A Low Distortion Map Between Disk and Square

We present a map between squares and disks that associates concentric squares with concentric circles. This map preserves adjacency and fractional area, and has proven useful in many sampling applications where correspondences must be maintained between the two shapes. We also provide code to compute the map that minimizes branching and is robust for all inputs. Finally, we extend the map to the hemisphere. Though this map has been used before in publications, details of its computation have never previously been published. Background Many graphics applications map points on the unit square S to points on the unit disk D f x y j x y g. Sampling disk-shaped camera lenses is one such application. Though each application can have its own unique requirements, experience has shown that a good, general-purpose map should have the following properties. Preserve fractional area. Let R M be a region in set M . The fractional area of R is defined as a R a M , where the function a denotes area. Then a map m S D preserves fractional area if the fractional area of R is the same as the fractional area a m R a D for all R S (Figure 1). This property ensures that a “fair” set of points on the square will map to a fair set on the disk. For distribution ray tracers this means a jittered set of points on the square will transform to a jittered set of points on the disk. Bicontinuous. A map is bicontinuous if the map and its inverse are both continuous. Such a map will preserve adjacency; that is, points that are close on the disk came from points that are close on the square, and vice versa. This property is useful primarily when working on the hemisphere. It is necessary, for example, if we wish to use linear distance on the square as an estimate of angular distance on the hemisphere. Low distortion. By low distortion, we mean that shapes are reasonably well preserved. Defining distortion more formally is possible [1], but probably of limited benefit because no single definition is clearly suitable for a wide range of applications. Figure 2 shows the polar map, probably the most obvious way to map the square to the disk: r is a