Two Area Preserving Maps from the Square to the p -Ball

. We construct two area preserving maps from the square to the p -ball (the domain deﬁned by | x | p + | y | p ≤ | a | p ) and use them to obtain low distorted uniform grids on the p -ball.


Introduction
In many applications in computer graphics, medical imaging, geosciences, one requires uniform and refinable grids on a given planar domain. In particular, one is interested in partitions having equal area cells with small diameter in order that no size distortions occur, but only small shape distortions. This goal can be achieved if we construct an area preserving map from the square to the given domain.
One of the most studied problem is how to map a square into a disc. There are many such maps (see [1], [5]) with different properties related to the distortions they introduce. One of these maps is the map introduced in [7], which is called the concentric map, because concentric squares are mapped to concentric circles. This map was reobtained in [2] in the study of partial differential equations on Lipschitz domains and also in [6] in the study of a new method to obtain uniform grids on the sphere. The concentric map is useful because it preserves the area, so a uniform distribution of points in the square will map to a uniform distribution of points in the disc. This fact was used in some sampling computations in computer graphics [4].
Our paper generalizes the concentric map. We obtain two area preserving maps from the square to the p-ball, i.e. two maps such that every subdomain of the square [−r, r] 2 is mapped into a subdomain of B p (a) = (x, y) ∈ R 2 , |x| p + |y| p ≤ a p (1.1) of the same area. These maps have the same good properties as the concentric map: equal-area preservation, bicontinuity (the map and its inverse are continuous) and low distortion. The paper is structured as follows: in Section 2 we construct an area preserving map from R 2 to R 2 and its inverse, which will be used in Section 3 to obtain two maps from the square to the p-ball. Finally, in Section 4 we present some uniform and refinable grids on the p-ball obtained from uniform and refinable grids on the square.

An area preserving map and its inverse
For p > 0 let B p (a) be the p-ball of radius a > 0, centered at the origin, defined by (1.1). The area of this p-ball can be computed using Beta and Gamma functions with c = √ 2 · Γ (1/p + 1)/ Γ (2/p + 1). We notice that for p = 1 the value of c is 1 and the radius of the ball B 1 (a ) must be a = ca in order to have the same area as B p (a).
Next, we construct the map U p : B p (a) → B 1 (ca) with the property that every subset of B p (a) is mapped into a subset of B 1 (ca) of the same area. For (x, y) ∈ B p (a) let us denote (X, Y ) = U p (x, y).
has the area preserving property.
Proof. It is sufficient to consider the case x, y ≥ 0. The area of the domain B p (a) must be equal to the area of the domain B 1 (a ), so we obtain the relation a = ca. We will impose the condition that concentric p-circles are mapped into concentric 1-circles. The relation a = ca implies X + Y = c(x p + y p ) 1/p . Now, because U p will be an area preserving map, its Jacobian is 1. So, we have ∂X ∂x Replacing Y with c(x p + y p ) 1/p − X we obtain the following first order partial differential equation: We will impose also the condition X(0, y) = 0. This means that the line x = 0 is mapped into the line X = 0. We have to solve the following symmetric system dx The first equation gives us x p + y p = C 1 . Using this we obtain dx Using the condition X(0, y) = 0 and the relation (obtained by the change of variable we get Replacing x p + y p = C 1 in this last equality we obtain Finally, from the relation X + Y = c(x p + y p ) 1/p we obtain the value of Y given in (2.1).

Remark 1.
Consider some particular cases for the map U p . In the case p = 1, we have F (t) = t and c = 1. Thus U 1 is the identity map (X, Y ) = (x, y).
For the case |y| < |x| we use the relation , when |y| < |x|.
Theorem 2. The inverse of the map defined in Theorem 1 is given by Proof. From the definition of the map U p we have We obtain From this, because sgn(x) = sgn(X), we finally get we similarly obtain Remark 2. For the map U −1 p consider the same particular cases as for the map U p . In the case p = 1 we have the identity map.
In the case p = 2 the function G is defined by G(t) = cos 2 π(1−t)

2
. The map U −1 2 is given by .

