Uniform spherical grids via equal area projection from the cube to the sphere

Transcription

1 Uniform spherical grids via equal area projection from the cube to the sphere Daniela Roşca Gerlind Plonka April 4, 0 Abstract We construct an area preserving map from the cube to the unit sphere S, both centered at the origin. More precisely, each face F i of the cube is first projected to a curved square S i of the same area, and then each S i is projected onto the sphere by inverse Lambert azimuthal equal area projection, ith respect to the points situated at the intersection of the coordinate axes ith S. This map is then used to construct uniform and refinable grids on a sphere, starting from any grid on a square. Key ords: equal area projection, uniform spherical grid, refinable grid, hierarchical grid. Introduction The generation of suitable finite subsets of the unit sphere is closely related to all kinds of data approximation of the sphere. Generic approximation problems on the unit sphere include the question of ho to find the best element from a defined function set that interpolates or approximates given function values on points of the sphere. Here the question that arises, is ho the given function values should be distributed on the sphere in order to achieve optimal approximation results. In many applications, especially in geosciences and astronomy, but also in medical imaging and computer vision, one requires simple, uniform and refinable grids on the sphere. One simple method to construct such grids is to transfer existing planar grids. A complete description of all knon spherical projections from a sphere or parts of a sphere to the plane, used in cartography, is realized in [, 8]. Hoever, most grid constructions given so far do not provide an equal area partition. But this property is crucial for a series of applications including statistical computations and avelet constructions, since non-equal area partitions can generate severe distortions at large distances. The existence of partitions of S into regions of equal area and small diameter has been already used by Alexander [], ho derives loer bounds for the maximum sum of distances beteen points on the sphere. Based on the construction by Zhou [], Leopardi derives a recursive zonal equal area sphere partitioning algorithm for the unit sphere S d embedded in R n+, see [4]. The constructed partition in [4] for S consists of polar cups and rectilinear regions that are arranged in zonal collars. Besides the problem that e have to deal ith different kinds of areas, the obtained partition is not suitable for various applications here one needs to avoid that vertices of spherical rectangles lie on edges of neighboring rectangles. Technical University of Cluj-Napoca, Department of Mathematics, str. Memorandumului 8, RO-4004 Cluj- Napoca, Romania. Daniela.Rosca@math.utcluj.ro University of Göttingen, Institute for Numerical and Applied Mathematics, Lotzestr. 6-8, Göttingen, Germany. plonka@math.uni-goettingen.de

2 Other constructions for equal area partitions of S used in astronomy include the HEALPix grid [3], providing a hierarchical equal area iso-latitude pixelation, and the icosahedron-based method by Tegmark [0], see also [9]. In [7], an equal area global partition method based upon circle edges is presented. Starting ith a spherical triangulation, obtained e.g. by projecting the faces of an icosahedron to the sphere, a subdivision method is proposed to partition each spherical triangle into four equal area subtriangles. In [5], one of the authors suggested a ne area preserving projection method based on a mapping of the square onto a disc in a first step, folloed by a lifting to the sphere by the inverse Lambert projection. This idea can also be generalized to construct uniform and refinable grids on elliptic domains and on some surfaces of revolution, see [6]. In this paper e construct an area preserving map from a cube to the unit sphere S. Thus, any grid on the cube can be transported to the sphere. The ne construction admits to transform ell-knon techniques from the square to (parts of) the sphere. For example, the ell-knon tensor-product avelet transform for data analysis and denoising can be simply transferred to the sphere, here distortions are negligible because of the equal area property. Further, since arbitrary grids on the cube resp. on the square can no simply be transported to the sphere, e believe that this construction may achieve an essential impact for different applications in geosciences. Since e give explicit formulas both for the map from the cube to the sphere, and from the sphere to the cube, the method is easy to implement. Our ne construction scheme consists of to steps. In the first step, e construct in Section 3 a bijection T from each face F i of the cube, onto a curved square S i. In Section 4 e construct the inverse T. In the second step, e combine T ith the inverse Lambert azimuthal projection, in order to map each face F i of the cube onto a subset F i of the sphere, such that 6 i= F i = S. Finally, e present some examples of the obtained spherical grids. Preliminaries Consider the unit sphere S centered at the origin O and the cube K centered at O, ith the same area. Thus, the edge of the cube has the length a = π/3. Denote = a/ = π/6. We cut the sphere ith the six diagonal planes z = ±x, z = ±y, y = ±x and obtain the curves in Figure. Let us focus on one of these curves C, given by the equations x + y + z =, z = [ x > 0, y 3, 3 ]. Using the parameter t [ 3, 3 ], the curve C is described by (x, y, z) = ( ( t ), t, ( t )). The Lambert azimuthal equal-area projection of the sphere x + y + z = onto the plane z = is given by ( ) (x L, y L ) = + z x, + z y. () Hence, the Lambert projection of the curve C onto the plane z = has the equations t x L =, () + t y L = z L =, t + t, (3)

