Uniform and refinable grids on elliptic domains and on some surfaces of revolution

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Abstract

We construct a bijection from R2 to R2, which maps rectangles centered at the origin O onto ellipses centered at O, and preserves area. This bijection allows us to construct uniform and refinable grids on elliptic domains. Then, we combine a particular case of this bijection (i.e. that bijection that maps squares into circles) with another area preserving projection from R2 to a surface of revolution around Oz. This yields uniform and refinable grids on this surface of revolution. The lines of these grids are situated in horizontal planes, if they are images of squares centered at O. We consider the particular case of the hemisphere and show how the northern hemisphere of the Earth is projected onto a square. Thus, our equiareal maps can be useful for constructing geographical maps of one hemisphere of the Earth onto rectangles.

Introduction

Uniform and refinable grids (UR) are useful in many applications, like construction of multiresolution analysis and wavelets, or for solving numerically partial differential equations. While on a rectangle or on other polygonal domains the construction of UR grids is trivial, it is not immediate on an elliptic domain or on a disc.

In this paper we construct a bijection from R2 to R2, which maps rectangles onto ellipses and preserves area. This allows us to transport a rectangular grid to an elliptic grid, preserving the area of the cells. In particular, any uniform1 rectangular grid is mapped into a uniform elliptic grid. A refinement process is needed when a grid is not fine enough to solve a problem accurately. A uniform refinement consists in dividing a cell into a given number of smaller cells with the same area. With the procedure described here, any uniform refinement of a rectangular grid leads to a uniform refinement of the corresponding elliptic grid.

In the particular case of the disc, such a bijection was constructed in a previous paper [2] and combined with Lambert azimuthal projection, helped us to construct uniform grids on the sphere. Also, the inverse projection, mapping discs onto squares, can be used to map a hemisphere of the Earth onto a square domain, more suitable for computer manipulations. The main advantage over the existing map projections is the area preserving property, desirable in statistical mapping.

We mention that the problem of finding an analytic function that maps a disc onto the interior of any convex polygon was first solved independently by E.B. Christoffel (1867) and H.A. Schwarz (1869). Their map is a complex integral which is conformal (i.e. preserves angles), but it does not preserves areas. In fact, it is known that no conformal map can preserve areas. Another drawback is that it has no explicit formula, requiring solving a system of nonlinear equations.

Some well-known projections mapping a hemi-sphere onto a square are Pierce quincuncial (1879 - conformal except four points), Guyou (1886 - conformal), Adams (1925 - conformal), Collignon (1865 - area preserving). An area preserving projection from a sphere to a rectangle is Lambert Cylindrical projection. A complete description of all known map projections used in cartography can be found in [1], [4].

Section snippets

Construction of an area preserving bijection in R2

Consider the ellipse Ea,b of semi-axes a and b, a, b > 0, of equationx2a2+y2b2=1and the rectangle RL1,L2 with edges 2L1 and 2L2, defined asRL1,L2={(x,y)R2,|x|=L1,|y|=L2}.The domains enclosed by Ea,b and RL1,L2 will be denoted by E¯a,b and R¯L1,L2, respectively.

We will construct a bijection TL1,L2a,b:R2R2 which maps each rectangle RαL1,αL2 onto the ellipse Eαa,αb and has the area preserving propertyA(D)=A(TL1,L2a,b(D)),for every domainDR2.Here A(D) denotes the area of D. Thus, A(R¯L1,L2)=A(E¯a,b

Uniform grids on surfaces of revolution

We have presented in [3] a method for constructing a projection from a surface of revolution onto a plane perpendicular to the rotation axis, which preserves area. The method works for a surface of revolution M generated by a plane curve of equation z = φ(x), with φ a nonnegative and increasing piecewise smooth function, defined on the interval I = [0, β) or I = [0, ∞). This projection allows us to construct a continuous wavelet transform and a multiresolution analysis, in the case when the curve φ has

Acknowledgements

I thank to the anonymous referees for their constructive suggestions for improving the paper. The work has been co-funded by the Sectorial Operational Programme Human Resources Development 2007-2013 of the Romanian Ministry of Labor, Family and Social Protection through the Financial AgreementPOSDRU/89/1.5/S/62557.

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