The Extended Oloid and Its Contacting Quadrics

Journal for Geometry and Graphics Volume 19 (2015), No. 2, 161–177.

Uwe B¨asel1, Hans Dirnb¨ock2

1Faculty of Mechanical and Energy Engineering, HTWK Leipzig, Karl-Liebknecht-Straße 134, 04277 Leipzig, Germany email: [email protected]

2Nussberg 22, 9062 Moosburg, Austria

Abstract. The oloid is the convex hull of two circles with equal radius in perpendicular planes so that the center of each circle lies on the other circle. It is part of a developable surface which we call extended oloid. We determine the tangential system of all contacting quadrics λ of the extended oloid where λ is the system parameter. From this result weQ conclude parameter equationsO of the touching curve λ between and λ, and of the edge of regression of . Properties of the curvesC are investigated,O Q including the case that λ . TheO self-polar tetra- Cλ → ∞ hedron of the tangential system is obtained. The common generating lines of Qλ and any ruled surface λ are determined. Furthermore, we derive the curves whichO are the images of Qwhen is developed onto the plane. Cλ O Key Words: Oloid, developable surface, torse, tangential system of quadrics, touching curve, edge of regression, self-polar tetrahedron, ruled surface MSC 2010: 53A05, 51N20, 51N15

1. Introduction

The oloid was discovered by Paul Schatz in 1929. It is the convex hull of two circles with equal radius r in perpendicular planes so that the center of each circle lies on the other circle. The oloid has the remarkable properties that its surface area is equal to 4πr2 and the two circles terminate on each generator a segment of the same length r√3. The boundary of the oloid is part of a developable surface [3, 9]. In the following this developable surface is called extended oloid. According to [3, pp. 105- 106], in an appropriate cartesian coordinate frame the circles with r = 1 can be deﬁned by

k := (x, y, z) R3 x2 +(y +1/2)2 =1 z =0 , A ∈ ∧ R3 2 2 kB := (x, y, z) (y 1/2) + z =1 x =0. ∈ − ∧ ISSN 1433-8157/$ 2.50 c 2015 Heldermann Verlag 162 U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics

Figure 1: The extended oloid , the circles kA, kB (dashed lines), and the edge of regression (solidO lines) in the box 2.5 x, y, z 2.5 R − ≤ ≤

In this case we denote the extended oloid by (see Figure 1). O Now we introduce homogeneous coordinates x0, x1, x2, x3 with x = x1/x0, y = x2/x0, z = x3/x0. Then the real projective space — as a set of points P = [x0, x1, x2, x3] — is given by P (R)= [x , x , x , x ] (x , x , x , x ) R4 0 , 3 0 1 2 3 | 0 1 2 3 ∈ \{ } R where [x0, x1, x2, x3] = [y0,y1,y2,y3] if there exist a µ 0 such that xj = µyj for j =0, 1, 2, 3. If necessary, complex coordinates will be used∈ instead\{ } of the real ones. For the description of the corresponding projective circles A and B to kA and kB, respectively, we write K K

= [x , x , x , x ] P (R) 3x2 4x x 4x2 4x2 =0 x =0 , KA 0 1 2 3 ∈ 3 0 − 0 2 − 1 − 2 ∧ 3 P R 2 2 2 B = [x0, x1, x2, x3] 3( ) 3x0 +4x0x2 4x2 4x3 =0 x1 =0 . K ∈ − − ∧ 2. Contacting quadrics

Lemma 1. The sets of the tangent planes u¯ = [u ,...,u ] of the circles and satisfy 0 3 KA KB the respective equations

F (¯u)=4u2 4u u 4u2 3u2 =0 and F (¯u)=4u2 +4u u 3u2 4u2 =0 . 0 0 − 0 2 − 1 − 2 1 0 0 2 − 2 − 3 Proof. Instead of following [4, pp. 41–42] or [5, pp. 160, 164–165], we conclude as follows: The plane [u ,u ,u ,u ] is tangent to the circle if and only if its intersection with the plane 0 1 2 3 KA U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics 163 x = 0 satisfies the tangential equation of . Therefore, the coefficient matrix of F (¯u) is 3 KA 0 obtained — up to a real factor = 0 — by inverting the symmetric 3 3 coefficient matrix of × the equation of A, K 1 3 0 2 − 4 0 2 − 1 − 0 4 0 = 0 4 0 ,  −  16 −  2 0 4 2 0 3 − − − − hence F (¯u)=4u2 4u u 4u2 3u2. Analogously, one finds F (¯u) from the coefficient 0 0 − 0 2 − 1 − 2 1 matrix of . KB

Remark 1. If the coordinates u0,...,u3 are considered as homogenized cartesian coordinates x0,...,x3 (cf. [1, p. 139]), then the equations in Lemma 1 deﬁne elliptic cylinders. Their inhomogeneous equations are x2 (y +2/3)2 (y 2/3)2 z2 2 + = 1 and − + 2 =1 . 2/√3 (4/3)2 (4/3)2 2/√3