3 Figure : The curves of intersection of the cube ith the diagonal plans. ith t [ / 3, / 3]. The calculations sho that x L + yl = t, (4) y L t = x. (5) L t We denote by S a the curved square in the tangent plane z =, formed by the curved edge of equations ()-(3) and three other curved edges obtained in the same manner by intersecting the sphere ith the half-planes z = x > 0, z = y > 0 and z = y > 0, respectively (Figure ). Because of the area-preserving property of the Lambert projection, the area A(S a ) of the obtained curved square S a is exactly π 3 = a. With the help of this projection, e have simplified the problem of finding an area preserving map from the cube to the sphere S to a problem of finding a to-dimensional map from a square, i.e., a face of the cube, to the curved square S a. 3 Mapping a square onto a curved square In this section e derive an area preserving bijection T : R R, hich maps the square S a = {(x, y) R, x a, y a} onto the curved square S a that has been constructed in Section. Here, e say that T is area-preserving, if it has the property A(D) = A(T (D)), for every domain D R. (6) Since e consider only the to-dimensional problem in this section, e ill ork in the (x, y)- plane for simplicity, i.e., the curved boundary C a of S a is given by () and (3) ith the notation x = x L and y = y L, and analogously for the three other curves. We focus for the moment on the first octant of the (x, y)-plane I = {(x, y) R, 0 < y x} ith origin O = (0, 0) and take a point M = (x M, y M ) = (x M, mx M ) I, here m is a parameter ith 0 m, see Figure. The map T ill be defined in such a ay that each half-line d m of equation y = m x (0 m ) is mapped onto the half-line d φ(m) of equation y = φ(m)x, here φ : [0, ] R is a differentiable function that satisfies φ(0) = 0, φ() = and 0 φ(m). (7) 3

4 M N P Q O Figure : The square S a, the curved square S a and the action of the transform T. Points N and P are the images of M and Q. This condition ensures that the diagonal of the square S a and the axis x are invariant under φ. We denote by (x N, y N ) the coordinates of the point N = T (M). Let Q be the intersection of OM ith the square S a (see Figure ). The point Q has the coordinates (x Q, y Q ) = (, m), here = a = π 6, and the line ON has the equation y = φ(m)x. Further, let the point P = (x P, y P ) = P (x P, φ(m)x P ) be the intersection of ON ith the curved square S a. Thus, the coordinates of P satisfy the equations (4) and (5) ith some t P [0, 3 ], and from (5) e have φ(m) = tp φ(m), hence t t P = P + φ (m). Replacing t P in () and (3), e obtain the coordinates of P in the form x P = y P = + φ (m), + +φ (m) φ(m) + φ (m). + +φ (m) Some simple calculations yield the distances OM = x M + m, OQ = + m, ON = x N + φ (m), ( / OP =. + φ (m)) We ant to determine the map T such that ON OP = OM OQ. 4

