (6.4).Further:In t∗ we have angles σ = σ. (6.3),Therefore, on t we have distancesσ = σ.P bisects the distance_v1v2 (6.5).Annotation: An EUKLIDean hyperbola xy = 1 has analogous properties as(6.4) and (6.5).7. LoxodromeFigure 7.1 loxodrome qAs in the figure a loxodrome q is defined as an isogonal trajectory of the merid-Absolute polarity on the sphere 237ians of a globe; let this angle be denoted as pi2 − θ,k = tan θ (7.1)k =dPdL cosP(7.2)we integrate and getln tan(P2+pi4)= k(L− LD) (7.3)For our task we always let LD = 0, then we havethe loxodrome 3 U...L = ln tan(P2 +pi4)k· · · q (7.4)The arclength is easy: s|P0 =1sin θ· P (7.5)In CARTESian coordinates we haveQ = ~X =cosLCh(kL)sin LCh(kL)Th (kL) · · · q (7.6)When derivating with respect to L, we write “ . ”,when derivating with respect to s, we write “ ′ ”.Useful are the following vectors:~X ′ =1√1+k2−Ch (kL) sin L−Sh (kL)k cosLCh (kL)1√1+k2Ch (kL)cosL−Sh (kL)k sinLCh(kL)1√1+k2kCh (kL) (7.7)~X ′′ =−Ch2(kL) cosL+Sh (kL)Ch (kL)k sin L−k2 cosL(1+k2)Ch (kL)−Ch2(kL) sinL−Sh (kL)Ch (kL)k cosL−k2 sin L(1+k2)Ch (kL)−k2Sh (kL)(1+k2)Ch (kL) (7.8)~X ′′e = unit vector of (7.8) (7.9)8. Spherical evolute curve es1,2 of the loxodrome q(7.7)× (7.9) is the unit vector orthogonal to (7.7) and orthogonal to (7.9), it is~be = ~X ′ × ~X ′′e =k sinL√k2+Ch2(kL)−k cosL√k2+Ch2(kL)Ch (kL)√k2+Ch2(kL) (8.1)238 H.Dirnbo¨ckalso it is the representation of the spherical center of curvature M s1,2; while L isvariable, it represents the evolute curve es1,2 of q. The + sign gives es1, the − signgives es2 in Figure 8.1.ψ = spherical radius of curvature1 of q,pi − ψ = spherical radius of curvature2 of q,cosψ =Sh (kL)√k2 +Ch2(kL)(8.2)Figure 8.1. Loxodrome q and spherical evolute curve es1,2Absolute polarity on the sphere 2399. Spherical tractrix fNow we polarize q. We will show that we get a spherical tractrix f = P ◦ q. Wepolarize F ∗, which is the spherical tangent line of q, then we get F ∈ f :F = Q× ~X ′ = (7.6)× (7.7),F =1√1 + k2Ch(kL)k sinL−Sh (kL) cosLCh (kL)−Ch(kL)k cosL−Sh (kL) sin LCh (kL)1Ch (kL) (9.1)Polarization of the elements and properties of q:P ◦ point Q = tangent line Q∗ of fP ◦ tangent line F ∗ = point F of fP ◦ meridian in Q = A¨P ◦ ( ̂angle meridian F ∗ = pi2− θ) = distance_A¨F = pi2− θ(9.2)Therefore F draws a tractrix f , while A¨ is running on the equator a¨; a¨ is anasymptotic line of f . See Figure 9.1, sometimes Figure7.1.Figure 9.1. Loxodrome q, evolute curve es1, tractrix f , path h240 H.Dirnbo¨ck10. Path hThe last step: we polarize es1,2. The result is the path h:h = P ◦ es1,2n = tangent line of es1,2 = [Q F ]H = P ◦ nH = Q× F = (7.6)× (9.1) = − ~X ′ (7.7)See Figure 9.1.11. Result from 7. to 10.Q F H form a spherical octant with three angles of pi2 and three edges ofpi2 . n =[Q F ] rolls along the evolute curve es1,2.Q draws the loxodrome q,F draws the tractrix f ,H draws the path h.es1,2 has two special evolvent curves q and f , they are equidistant curves from eachother, the distance is pi2. See Figure9.1.References[1] H.Dirnbo¨ck, Die Syntrepenz zweier Loxodromen auf der Kugel (Syntrepenceof two loxodromes on the sphere), paper for a lecture, Austrian GeometryCongress, April 27–May 2, 1992., Schloss Seggauberg, A-8430 Leibnitz.[2] R. Sigl, Ebene und spha¨rische Trigonometrie, Akademische Verlagsge-sellschaft, Frankfurt am Main, 1969.

Absolute polarity on the sphere; conics; loxodrome, tractrix

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