This means that the extended oloid, which is the connecting torse of the two circles kA and kB, is dual to the curve of intersection of two elliptic cylinders. The equations in Lemma 1 were already given in [3, p. 115]. Theorem 2. All regular quadrics which contact the extended oloid along a curve are given by O = (x, y, z) R3 f (x, y, z)=0 , Qλ ∈ λ where x2 (y λ + 1/2) 2 z2 + − + 1 if λ R 0, 1 , f (x, y, z)= 1 λ 1 λ + λ2 λ − ∈ \{ } λ  − − x2 z2 +2y if λ = .  − ∞ Proof. Because of the homogeneity of the coordinates we may write the tangential equations of all contacting quadrics in question as linear combinations of F0(¯u) and F1(¯u) (see Lemma 1),

Fλ(¯u) := (1 λ) F0(¯u)+ λ F1(¯u) or F (¯u) := F0(¯u)+ F1(¯u). − ∞ −

The formula for F (¯u) is the limit of Fλ(¯u)/λ for λ . The homogeneous point equation ∞ f˜ (x , x , x , x ) = 0 of any regular is obtained —→∞ up to a real factor = 0 — by inverting λ 0 1 2 3 Qλ the symmetric coefficent matrix of Fλ(¯u), 3 1 2λ 1 2 0 − 2 0 4 0 2(2λ 1) 0 − −4(1 λ + λ ) 2(1 λ + λ ) − − 1 − 0 4(λ 1) 0 0 1  0 0 0   −  = 1 λ , 2(2λ 1) 0 3 0 −4  1 2λ − 1  − −  − 2 0 2 0   0 0 0 4λ   2(1 λ + λ ) 1 λ + λ   −   − − 1     0 0 0   λ    hence 3x2 x2 x2 x2 (1 2λ)x x f˜ (x , x , x , x )= 0 + 1 + 2 + 3 + − 0 2 λ 0 1 2 3 −4(1 λ + λ2) 1 λ 1 λ + λ2 λ 1 λ + λ2 − − − − 2 x2 x + 1 λ x x2 = 1 + 2 2 − 0 + 3 x2 , (1) 1 λ 1 λ + λ2 λ − 0 − − 164 U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics and ˜ ˜ 2 2 f (x0, x1, x2, x3) = lim λfλ(x0, x1, x2, x3)= (x1 +2x0x2 x3) . ∞ λ →∞ − − We write the equations in inhomogeneous coordinates as x2 (y λ +1/2)2 z2 f (x, y, z)= f˜ (1, x, y, z)= + − + 1 , λ λ 1 λ 1 λ + λ2 λ − − − f (x, y, z)= f˜ (1, x, y, z)= x2 z2 +2y . ∞ − ∞ − Corollary 3. a) Every quadric λ and the extended oloid are symmetric with respect to the planes x =0 and z =0. Q O b) The axial symmetries (x, y, z) (z, y, x) and (x, y, z) ( z, y, x) exchange the → − → − − − quadrics λ and 1 λ. Q Q − Proof. a) Since f ( x, y, z)= f (x, y, z) and f (x, y, z)= f (x, y, z), every quadric is sym- λ − λ λ − λ metric with respect to x = 0 and z = 0. The symmetry of with respect to these planes follows immediately. O b) From x2 (y (1 λ)+1/2)2 z2 f1 λ(x, y, z)= + − − + 1 − 1 (1 λ) 1 (1 λ)+(1 λ)2 1 λ − − − − − − − z2 ( y λ +1/2)2 x2 = + − − + 1= f (z, y, x) 1 λ 1 λ + λ2 λ − λ − − − and fλ(x, y, z)= f1 λ(z, y, x) follows that the isometry (x, y, z) (z, y, x) exchanges the − − → − quadrics λ and 1 λ, while all points (t, 0, t) with t R remain fixed. The same holds for Q Q − ∈ (x, y, z) ( z, y, x), which fixes all points (t, 0, t). → − − − −

Table 1: Types of contacting quadrics Qλ Parameter Equation of Type of Qλ Qλ x2 (y λ + 1/2)2 z2 λ< 0 + − = 1 Hyperboloid of one sheet a2 b2 − c2 λ =0 x2 +(y +1/2)2 =1 z =0 Circle k ∧ A x2 (y λ + 1/2)2 z2 0 <λ< 1 + − + = 1 Ellipsoid a2 b2 c2 λ =1 (y 1/2)2 + z2 =1 x =0 Circle k − ∧ B (y λ + 1/2)2 z2 x2 λ> 1 − + = 1 Hyperboloid of one sheet b2 c2 − a2 λ = x2 z2 +2y =0 Hyperbolic paraboloid ∞ −

Table 1 shows the classification of the quadrics λ, λ R , in the Euclidean space. In this table the abbreviations a2 := 1 λ , b2 := 1Q λ +∈λ2 ∪{∞}> 0 and c2 := λ are used. | − | − | | A point of the circle kA is given by 1 A = α (t),α (t),α (t) with α (t) = sin t, α (t)= cos t, α (t)=0 . 1 2 3 1 2 −2 − 3 U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics 165

There are two points B1, B2 of the circle kB such that the connecting lines AB1 and AB2 are generators of : O B = β (t), β (t), β (t) , B = β (t), β (t), β (t) , 1 1 2 3 2 1 2 − 3 where 1 cos t √1 + 2 cos t β (t)=0 , β (t)= , β (t)= 1 2 2 − 1 + cos t 3 1 + cos t