5 From the above calculations e then obtain x N = x + φ (m) y N = φ(m)x N = x + φ (m), φ(m) + φ (m) + φ (m). With this assertion, the map T is no completely described by means of the function φ, and e obtain that T maps the point (x, y) I onto the point (X, Y ) given by X = x Y = x + φ ( y x ) + φ ( y (8) x ), φ( y x ) + φ ( y x ) + φ ( y (9) x ). Next e impose the area preserving property by a suitable determination of φ. For this purpose, e define the function φ such that the Jacobian of T is. After simplification, the Jacobian rites as J(T ) = det ( X x Y x X y Y y ) = ( φ y x) + φ ( y x ) + + φ ( ). y For solving the equation J(T ) = e substitute v := y x, and thus, in the considered case 0 < y x e have v (0, ]. Hence, ith the simplified notation φ = φ(v), e get Integration gives φ + φ + + φ =. arctan φ arctan φ + φ = v + C. The condition φ(0) = 0 yields C = 0. Next, in order to determine φ e use the formula x arctan a arctan b = arctan a b + ab a, b R, ab > and e further obtain φ ( + φ ) + φ + φ = tan v. (0) To simplify this term, e introduce the notation γ = tan v = tan π v, and equality (0) yields + φ (φ γ) = φ(γφ + ). () From the requirements (7) e can deduce that, for (x, y) I ith y > 0, both sides of equality () are positive. Thus, () is equivalent ith ( + φ )(φ γ) = φ (γφ + ) (φ + )(( γ )φ 4φγ + γ ) = 0, hich gives φ, (v) = γ ± γ + γ γ. 5

6 We denote α = v = y x. Thus γ = tan α and some calculations sho that the function φ becomes ( y ) sin α φ, (v) = φ, =, ith α = πy x cos α ± x. If e take into account the condition φ() = and the equality sin π = cos π, the only convenient solution is ( y sin α φ =, x) cos α ith α = πy x. Finally, e need to replace φ in the equations (8)-(9). Some straightforard calculations sho that φ + = cos α cos α, φ + = 3 cos α cos α, φ φ + = sin α 3 cos α. Hence, for (x, y) I, the formulas for the desired area preserving map T (x, y) = (X, Y ) are given by X = /4 x cos α, cos α Y = /4 x sin α, ith α = yπ π cos α x, = 6. Similar arguments for the other seven octants sho that the area preserving map T : R R hich maps squares into curved squares is defined as follos: For y x, For x y, (x, y) (X, Y ) = (x, y) (X, Y ) = /4 x /4 y Figure 3 shos to grids on the curved square. 4 The inverse map cos yπ x, /4 x cos yπ x sin xπ y, /4 y cos xπ y sin yπ x ; () cos yπ x cos xπ y. (3) cos xπ y To make the area preserving map T applicable in practice, e need also to derive a closed simple form for the inverse mapping T. Let us consider first the case hen x y > 0, hen the map T is given by formula (). With the notation η = cos yπ x, (4) 6

7 Figure 3: To grids on the curved square, ith seven and eight equidistant horizontal and vertical lines. e have In particular, X = x ( η ) x, Y ( η ) = η. η X + Y = X + Y = X Y = x 3 η, η x ( η), (5) x η(3η ). η We define Hence, e obtain the equation in η ith the solutions B := X + Y X + Y = ( η) 3 η. η ( B)η + 3B = 0 η, = ( B) ± B(B ). In our case, η (cos π, ), therefore the only convenient solution is η = ( B) + B(B ). Then, from it follos that B = Y (X + Y ) and B = X X + Y η = Y + X X + Y. (6) (X + Y ) For simplicity e introduce the notation := Y/X. Thus, e have [0, ] and ( ) π y + = arccos η = arccos x ( + ) +. ( + ) 7

