After all the waveform derivations, applications, variations, rules and options, one might guess that simply attaining a flat surface would be easy. However, IMO, flat contouring actually poses the most complicated challenge for more than one reason.
In the context of the code, there are 2 definition of “flat.” First is the natural flat that is more or less perpendicular to lines of gravitational force, and as such is the local flat as determined by levels, plumb lines, and lasers. Then there is the abstract flat plane presented by the projection of the celestial co-cube. The discrepancy between the 2 flats arises from earth’s deviation from a perfect sphere, and as such varies with latitude.
A related problem posed by the quest for a flat surface arises from the question of what exactly is sought in the contouring. If one seeks conformity continuity by merging the tough lines and arcs of waveform constructs, flat contouring should be of the local natural works because such also happens to be the reference plane utilized in the construction of mounds and berms. However, this approach leads to water drainage problems in proportion to surface area graded thus.
Resolving these conflicts as simply and sensibly as possible entails using each plane definition according to its attributes; using the virtue of flat planes in general to afford a clean interface with natural terrain or intuitive freeform contouring; and formulating a kind of compromise that speaks to both flats.
As noted above, natural flat is the easiest to attain with conventional tools and methods. If the waveform construct does not have a vertical or max slope interface with natural terrain, it should be surrounded by this version of flat grading. The abstract flat on the other hand is restricted to one direction only, downward relative to the local flat toward the equator – due south in the northern hemisphere and due north in the southern hemisphere. Utilizing the abstract flat is not as easy to apply mounds and berms as it is with “flat” roof sections.
The steepness of the abstract slope varies with latitude (Θ) according to the indicated formula. As with flat roofs, this slope provides some water drainage and can thus be applied to a larger area than the natural flat, but the slope is not that great (about 3 in 1000 in latitudes 30° to 60°) and should not be much larger. If both natural and abstract flats are used together, the interface is a line that coincides with latitude. Along the other 3 sides, the deepening break between the planes may be smoothed with intuitive freeform contouring. Otherwise, either flat plane provides a clean interface with natural terrain.
The reasoning behind the compromise formulation starts by keying the differing flat plane versions to the abstract square and natural triangle in the context of the vertex-up prism. This prism is apropos because its mundane pole affords an axis by which a circular area may be swept. Of course the area swept will also be disciplined by a right-angled rectilinear grid and for this reason an axis orthogonal to the mundane pole spanning the opposing edges is utilized. [Note: The above rationale for employing an orthogonal prism axis is a bit of a stretch. There might be a logical connection between orthogonal axes and rectilinear grid contouring, but if there is, I suspect the connection would have many steps. Fortunately there is a much simpler and sound connection afforded by the fact of the differing flats having their origins in oblate and perfect spherical earths. The flat cross-sections of either of these may conform to the other by rotation about either polar or orthogonal axes – RRW, 3-4-2015]. So established, attention is focused on the element common to the square and triangle – the edge.
At this point it is interesting to note that in all orthogonal (secondary) rotations of the cubodal shell, the edge is the only dimensional element, viewed directly in and of itself regardless of what it borders, that has neither up/down or left/right symmetry. For the edge to participate in such the prism is rotated about the opposing edge axis until the vertically-aligned triangle is faced directly. The angular difference between the edge intersection of the 2 different planes – from symmetry to asymmetry – is about 1.5°.
This angle can then keyed to the tangent of an omnidirectional spherical segment or mono-directed cylindrical segment as with a football field whose maximum slope is about the same 1.5°. This angle can also be keyed to the maximum slope of a waveform used to create a mound or extended berm, either convexly upward or concavely downward.
Rectilinear areas can be reached with an approach that is the reverse of the inside corner contouring method. The (half) wave’s size and proportion are kept equal by shifting the pivot point along a diagonal ridge. Thus may more effective water drainage be attained in the context of grid constructs.