Epihypocurves and epihypocyclic surfaces with arbitrary base curve

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Introduction Geometry of epi- and hypocycloids formed by the point bind with the circle which is rolling inside or outside the motionless circle is given in various textbooks, references books, monographs [1-9]. If the moving circle is moving along another motionless circle with some constant slope, then the point bind with the moving circle forms the space curve. If the moving circle is rotating around the tangent at the point of contact of the circles, the point bind with moving circle is arising a circle normal the tangent line. With rotation at 180° the point of epicycloid goes to point of hypocycloid and if the moving circle is rolling around the motionless circle and rotating at 180° the epicycloid curve goes to the hypocycloid curve. The space curves formed with constant slope of moving circle can be called the space epihypocycloids. If the moving circle is rolling along the motionless circle with rotating around tangents, the cyclic surfaces are formed. The geometry of the epihypocycloids and epihypocycloidal cycle surfaces there were analyzed in [10; 11]. It is shown in this paper that the epihypocycloid cycle surfaces are the canal surfaces [3; 12-15]. This article discusses the geometry of the epihypocycloid and epihypocycloidal surfaces formed when rolling a circle along an arbitrary base curve and cyclic surfaces formed when rolling and rotating a circle. Geometry of the epihypocycloids of the common type Let us consider the directrix base line rн(u) along which the circle of radius a is rolling (Figure 1). The point d which is linked with the moving circle is arising the curve r(u). To obtain the equation of the curve it’s necessary to determine the length of the base line, covered by the moving circle from the initial point u0 to the contact point u of the motionless cycle: , где . The radius of the moving circle will be rotated relative the normal of the base curve at the angle . For many curves the integral of the length doesn’t have an analytical solution, but the integrals can be calculated by numerical methods, for example using MathCad software. The vector equation of the formed curve can be written as , (1) where - the vectors of the tangent and normal of the base curve. б · q(u) macosq 01 masinq d r(u) rн(u) a (1-mcosq) nн tн · · ma a d r(u) rн(u) u0 0 i j k nн tн · u1 q(u) d 01 01 a Figure 1. Rolling of the circle along base directrix If the there is rolling along the base curve at the angle v to the plane of the base curve, then the point d forms the space curve, the vector equation of which can be obtained as following: , (2) where β - binormal of the base curve. If the circle rolls from point u0 towards point u1 of the base curve and the radius of the rolling circle is determined as where , then the moving circle makes k complete turns of 2π. For plane base curve β = k is an ort of rectangular coordinate system. If angle v is 0, then the circle moves at concave side of the base curve, the hypocycloid is arised, if v = π, epicycloid is formed at the convex side of the base curve. If parameter v changes from 0 to 2π, then point d describes a circle of radius around the tangent to the base curve. If parameter v changes from 0 to π then point of the hypocycloid transforms into the point of the epicycloid. If the moving circle is rolling along a base curve and rotating v = (0 ÷ 2π) the cycle surface is forming at all points of the contacts with the base curve. . (3) The equations (2), (3) are correct for forming cycloid surfaces by rolling the circle around both plane and space curve. Epihypocycloids At the Figure 2 there is shown the figures of the epihypocycloids with the base ellipse, given by the equation , with parameters b = 2, с = 1 with point d on the moving circle (μ = 1) and with different radius of the moving circle and initial point u0 at the base curve. f k = 7 а = 0.220 d k = 4 а = 0.386 b k = 2 а = 0.771 u0 = 0 u0 = π/4 u0 = π/2 c k = 3 а = 0.514 e k = 5 а = 0.308 u0 = π/4 u0 = π/2 a u0 = 0 k = 1 а = 1.542 Figure 2. Epihypocycloids with the base ellipse b = 2, c = 1 formatting point is on the circle, μ = 1 e k = 5 а = 0.308 a k = 1 а = 1.542 u0 = π/2 u0 = π/4 u0 = 0 c k = 3 а = 0.514 b k = 2 а = 0.771 f k = 7 а = 0.220 d k = 4 а = 0.386 Figure 3. Epihypocycloids with the base ellipse b = 2, c = 1 formatting point is out of the circle, μ = 2 a k = 1 а = 1.542 u0 = 0 u0 = π/2 u0 = π/4 c k = 3 а = 0.514 e k = 5 а = 0.514 b k = 2 а = 0.771 d k = 4 а = 0.386 f k = 7 а = 0.220 Figure 4. Epihypocycloids with the base ellipse b = 2, c = 1 formatting point is inside the circle, μ = 0.5 As it’s seen from the Figure 2 the shapes of the epi- and hypocycloids depend on the initial point u0 on the base curve. At the Figure 3 the figures of the epihypocycloids are shown, with the on the base ellipse with parameters b = 2, c = 1 and the forming point d outside the rolling circle (μ = 2) and with different parameters of rolling circle. At the Figure 4 the figures of the epihypocycloids are shown, with the base ellipse with parameters b = 2, c = 1 and the forming point d inside the rolling circle (μ = 0.5) and with different parameters of rolling circle. At the Figure 5 the figures of the epihypocycloids are shown with the base sine curve у = аsinpx/L, a = 0.65, L = 1, x = 0 ¸ 2L; with various parameter of μ: а) μ =1; b) μ = 0.5; c) μ = 2. 1 2 3 a c b Figure 5. Epihypocycloids with the base sinus curve on two halfwaves: а - μ = 1; b - μ = 0.5; c - μ = 2; 1 - top row к = 1; 2 - middle row к = 2; 3 - lower row к = 3 At the Figures 5, а1, b1, c2 the hypocycloids are shown. At another figures epi- and hypocycloids together are shown. It’s seen from the combined figures that hypocycloids on the base sine are antisymmetrical to the epicycloids. Epihypocycloid cycle surfaces The figures of epihypocycloid cycle surfaces on the base ellipse are given with parameters к = 1 (Figure 6), к = 2 (Figure 7), к = 3 (Figure 8). At the Figure 9 there are shown epihypocycloid cycle surfaces on base sine with different numbers of half-waves. a c b Figure 6. Epihypocycloid cycle surfaces with the base ellipse, k = 1: a - μ = 1; b - μ = 0.5; c - μ = 2 а b c Figure 7. Epihypocycloid cycle surfaces with the base ellipse, k = 2: а - μ = 1; b - μ = 0.5; c - μ = 2 a b c Figure 8. Epihypocycloid cycle surfaces with the base ellipse, k = 3: а - μ = 1; b - μ = 0.5; c - μ = 2 a b c k = 1 k = 3 k = 2 Figure 9. Epihypocycloid cyclic surfaces with the base sine: а - μ = 1; b - μ = 0.5; c - μ = 2 Conclusion The article shows the possibility of constructing generalized epihypocycloidal curves, formed by the cycle with bind point rolling along any base curve, and the epihypocycloid cyclic surfaces, formed by rotating the moving circle around the tangent of the base curve. The figures are shown of the variants of the epihypocycloids and epihypocycloid cycle surfaces with rolling cycle at the base ellipse and sine. The figures of curves and surfaces are made with the help of MathCad software, on the base of the vector equations (1), (3). Using the equations with the different base curves including space ones it is possible to create various shapes of cyclic surfaces. The given examples of surface visualization show great opportunities for creating new forms of spatial structures and their use in architecture and modern urban planning.

About the authors

Vyacheslav N. Ivanov

Author for correspondence.
Email: i.v.ivn@mail.ru
ORCID iD: 0000-0003-4023-156X

Doctor of Technical Sciences, Professor of the Department of Civil Engineering, Academy of Engineering

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