8 No e use the identity for a = arccos(ab + a b ) = arccos a arccos b, 0 a b, and b = (+ ) + π y x = arccos and obtain ( + ) arccos Other possible expressions are π y + x = arctan arctan = arctan arctan + +. = arctan Y X arctan Y X + Y. (7) For the calculation of x from X and Y, e use the second equality in (5) and η = + + (+ ) in order to find x = X + Y = X η = X Finally, from (4) and (7) e find y = x π x arccos η = π ( + ) η = X + ( + + ) = X + Y ( X + Y + X). ( arctan arctan ) + = X 4 + π + + (arctan arctan 4 = X + Y X + ( X + Y arctan Y X arctan ) + ) Y. X + Y Using similar arguments for the other seven octants, e obtain that the inverse T : R R has the expressions For Y X, ( (X, Y ) (x, y) = sign(x) 4 X + Y X + X + Y, For X Y, 4 X + Y X + X + Y (sign(x) arctan Y X arctan Y X +Y ) ) ; (X, Y ) (x, y)=( 4 X + Y Y + X + Y (sign(y ) arctan X Y arctan X X ), +Y sign(y ) 4 X + Y Y + ) X + Y. 8

9 5 Mapping the curved squares onto the sphere The complete mapping from the cube to the sphere S is no described in to steps. In the first step, each face F i of the cube K ill be mapped onto a domain F i, bounded by a curved square, using the transformation T. In the second step, each F i ill be mapped onto F i S by the inverse Lambert azimuthal projection, ith respect to the center of F i. Obviously 6 F i = i= and 6 F i = S. i= We denote by L (0,0,) the Lambert azimuthal area preserving projection ith respect to the North Pole N = (0, 0, ). Remember that L (0,0,) maps a point (x, y, z) S onto the point (X L, Y L, ) situated in the tangent plane at N, as given in (). Applying the inverse Lambert projection L (0,0,), the point (X, Y, ), situated in the tangent plane to S at the pole N, maps onto (x L, y L, z L ) S given by x L = X + Y X, (8) 4 y L = X + Y Y, (9) 4 z L = X + Y. (0) Thus, the application T L (0,0,) maps the upper face of the cube projects onto F S. The formulas for T L (0,0,)(x, y) are given by (8), (9) and (0), here X, Y are calculated ith formulas (), (3). One can easily obtain similar formulas for T L (0,0, ), T L (0,,0), T L (0,,0), T L (,0,0) and T L (,0,0). Figure 4 shos some grids of the sphere, here a regular partition of the faces of the cube into 49, 64, and 44 equal area squares has been applied. Acknoledgement The ork has been funded by the grant PL 70/4- of the German Research Foundation (DFG). This is gratefully acknoledged. References [] R. Alexander, On the sume of distances beteen N points on the sphere, Acta Mathematica 3 (97), [] E.W. Grafared, F.W. Krumm, Map Projections, Cartographic Information Systems, Springer-Verlag, Berlin, 006. [3] K.M. Górski, B.D. Wandelt, E. Hivon, A.J. Banday, B.D. Wandelt, F.K. Hansen, M. Reinecke, M. Bartelmann, HEALPix: A frameork for high-resolution discretization and fast analysis of data distributed on the sphere, The Astrophysical Journal, 6, (005), p [4] P. Leopardi, A partition of the unit sphere into regions of equal area and small diameter, Electron. Trans. on Numer. Anal. 5 (006), [5] D. Roşca, Ne uniform grids on the sphere, Astronomy & Astrophysics, 50, A63 (00). [6] D. Roşca, Uniform and refinable grids on elliptic domains and on some surfaces of revolution, Appl. Math. Comput, accepted doi:0.06/j.amc

10 Figure 4: Grids on the sphere. 0

11 [7] L. Song, A.J. Kimerling, K. Sahr, Developing an equal area global grid by small circle subdivision, in Discrete Global Grids, M. Goodchild and A.J. Kimerling, eds., National Center for Geographic Information & Analysis, Santa Barbara, CA, USA, 00. [8] J.P. Snyder, Flattening the Earth, University of Chicago Press, 990. [9] N.A. Teanby, An icosahedron-based method for even binning of globally distributed remote sensing data, Computers & Geosciences 3(9) (006), [0] M. Tegmark, An icosahedron-based method for pixelizing the celestial sphere, The Astrophysical Journal 470 (996), L8-L84. [] Y.M. Zhou, Arrangements of points on the sphere, PhD thesis, Mathematics, Tampa, FL, 995.