(see [3, pp. 106–107]). Hence, for ﬁxed t [ 2π/3, 2π/3], a parametrization of the line AB1 is ∈ − ω (m, t) := (1 m)α (t)+ mβ (t), i =1, 2, 3, with m R . i − i i ∈ We obtain ω (m, t)= (1 m) sin t, 1 − 2(m 1) cos2 t + (2m 3) cos t +2m 1 ω2(m, t)= − − − ,  2(1 + cos t)  (2)  m √1 + 2 cos t  ω (m, t)= . 3 1 + cos t   Consequently, 

1) x = ω1(m, t), y = ω2(m, t), z = ω3(m, t) , t [ 2π/3, 2π/3], m R , 2) x = ω (m, t), y = ω (m, t), z = ω (m, t) , ∈ − ∈ 1 2 − 3 is a parametrization of with the generators as m-lines. The restriction of m to the interval O [0, 1] yields the oloid in the narrow sense, which is the convex hull of kA and kB.

Corollary 4. For a ﬁxed value of λ R, we can parametrize the touching curve λ between and as ∈ C O Qλ

γ1(λ, t) if t [ 2π/3, 2π/3] , γ(λ, ):[ 2π/3, 2π] R3 , t γ(λ, t)= ∈ − − → → γ2(λ, t) if t (2π/3, 2π] , ∈ where

γ1(λ, t)= κ1(λ, t), κ2(λ, t), κ3(λ, t) ,

γ2(λ, t)= κ1(λ, 4π/3 t), κ2(λ, 4π/ 3 t), κ3(λ, 4π/3 t) , − − − − and (1 λ) sin t 2λ 1+(λ 2) cos t λ √1 + 2 cos t κ (λ, t)= − , κ (λ, t)= − − , κ (λ, t)= . 1 1+ λ cos t 2 2(1 + λ cos t) 3 1+ λ cos t

Proof. For a ﬁxed t the generator

:= (x, y, z) R3 x = ω (m, t), y = ω (m, t), z = ω (m, t); m R Lt ∈ 1 2 3 ∈ of is tangent to for one value m of m. As a double root of the equation O Qλ fλ(ω1(m, t),ω2(m, t),ω3(m, t))=0 166 U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics we find λ(1 + cos t) m = ψ(λ, t) := , 1+ λ cos t and consequently (1 λ) sin t 2λ 1+(λ 2) cos t ω (ψ(λ, t), t)= − , ω (ψ(λ, t), t)= − − , 1 1+ λ cos t 2 2(1 + λ cos t) λ √1 + 2 cos t ω (ψ(λ, t), t)= . 3 1+ λ cos t

We put κj(λ, t) := ωj(ψ(λ, t), t) for j =1, 2, 3. This yields

γ1(λ, t) := κ1(λ, t), κ2(λ, t), κ3(λ, t) as the contact point between t and λ for all lines t with t [ 2π/3, 2π/3]. Due to the symmetry of with respect toL the planeQ z = 0, we haveL ∈ − O 4π 4π 4π γ (λ, t) := κ λ, t , κ λ, t , κ λ, t 2 1 3 − 2 3 − − 3 3 − if t (2π/3, 2π]. Obviously, ∈ γ (λ, 2π/3) = γ (λ, 2π/3) and γ (λ, 2π)= γ (λ, 2π/3) 2 1 2 1 − for every λ R. ∈ Examples with contacting quadric and touching curves are shown in the Figures 2 and 3. Remark 2. In the special case λ =1/2 one gets the equations of the central inscribed ellipsoid, as mentioned in [3, p. 115, Eq. (27)].

3. Properties of the touching curves Cλ From Corollary 3 follows that all touching curves λ are symmetric with respect to the planes x = 0 and z = 0. We denote by X , X the intersectionC points of with the plane x = 0, 1 2 Cλ and by Z1, Z2 those with the plane z = 0, and ﬁnd

3(1 λ) √3 λ 2π √3 (1 λ) 3λ X (λ) = γ(λ, 0) = 0, − , , Z (λ)= γ λ, = − , , 0 , 1 −2(1 + λ) 1+ λ 1 3 2 λ 2(2 λ) − −  4π 3(1 λ) √3 λ √3 (1 λ) 3λ X (λ) = γ λ, = 0, − , ,Z (λ)= γ(λ, 2π) = − , , 0 . 2 3 2(1 + λ) 1+ λ 2 2 λ 2(2 λ)  − − − − − (3)  One easily finds a parametrization of the tangent to at the point γ(λ, t):  Tλ Cλ (x, y, z) R3 x = τ (λ,t,µ), y = τ (λ,t,µ), ∈ 1 2 z = τ (λ,t,µ); µ R if t [ 2π/3, 2π/3],  3 ∈ } ∈ −  λ(t)=   (4) T  (x, y, z) R3 x = τ (λ, 4π t, µ), y = τ (λ, 4π t, µ),   ∈ 1 3 − 2 3 −  z = τ (λ, 4π t, µ); µ R if t (2π/3, 2π], 3 3  − − ∈ ∈    with   dκ (λ, t) τ (λ,t,µ)= κ (λ, t)+ µ κ˙ (λ, t) , κ˙ = j for j =1, 2, 3 , j j j j dt U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics 167

