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Spirals on Surfaces of Revolution

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Spirals on Surfaces of Revolution Spirals on surfaces of revolution Cristian L˘azureanu Department of Mathematics, Politehnica University of Timi¸soara Piat¸a Victoriei nr. 2, Timi¸soara, 300006, Rom^ania [email protected] Abstract In this paper some spirals on surfaces of revolution and the corresponding helicoids are presented. Keywords: helix; spiral; surface of revolution; ruled surface; helicoid MSC 2010: 14H50, 53A04, 53A05 1 Introduction One of the most known spiral in space is the cylindrical helix (Figure 1). It is a a curve for which the tangent makes a constant angle with a fixed line (Oz-axis in our case) [1]. The parametric equations of the helix are given by: 8 < x = r cos t y = r sin t ; t 2 R; (1) : z = ct where r is the radius of the cylinder and c is a positive parameter such that 2πc is a constant giving the vertical separation of the helix's loops and controls the number of the rotations. Figure 1: A cylindrical helix Figure 2: A circular helicoid The circular helicoid (Figure 2) is a ruled surface having a cylindrical helix as its boundary [1]. In fact, the cylindrical helix is the curve of intersection between the cylinder and the helicoid. The circular helicoid bounded by the planes z = a and z = b is given in parametric form by: 8 x = u cos v < a b y = u sin v ; (u; v) 2 [0; r] × ; (2) c c : z = cv The aim of this work is to present similar curves on the other surfaces of revolution, the similar helicoids and their equations. Also, some planar curves which looks similar with known plane curves are obtained. 1 2 Spirals, helicoids and surfaces of revolution Let f be a positive continuous function. A surface of revolution generated by rotating the curve y = f(z); z 2 [a; b] around the Oz-axis has the equation [1]: x2 + y2 = f 2(z) (3) and a parametric representation can be given by 8 < x = f(t) cos s y = f(t) sin s ; (t; s) 2 [a; b] × [0; 2π): (4) : z = t As the intersection of the circular helicoid and cylinder is a cylindrical helix as intersection of the helicoid and another surface of revolution gives rise to a three-dimensional spiral. Equating x; y, respectively z from equations (2) and (4) one gets 8 < u cos v = f(t) cos s u sin v = f(t) sin s (5) : cv = t Imposing the condition f(t) 2 [0; r] it follows u = f(t), whence the equations of the three-dimensional spiral are given by: 8 t > x = f(t) cos > c <> t y = f(t) sin ; t 2 [a; b] (6) > c > :> z = t or equivalent 8 x = f (ct) cos t < a b y = f (ct) sin t ; t 2 ; : (7) c c : z = ct Spiral (6) and Oz-axis generate a ruled surface. Taking into account the vectorial equation of a ruled surface [1], the equations of a ruled surface having the aforementioned spiral as its boundary are given by : 8 v > x = uf(v) cos > c < v y = uf(v) sin ; (u; v) 2 [0; 1] × [a; b] (8) > c > : z = v or in equivalent form 8 x = uf (cv) cos v < a b y = uf (cv) sin v ; (u; v) 2 [0; 1] × ; : (9) c c : z = cv The projection of spiral (6) on xOy plane is given by 8 t <> x = f(t) cos c ; t 2 [a; b] (10) t :> y = f(t) sin c or in equivalent form x = f (ct) cos t a b ; t 2 ; : (11) y = f (ct) sin t c c Equation (11) can be written in polar coordinates as follows: a b ρ = f (cθ) ; θ 2 ; ; (12) c c 2 which sometimes represents the equation of a planar spiral [1]. This property allows us to consider some particular planar spirals [2] and then to obtain the corresponding three-dimensional spirals. Pappus' conical spiral The first considered planar spiral is the Archimedean spiral (Figure 3) described by the equation ρ = β + αθ; θ 2 R; α where α and β are real numbers. In this case f(z) = β + z, z 2 (red line in Figure 4) and the surface c R α 2 of revolution around Oz-axis (black line in Figure 4) is a cone: x2 + y2 = β + z . The equations of c spiral (7), known as Pappus' conical spiral [3] (Figure 4, just a half of the cone is shown), and of ruled surface (9), called conical helicoid (Figure 5) are respectively: 8 x = (β + αt) cos t 8 x = u(β + αv) cos v <> <> y = (β + αt) sin t ; t 2 R and y = u(β + αv) sin v ; (u; v) 2 [0; 1] × R: > > : z = ct : z = cv Figure 3: An Archimedean spiral Figure 4: A conical spiral Figure 5: A conical helicoid Three-dimensional Fermat's spiral Considering Fermat's spiral or the parabolic spiral (Figure 6) described by the equation p ρ = α θ; θ 2 [0; 1); r1 where α is a real number, the function f(z) = α z; z 2 [0; 1) generates a paraboloid of revolution: c α2 x2+y2 = z. The equations of paraboloidal spiral (three-dimensional parabolic spiral, three-dimensional c Fermat's spiral) (7) and paraboloidal helicoid (9) are given by: p p 8 x = α t cos t 8 x = αu v cos v <> p <> p y = α t sin t ; t 2 [0; 1) and y = αu v sin v ; (u; v) 2 [0; 1] × [0; 1) > > : z = ct : z = cv p Analogously we can consider the function f(z) = α b − z; z 2 [a; b] (red line in Figure 7). One obtain a paraboloid of revolution: x2 + y2 = α2(b − z). The equations of paraboloidal spiral (Figure 7) 3 and paraboloidal helicoid (Figure 8) are given by: 8 p t 8 p v > x = α b − t cos x = αu b − v cos > c > c <> <> p t p v y = α b − t sin ; t 2 [a; b] and y = αu b − v sin ; (u; v) 2 [0; 1] × [a; b]: > c > c > > :> z = t : z = v Figure 6: A Fermat's spiral Figure 7: A paraboloidal spiral Figure 8: A paraboloidal helicoid Three-dimensional hyperbolic spiral Next planar spiral is the hyperbolic spiral (Figure 9) given by the equation α ρ = ; θ 2 (0; 1); θ where α is a real numbers. Figure 9: A hyperbolic spiral Figure 10: 3D hyperbolic spiral Figure 11: A hyperbolic helicoid cα In this case f(z) = , z 2 (0; 1) (red line in Figure 10) generates the surface of revolution around z c2α2 Oz-axis given by x2 +y2 = (Figure 10, a rectangular hyperbola with horizontal/vertical asymptotes z2 rotates around its vertical asymptote). The three-dimensional hyperbolic spiral (Figure 10) and the hyperbolic helicoid (Figure 11) are given by the following equations: 8 α 8 u > x = cos t > x = α cos v > t > v < α < u y = sin t ; t 2 (0; 1) and y = α sin v ; (u; v) 2 [0; 1] × (0; 1): > t > v > > : z = ct : z = cv 4 Three-dimensional lituus spiral Let us consider the lituus spiral (Figure 12) given by the equation α ρ = p ; θ 2 (0; 1); θ p α c where α is a real numbers. In this case f(z) = p , z 2 (0; 1) (red line in Figure 13) generates the z cα2 surface of revolution around Oz-axis given by x2 + y2 = (Figure 13). The three-dimensional lituus z spiral (Figure 13) and the lituus helicoid (Figure 14) are given by the following equations: 8 α 8 u x = p cos t x = αp cos v > > > t > v < α < u y = p sin t ; t 2 (0; 1) and y = αp sin v ; (u; v) 2 [0; 1] × (0; 1): > t > v > > :> z = ct :> z = cv Figure 12: A lituus spiral Figure 13: 3D lituus spiral Figure 14: A lituus helicoid Three-dimensional logarithmic spiral The last considered planar spiral is the logarithmic spiral (Figure 15) described by the equation βθ ρ = αe ; θ 2 R; where α and β are real numbers. Figure 15: A logarithmic spiral Figure 16: 3D logarithmic spiral Figure 17: A logarithmic helicoid β 1 z In this case f(z) = αe c ; z 2 R (red line in Figure 16) and the surface of revolution around Oz-axis 2 2 2 2β z has the equation x + y = α e c . The equations of three-dimensional logarithmic spiral (Figure 16) and logarithmic helicoid (Figure 17) are given by: 8 x = αeβt cos t 8 x = αueβv cos v <> <> y = αeβt sin t ; t 2 R and y = αueβv sin v ; (u; v) 2 [0; 1] × R: > > : z = ct : z = cv 5 In the following we consider other important surfaces of revolution and we present the corresponding three-dimensional spirals (6) and ruled surfaces (8). Spherical helicoid p The sphere of radius r is generated by the function f :[−r; r] ! [0; r]; f(z) = r2 − z2 (red line in Figure 18). Then the spherical helical curve (Figure 18) and the spherical helicoid (Figure 20) are given by: 8 p t 8 p v > x = r2 − t2 cos x = u r2 − v2 cos > c > c <> <> p t p v y = r2 − t2 sin ; t 2 [−r; r] and y = u r2 − v2 sin ; (u; v) 2 [0; 1] × [−r; r]: > c > c > > :> z = t : z = v Figure 18: A spherical helical Figure 19: x − y projection of Figure 20: A spherical helicoid curve (c = 1) the spiral (c = 1) (c = 1) Figure 21: A spherical helical Figure 22: x − y projection of Figure 23: A spherical 1 1 1 curve (c = 2 ) the spiral (c = 2 ) helicoid(c = 2 ) Let us observe that if we take different values of c, then we obtain different shapes of the spherical helical curve (Figures 18,21,24).
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