Solar Highbeams: Center-coordinated packings of circles and ellipses

Certain "magic" aspect ratios allow a close packing of ellipses to be center-coordinated with a hexagonal close-packing of circles. The aspect ratios that work are central polygonal numbers; in this case 3.

Center-coordinated tessellations or packings are relevant in two ways to telescopic heliostats: they describe a way to simultaneously close-pack the primary mirrors (as seen from the sun's perspective) and the secondary mirrors (as seen from the perspective of the beam-down optics,) and they also describe ways that the lenslet arrays in the beam-down optics can be paired in registration with each other. A previous post showed that a tessellation of rectangles can be centroid-coordinated with a tessellation of squares if the aspect ratio of the rectangle is drawn from the set of integers. This post considers packings of ellipses that are center-coordinated with hexagonal close-packings of circles.

In the figure above, an affine transformation that transforms the bold equilateral triangle into the dashed green triangle will also transform the packed circles into the packed ellipses, and use the same set of points for centers. If the bold triangle, together with its decoration of three circular arcs, is considered a truchet tile that generates the packing of circles, shearing and stretching this truchet triangle (along with its decorations) yields a new truchet triangle that generates the packing of ellipses.

The example illustrated in the figure is just the first in a series of possibilities: for instance, we could have translated the apex of the triangle rightward two steps instead of just one. Taking the sides of the equilateral triangle to be unit length, the transformed left side of the triangle has length √3, and this is identically the length of the major axis of the ellipses. Since the area of the ellipses must be equal to that of the original circles (in order to give a center-coordinated packing) the minor axis of the ellipses is therefore 1/√3, and the aspect ratio of the ellipses is 3—or simply the square of the length of the longest side of the new triangle. From the Pythagorean Theorem, the square of that length is, where n is the number of steps the apex has translated:

(1/2 + n)² + (√3/2)²

= 1/4 + n + n2 + 3/4

= n + n2 + 1

= {1, 3, 7, 13, 21, 31, 43, 57, 73, 91...}.

These numbers are known as the central polygonal numbers.

Therefore, a close-packing of ellipses can be center-coordinated with a hexagonal close-packing of circles provided the aspect ratio of the ellipses is drawn from the set of central polygonal numbers.

The angle of the major axis is rotated by an angle θ from vertical where

θ = arctan((1 + 2n)/√3)).

Any two close-packings of ellipses can be center-coordinated provided that the two varieties of ellipses have equal areas and their respective aspect ratios are drawn from the central polygonal numbers. As mentioned in the previous post, given a centroid-coordinated pattern, it can be transformed by any affine transformation into another centroid-coordinated pattern. In particular, ellipse/ellipse patterns can be transformed to a center-coordinated pattern where one variety of the ellipses are circles. Thus the circle/ellipse patterns are the fundamental ones, the rest can be generated from these by affine transformation.