Figure 2: Touching curves λ for λ =0, 0.1, 0.2,..., 0.9, 1, and the ellipsoid 0.3 in the box 1.5 Cx, y, z 1.5; the dashed line belongs to . Q − ≤ ≤ C0.3 where (1 λ)(λ + cos t) (1 λ + λ2) sin t κ˙ (λ, t)= − , κ˙ (λ, t)= − , 1 (1 + λ cos t)2 2 (1 + λ cos t)2 λ[λ(1 + cos t) 1] sin t κ˙ 3(λ, t)= − . (1 + λ cos t)2 √1 + 2 cos t

Now we consider the function 1 + λ cos t in the denominators of κ1, κ2, κ3. It vanishes in the interval [ 2π/3, 2π/3] for t = arccos( 1/λ) if λ R ( 1, 2). (For λ = 1, we have t = 0; and for λ−= 2, t = 2π/3.) 1+±λ cos t has− no zeros in∈ [ \2π/−3, 2π/3] if λ (− 1, 2). Therefore, κ , κ , κ are continuous± functions if λ ( 1, 2); they− are not continuous∈ if−λ R ( 1, 2). 1 2 3 ∈ − ∈ \ − So we have to distinguish the following cases: Case 1, 1 <λ< 2 : Since κ , κ , κ are continuous functions, and − 1 2 3 γ (λ, 2π/3) = Z (λ)= γ (λ, 2π/3), γ (λ, 2π)= Z (λ)= γ (λ, 2π/3), 2 1 1 2 2 1 − the curve is closed (see Figure 2). Cλ Case 2, λ R [ 1, 2] : The parametrization γ(λ, t) of has poles for ∈ \ − Cλ 1 1 4π 1 4π 1 t = arccos , t = arccos , t = arccos , t = + arccos . 1 − −λ 2 −λ 3 3 − −λ 4 3 −λ Therefore, λ consists of four branches. We calculate the asymptotes of λ at the poles. For t [ 2Cπ/3, 2π/3] the tangent (t) intersects the plane x = 0 at theC point with the ∈ − Tλ 168 U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics

Figure 3: Extended oloid (left) and hyperboloid (right) together with the touching O Q4 curve 4 (solid lines) and the four common generators j(4), j =1, 2, 3, 4, (dashed lines) withinC the box 4 x, y, z 4. G − ≤ ≤ coordinates

κ˙ 2(λ, t) λ 2+(2λ 1) cos t y = κ2(λ, t) κ1(λ, t)= − − , − κ˙ 1(λ, t) 2(λ + cos t) 2 κ˙ 3(λ, t) λ(1 + cos t + cos t) z = κ3(λ, t) κ1(λ, t)= , − κ˙ 1(λ, t) (λ + cos t) √1 + 2 cos t and z =0 at κ˙ (λ, t) (λ 1)(1 + cos t + cos2 t) x = κ (λ, t) 1 κ (λ, t)= − , 1 − κ˙ (λ, t) 3 [λ(1 + cos t) 1] sin t 3 − κ˙ (λ, t) 1+ λ + (2 λ) cos t y = κ (λ, t) 2 κ (λ, t)= − . 2 − κ˙ (λ, t) 3 − 2[λ(1 + cos t) 1] 3 − For t = t one ﬁnds that the asymptote (t ) intersects the plane z = 0 at the point 1 Tλ 1 1 λ + λ2 1 2 2λ λ2 (x ,y , z )= − 1 , − − , 0 , 1 1 1 2+ λ λ2 − λ2 2λ(λ 2) − − and the plane x =0 at

1 4λ + λ2 1 λ + λ2 λ (˜x , y˜ , z˜ )= 0 , − , − . 1 1 1 2(λ2 1) λ2 1 λ 2 − − − Hence, a parametrization (λ) of the asymptote (t ) is A1 Tλ 1 (λ)= (x, y, z) R3 x =τ ˜ (λ, ν), y =τ ˜ (λ, ν), z =τ ˜ (λ, ν); ν R A1 { ∈ 1 2 3 ∈ } U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics 169 with

1 λ + λ2 1 τ˜ (λ, ν) := x + ν (˜x x )= − 1 (1 ν) , 1 1 1 − 1 2+ λ λ2 − λ2 − − 2 2λ λ2 (1 λ + λ2)2 τ˜ (λ, ν) := y + ν (˜y y )= − − + ν − , 2 1 1 − 1 2λ(λ 2) λ(λ 2)(λ2 1) − − − 1 λ + λ2 λ τ˜ (λ, ν) := z + ν (˜z z )= ν − . 3 1 1 − 1 λ2 1 λ 2 − − In order to abbreviate the notation, we set

(λ)= (˜τ (λ, ν), τ˜ (λ, ν), τ˜ (λ, ν)) ν R . A1 { 1 2 3 | ∈ } The remaining asymptotes are

t = t : (λ)= ( τ˜ (λ, ν), τ˜ (λ, ν), τ˜ (λ, ν)) ν R , 2 A2 { − 1 2 3 | ∈ } t = t : (λ)= ( τ˜ (λ, ν), τ˜ (λ, ν), τ˜ (λ, ν)) ν R , 3 A3 { − 1 2 − 3 | ∈ } t = t : (λ)= (τ ˜ (λ, ν), τ˜ (λ, ν), τ˜ (λ, ν)) ν R . 4 A4 { 1 2 − 3 | ∈ }