Two maps from the square to the p-ball
In this section we will obtain two maps from the square to the p-ball using the inverse of the map U p , which was defined in the previous section and we will study the distortions introduced by these maps. First, let us present how we will measure the distorsions of these two maps.

Distorsions measurements
We use the local isotropy distorsion [5], denoted D I , which is the ratio of the semi-axes of an ellipse, called Tissot's indicatrix [8]. Tissot introduced this tool to measure the distorsions of any map from a sphere to a planar domain. For any point on the sphere an infinitesimal circle is mapped onto an infinitesimal ellipse on the plane. This ellipse, with semi-major axis a and semi-minor axis b, describes the local characteristics of the map. For a map (x, y) −→ (U, V ) from a planar domain to a planar domain, the formulae for a and b are We have s = a · b and this is 1 for area preserving maps. The local isotropy distorsion D I is given by the relation When the map is area preserving the local isotropy distortion is given by The smallest value of D I , in this case, is 1 and this value is obtained when i.e. the map is conformal. Let us remark that a map from a planar domain to a planar domain is both area preserving and conformal if and only if the map is isometric, i.e. it is length-preserving (a rotation, a translation, a reflection, or a composition of them).

The first map from the square to the p-ball
We define T p : This map has the expression In the particular case p = 2 this map becomes which is the concentric map, introduced in [7], also obtained in [2,6]. The inverse T −1 p is given by In order to evaluate the distorsions introduced by the map T p we have computed the value of D I using formula (3.1), where with g = G(t/2) and t = min(|x|/|y|, |y|/|x|).

The second map from the square to the p-ball
The map we have constructed is not unique, there are many other area preserving maps from the square to the p-ball. For example, a different such area

A. Holhoş
preserving map can be constructed as follows: we start with the map from the square [−r, r] 2 onto the rhombus |X| + |Y | ≤ √ 2r given by which is actually a rotation with π/4 in the trigonometric sense, and we compose it with the inverse of U p . Thus, we obtain the map S p = U −1 p • S, given by . The map S p has the property of area preserving, being the composition of two area preserving maps. The inverse S −1 p is given by For the analysis of distorsions we use (3.1), with the same formula (3.2) for computing h 2 + k 2 , but with g = G 1+t 2 and t = −x/y, if |y| ≥ |x|, or t = −y/x, if |y| < |x|. In Figure 2 we have represented the value of D I for the map S p for some values of p. For p ∈ (0, 1) the value of D I grows indefinitely on the lines y = ±x. For p = 1, the value of D I is 1 everywhere. This is true, because, in this case, the map is just a rotation. For p > 1 the maximum value of D I is finite and depends on p. For p = 2 the distorsion is given by the same formula (3.3). First, let us consider the regular square grid. In Figure 3 we have represented the image of the square from left through the map T p for some values of p. In the case p = 4, we obtain a uniform grid on the squircle (we use the notion of "squircle", not in the sense of Fernández Guasti [3], but as a special case of a superellipse). For p = 2, we have a grid on the circle, for p = 1, a grid on the diamond(rhombus) and for p = 2 3 , a uniform and refinable grid on the astroid. We can remark that the shape of the little squared cells is better preserved in the cases p = 2 and p = 4 for those cells situated on the vertical and horizontal lines passing through the center of the ball. We can also remark the greatest distorsion of shape, in the case of the cells situated on the diagonals y = ±x. In Figure 4 we have represented the image of the square from the left through the map S p , for the same cases: squircle (p = 4), circle (p = 2), diamond(rhombus) (p = 1) and astroid (p = 2 3 ). We can remark the smallest distorsion on the lines y = ±x and the greatest distorsion on the vertical and horizontal lines passing through the center of the balls. We can also notice that the little "squared" cells are rotated anticlockwise with an angle of π/4 from the original ones. Now, by overlapping the uniform grids obtained through the maps T p and S p we get new grids on the p-ball. These new grids have triangular shaped cells. In Figure 5 we have represented these grids and it can be seen that the "triangular" cells look almost the same. Visually, the differences of shape between the cells are smaller than in the case of the square grids. Let us remark one more thing about these triangular grids: they can be obtained also, using only one of the maps we have introduced, either T p or S p , by mapping the