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The Extended Oloid and Its Contacting Quadrics

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The Extended Oloid and Its Contacting Quadrics Journal for Geometry and Graphics Volume 19 (2015), No. 2, 161–177. The Extended Oloid and Its Contacting Quadrics 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 perpen- dicular 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 sys- tem 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. B¨asel, H. Dirnb¨ock: 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. B¨asel, H. Dirnb¨ock: 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.
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