Figure 4 shows as an example two projections of the touching curve 1.4 (thick lines). The − projection onto the plane x = 0 (see left-hand picture) is part of a hyperbolaC with the center (y = 71/238 0.298, z = 0). ≈ Case 3, λ = : For γ∗(t) := limλ γ(λ, t) one easily ﬁnds ∞ →∞ γ (t) if t [ 2π/3, 2π/3] , 3 1∗ γ∗ :[ 2π/3, 2π] R , t γ∗(t)= ∈ − − → → γ2∗(t) if t (2π/3, 2π] , ∈ with

4π 4π 4π γ∗(t)=(κ∗(t), κ∗(t), κ∗(t)) , γ∗(t)= κ∗ t , κ∗ t , κ∗ t , 1 1 2 3 2 1 3 − 2 3 − − 3 3 −

Figure 4: Projections of 1.4 and its asymptotes k( 1.4), k =1, 2, 3, 4, onto the planes C− A − x = 0 and y = 0, within the box 12 x, y, z 12. − ≤ ≤ 170 U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics where 1 1 κ1∗(t) = limλ κ1(λ, t)= tan t, κ2∗(t) = limλ κ2(λ, t)= + , →∞ − →∞ 2 cos t √1 + 2 cos t κ3∗(t) = limλ κ3(λ, t)= . →∞ cos t

The parametrization γ∗(t) of has poles for C∞ π π 4π π 5π 4π π 11π t = , t = , t = = , t = + = , 1 − 2 2 2 3 3 − 2 6 4 3 2 6 and therefore, consists of four branches (see Figure 5). C∞

Figure 5: Projections of and its asymptotes k( ), k =1, 2, 3, 4, onto the planes ∞ x = 0 andC y = 0 within the boxA ∞5 x, y, z 5. − ≤ ≤ For the asymptotes of we ﬁnd C∞ lim 1(λ)= (ν 1, ν 1/2, ν) ν R , lim 2(λ)= (1 ν, ν 1/2, ν) ν R , λ λ →∞ A { − − | ∈ } →∞ A { − − | ∈ } lim 3(λ)= (1 ν, ν 1/2, ν) ν R , lim 4(λ)= (ν 1, ν 1/2, ν) ν R . λ λ →∞ A { − − − | ∈ } →∞ A { − − − | ∈ } By virtue of (3), the intersection points with the planes of symmetry are

lim X1(λ)= 0 , 3/2 , √3 , lim Z1(λ)= √3 , 3/2 , 0 , λ λ − →∞ →∞ lim X2(λ)= 0 , 3/2 , √3 , lim Z2(λ)= √3 , 3/2 , 0 . λ − λ − − →∞ →∞ From the parametrization γ∗(t) of we have C∞ 1 cos2 t 1 1 1 + 2 cos t x2 = tan2 t = − , y = + , z2 = . cos2 t 2 cos t cos2 t After the elimination of cos t we find the following algebraic equations of the projections of onto the planes x = 0, y = 0, and z = 0: C∞ x =0: (y +1/2)2 z2 =1 , (5) − y =0: (x2 z2)2 2(x2 + z2)=3 , (6) − − z =0: (y 1/2)2 x2 =1 . (7) − − U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics 171 The Eqs. (5) and (7) define hyperbolas with the respective centers (0, 1/2, 0) and (0, 1/2, 0). − The actual projection of onto the plane x = 0 (plotted with thick lines) is part of the ∞ hyperbola (5). C Case 4, λ 1, 2 : consists of two branches. From (3) follows for λ = 1 ∈ {− } Cλ −

lim X1( 1 ε)=(0, , ) , lim X1( 1+ ε)=(0, , ) , ε 0 ε 0 → − − ∞ ∞ → − −∞ −∞ lim X2( 1 ε)=(0, , ) , lim X2( 1+ ε)=(0, , ) , ε 0 ε 0 → − − ∞ −∞ → − −∞ ∞ Z ( 1) = 2/√3, 1/2, 0 , Z ( 1) = 2/√3, 1/2, 0 , 1 − − 2 − − − and for λ = 2, X (2) = 0, 1/2, 2/√3 , X (2) = 0, 1/2, 2/√3 , 1 2 − lim Z1(2 ε)=( , , 0) , lim Z1(2 + ε)=( , , 0) , ε 0 ε 0 → − −∞ ∞ → ∞ −∞ lim Z2(2 ε)=( , , 0) , lim Z2(2 + ε)=( , , 0) . ε 0 ε 0 → − ∞ ∞ → −∞ −∞ 4. The edge of regression

In the following we denote by the edge of regression of the developable surface (see [8, R O pp. 119–125], and Figure 1). Theorem 5. For t [ 2π/3, 2π/3], the parametric equations of are given by ∈ − R sin t tan t 2 + 3 cos t 3 cos2 t 2 cos3 t x = g (t)= − , y = g (t)= − − , 1 3 2 6(1 + cos t) cos t (1+2cos t)3/2 z = g (t)= . ± 3 ±3(1 + cos t) cos t Proof. is the solution of the system of equations R ∂ ∂2 f (x, y, z) =0 , f ′ (x, y, z)= f (x, y, z)=0 , f ′′(x, y, z)= f (x, y, z)=0 λ λ ∂λ λ λ ∂λ2 λ with variable λ (see [2, pp. 523–524]). Since the touching curve is the intersection curve Cλ of the quadric and the quadric deﬁned by f ′ (x, y, z) = 0, the parametric functions κ , κ , Qλ λ 1 2 κ of are not only solutions of 3 Cλ

fλ(κ1(λ, t), κ2(λ, t), κ3(λ, t))=0, but also of fλ′ (κ1(λ, t), κ2(λ, t), κ3(λ, t))=0 . Furthermore, one ﬁnds x2 3y2(λ 1)λ y (2 3λ 3λ2 +2λ3) z2 9(λ 1)λ f ′′(x, y, z)= + − − − − + − . λ (1 λ)3 (1 λ + λ2)3 λ3 − (1 λ + λ2)3 − − − We solve the equation fλ′′(κ1(λ, t), κ2(λ, t), κ3(λ, t))= 0 for λ and obtain 1 + 2 cos t λ = φ(t) := . (2 + cos t) cos t

This yields x = g1(t), y = g2(t), z = g3(t), where gj(t) := κj φ(t), t for j = 1, 2, 3 . Due to the symmetry of with respect to the plane z = 0, we also have x = g (t), y = g (t), 1 2 z = g (t). O − 3 172 U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics Corollary 6. In terms of the parameter λ, the edge of regression is given by

= (r (λ), r (λ), r (λ)) λ R [0, 1] ( r (λ), r (λ), r (λ)) λ R [0, 1] R { 1 2 3 | ∈ \ }∪{ − 1 2 3 | ∈ \ } ( r (λ), r (λ), r (λ)) λ R [0, 1] (r (λ), r (λ), r (λ)) λ R [0, 1] , ∪{ − 1 2 − 3 | ∈ \ }∪{ 1 2 − 3 | ∈ \ } where

(λ 1)3 [2 λ +2ρ(λ)] λ2 +2λ 2+(λ 2) ρ(λ) r (λ)= − − , r (λ)= − − , 1 λ [2 λ + ρ(λ)] 2 2λ [2 λ + ρ(λ)] − − sgn(λ) λ [2 λ +2ρ(λ)] r (λ)= − , ρ(λ) = sgn(λ) √1 λ + λ2 . 3 2 λ + ρ(λ) − − Proof. We consider the function 1 + 2 cos t φ:[ 2π/3, 2π/3] R , t φ(t)= , − → → (2 + cos t) cos t used in the proof of Theorem 5. We denote by φ the restriction of φ to the interval ( 2π/3, 0), 1 − and by φ2 the restriction of φ to (0, 2π/3). One easily ﬁnds the respective inverse functions

1 1 λ + ρ(λ) 1 1 λ + ρ(λ) φ− (λ)= arccos − , φ− (λ) = arccos − 1 − λ 2 λ with ρ(λ) := sgn(λ) √1 λ + λ2 and λ R [0, 1], hence − ∈ \ 3 1 (λ 1) [2 λ +2ρ(λ)] r (λ) := κ λ,φ− (λ) = − − , 1 1 1 λ [2 λ + ρ(λ)] − 2 1 λ +2λ 2+(λ 2) ρ(λ) r (λ) := κ λ,φ− (λ) = − − , 2 2 1 2λ [2 λ + ρ(λ)] − 1 sgn(λ) λ [2 λ +2ρ(λ)] r (λ) := κ λ,φ− (λ) = − , 3 3 1 2 λ + ρ(λ) − and 1 1 1 κ λ,φ− (λ) = r (λ) , κ λ,φ− (λ) = r (λ) , κ λ,φ− (λ) = r (λ) . 1 2 − 1 2 2 2 3 2 3 This implies x = r1(λ), y = r2(λ), z= r3(λ), and, due to the symmetry of with respect to the plane z = 0,± also x = r (λ), y = r (λ), z = r (λ). O ± 1 2 − 3 5. The self-polar tetrahedron

Theorem 7. The faces of the common self-polar tetrahedron of the inscribed quadrics λ are formed by the planes x =0, x =0, 2x = √3 ix , and 2xP = √3 ix . Q 1 3 2 0 2 − 0 Proof. In a tangential system of quadrics there are four which degenerate to conics, and their planes are the faces of the common self-polar tetrahedron [6, pp. 205], [7, p. 254], or [1, p. 136]. λ degenerates if in (1) one of the denominators in f˜λ(x0, x1, x2, x3) vanishes. Q 2 For λ = 0 follows x3 = 0, and therefore the plane x3 = 0 delivers the first face of the tetrahedron . For λ = 1 we have x2 = 0, and therefore x = 0 is the second face of . P 1 1 P U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics 173 The two remaining faces correspond to the roots of the equation 1 λ + λ2 = 0, i.e., to − λ = 1 + i √3 /2 and λ = 1 i √3 /2. So we have 1 2 − x + 1 λ x 2 = x √3 ix 2 =0 , x + 1 λ x 2 = x + √3 ix 2 =0 . 2 2 − 1 0 2 − 2 0 2 2 − 2 0 2 2 0 Hence, the planes 2x = √3 ix and 2x = √3 ix are the remaining two faces of . 2 0 2 − 0 P

The degenerate quadrics and are the circles k and k , respectively. The conics of the Q0 Q1 A B degenerate quadrics 1 and 2 have the equations Qλ Qλ x2 z2 x2 z2 fλ1 (x, y, z)= 2 + 2 1, fλ2 (x, y, z)= 2 + 2 1. 1 √3 i 1 + √3 i − 1 + √3 i 1 √3 i − 2 − 2 2 2 2 2 2 − 2 6. Common generating lines of and Qλ O

Only two of the conic sections , , 1 , 2 are real. Therefore (see [6, p. 206]), the Q0 Q1 Qλ Qλ quadrics λ are divided into two sets; one of these sets consists of ruled surfaces, each having four commonQ generating lines with the developable surface . Clearly, these ruled surfaces O are the one-sheeted hyperboloids λ, λ R [0, 1], and the hyperbolic paraboloid . Q ∈ \ Q∞ Theorem 8. (i) For ﬁxed value of λ R [0, 1], the four common generating lines of the ∈ \ one-sheeted hyperboloid and the extended oloid are Qλ O (λ) = ( ω (m, λ), ω (m, λ), ω (m, λ)) m R , G1 { 1 2 3 | ∈ } (λ) = ( ω (m, λ), ω (m, λ), ω (m, λ)) m R , G2 { − 1 2 3 | ∈ } (λ) = ( ω (m, λ), ω (m, λ), ω (m, λ)) m R , G3 { − 1 2 − 3 | ∈ } (λ) = ( ω (m, λ), ω (m, λ), ω (m, λ)) m R , G4 { 1 2 − 3 | ∈ } with (λ 1)(2 λ +2ρ(λ)) ω (m, λ)=(1 m) − − , 1 − λ −| | λ 2 2ρ(λ) 2λ 1 ρ(λ) ω (m, λ)=(1 m) − − + m − − , 2 − 2λ 2(1 + ρ(λ)) sgn(λ) λ(2 λ +2ρ(λ)) ω3(m, λ)= m − , 1+ ρ(λ) where ρ(λ) = sgn(λ) √1 λ + λ2. − (ii) (λ), (λ), (λ), (λ) are the tangents to , and to , in the respective points G1 G2 G3 G4 R Cλ P (λ)=(r (λ),r (λ),r (λ)) , P (λ)=( r (λ),r (λ),r (λ)) , 1 1 2 3 2 − 1 2 3 P (λ)=( r (λ),r (λ), r (λ)) , P (λ)=(r (λ),r (λ), r (λ)) 3 − 1 2 − 3 4 1 2 − 3

with rj(λ) according to Corollary 6 for j =1, 2, 3 . 174 U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics Proof. (i) As already known, the parametric equations of the generating lines of are O x = ω (m, t)=1 m) sin t, 1 − 1 1 cos t y = ω (m, t)= (1 m) cos t + m , 2 − −2 − 2 − 1 + cos t m √1 + 2 cos t z = ω (m, t)= . ± 3 ± 1 + cos t

We substitute x = ω1(m, t), y = ω2(m, t), z = ω3(m, t) in fλ(x, y, z) = 0 and solve this equation for t. Thus, we ﬁnd 1 λ √1 λ + λ2 t = t˜(λ) with t˜(λ) = arccos − ± − . ± λ Since we are only interested in real solutions, we can write 1 λ + ρ(λ) t˜(λ) = arccos − (8) λ with the function ρ from Corollary 6. This implies (λ 1)(2 λ +2ρ(λ)) ω (m, t˜(λ)) = (1 m) − − , 1 ± ± − λ | | λ 2 2ρ(λ) 2λ 1 ρ(λ) ω (m, t˜(λ)) = (1 m) − − + m − − , 2 ± − 2λ 2(1 + ρ(λ)) sgn(λ) λ(2 λ +2ρ(λ)) ω3(m, t˜(λ)) = m − . ± 1+ ρ(λ)

We put ωj(m, λ) := ωj(m, t˜(λ)) for j = 1, 2, 3. This yields 1(λ) and 2(λ). Due to the symmetry of and with− respect to the plane z = 0, the linesG (λ) andG (λ) follow. Qλ O G3 G4 (ii) By virtue of (4), the tangent (t)= (τ (λ,t,µ), τ (λ,t,µ), τ (λ,t,µ)) µ R , t [ 2π/3, 2π/3] , Tλ { 1 2 3 | ∈ } ∈ − to is a generating line of for all values of t that are solutions of Cλ Qλ fλ (τ1(λ,t,µ), τ2(λ,t,µ), τ3(λ,t,µ))=0 . One finds t = t˜(λ) with t˜(λ) from (8). At first we consider only t = t˜(λ). Calculation shows that ± − κ (λ, t˜(λ)) = r (λ) , j =1, 2, 3. j − j (1) Hence, the tangent (λ) := λ( t˜(λ)) touches λ at the point P1(λ)=(r1(λ), r2(λ), T T − (1) C r3(λ)) . According to [2, p. 489], (λ) is equal to the tangent to at this point. The common∈ R generating lines are tangentsT to the edge of regression [6, p. 206].R Thus one finds 1+ ρ(λ) ω (m(λ),λ)= r (λ) for j =1, 2, 3, where m(λ) := . j j 2 λ + ρ(λ) − (1) It follows that (λ)= 1(λ). Due to symmetry with respect to the planes x = 0 and z = 0, withτ ˜ (λ,µ) :=T τ (λ, t˜(Gλ),µ) we also have j j − (2)(λ) := ( τ˜ (λ,µ), τ˜ (λ,µ), τ˜ (λ,µ)) µ R = (λ) , T { − 1 2 3 | ∈ } G2 (3)(λ) := ( τ˜ (λ,µ), τ˜ (λ,µ), τ˜ (λ,µ)) µ R = (λ) , T { − 1 2 − 3 | ∈ } G3 (4)(λ) := (τ ˜ (λ,µ), τ˜ (λ,µ), τ˜ (λ,µ)) µ R = (λ) . T { 1 2 − 3 | ∈ } G4 U. Bäsel, H. Dirnböck: The Extended Oloid and Its Contacting Quadrics 175 As an example, Figure 3 shows the common generating lines (4), (4), (4), (4) of G1 G2 G3 G4 and . Q4 O

7. The development of O Now we consider the development of the extended oloid onto its tangent plane E. For O this, we deﬁne a cartesian (ξ, η)-coordinate system in E as follows: Let E touch along the generating line O = ω (m, 0), ω (m, 0), ω (m, 0) m R L0 { 1 2 − 3 | ∈ } (see (2) and the proof of Corollary 4). Then 0 is the η-axis, and the line perpendicular to at the point m = 0 is the ξ-axis. L L0 Any curve is developed onto a plane curve ∗ E. A parametrization of ∗ by the arc length t ofC the ⊂ Odouble circular arc can be obtainedC ⊂ from the vector transformationC in C0 [3, p. 114, Theorem 4]. For the sake of brevity, we set in the following c = cos t and s = sin t; while denotes the integer part of ‘ ’. ⌊⌋ Theorem 9. The development of the touching curve into the plane E is the curve Cλ

∗ = (κ∗(λ, t), κ∗(λ, t)) t R Cλ { 1 2 | ∈ } with the parametrization

3 t 1 4π κ∗(λ, t) = sgn(t) | | + + sgn(h(t)) κ˜ (λ, h(t)) , κ∗(λ, t)=˜κ (λ, h(t)) , 1 4π 2 √ 1 2 2 3 3 where

2 √3 √2 c (1 2λ) s 2(1+2c) κ˜1(λ, t)= arccos + − | | , 9 √1+ c (1 + λc)√1+ c √3 2 4+7λ + (11λ 4)c κ˜ (λ, t)= ln + − , 2 9 1+ c 1+ λc 3 t 1 4π h(t)= t sgn(t) | | + . − 4π 2 3

Proof. After the substitution x = κ1(λ, t), y = κ2(λ, t), z = κ3(λ, t) (see Corollary 4) in the vector transformation [3, p. 114, Theorem 4], a straight-forwar− d calculation yields

2 √3 √2 c (1 2λ) s 2(1+2c) ξ = κ˜1(λ, t)= arccos + − , 9 √1+ c (1 + λc) √1+ c √3 2 4+7λ + (11λ 4)c η =κ ˜ (λ, t)= ln + − . 2 9 1+ c 1+ λc

κ˜2(λ, t) is valid for t [ 2π/3, 2π/3]. The periodic continuation ofκ ˜2(λ, t) gives κ2∗(λ, t), which is valid for t R∈. − ˜ ∈ κ˜1(λ, t) is valid only for t [0, 2π/3]. The restriction of κ1∗(λ, t) to t [ 2π/3, 2π/3] must be an odd function. Replacing∈ sin t by sin t in κ˜ (λ, t), we get the even∈ − functionκ ˜ (λ, t), | | 1 1 176 U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics

Figure 6: The curves ∗ for λ =0, 0.1, 0.2,..., 0.9, 1 Cλ t [ 2π/3, 2π/3]. Now, κ⋄(λ, t) := sgn(t) κ˜ (λ, t) is the required restriction of κ∗(λ, t). We ∈ − 1 1 1 get 4π κ⋄(λ, 2π/3) κ⋄(λ, 2π/3) = , 1 − 1 − 3 √3 and therefore, using the step function

3 t 1 4π sgn(t) | | + , 4π 2 √ 3 3 we have found κ∗(λ, t), valid for t R. 1 ∈

Examples of curves λ∗ with λ [0, 1] are shown in Figure 6. The curve ∗ and the development of the edgeC of regression∈ are shown in Figure 7. C∞ R

Figure 7: ∗ (thick), development of (thin), 0∗ and 1∗ (dashed) C∞ R C C U. B¨asel, H. Dirnb¨ock: The Extended Oloid and Its Contacting Quadrics 177 Acknowledgments

We want to thank the reviewers for their valuable and detailed suggestions that helped to improve this paper.

References

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Received April 7, 2015; ﬁnal form November 9, 2015