It might be easy to view the transporter template as nothing more than a pattern of lines and planes whose angles are as set as a crystalline lattice, and that therefore anything chiseled from the pattern will be characterized by those same hard angles.
Although spheres’ inherent contribution to the template have been duly noted, how exactly they may be brought into play for practical applications needs some explanation.
To appreciate the scope of spheres’ potential uses, what bears repeating is that every intersection of lines – which are potentially anywhere desired – may represent the center-point of a sphere of any size. So conceptualized, an arc of such a sphere curving in the plane formed by intersecting lines effectively round that plane.
Taking this idea a step further, spheres may be centered on 2 such intersections having one line in common; or centered at each end of any line’s endpoints. In the simplest cases presented in this post involving more than one sphere, spheres must be of equal size. Other than that they may either be in contact, separated, or overlapping.
So specified, the sphere pair may be joined cylindrically with that form’s axis coinciding with the line and its ends melding diametrically with the sphere in one continuous surface. By extension, a plane defined by a minimum of 3 lines may be rounded 3 dimensionally by centering spheres at each corner, joining these with cylinders, and capping them with am identical plane that is parallel to the original to form the desired continuous surface.
Uses for such constructs can be found inside or out of the transporter, but the most obvious application of the technique lies externally with the transporter’s shell. Its vertices – formed where adjacent planes converge – afford the loci where spheres may be centered.
With spheres so located, cylinders (real or imaginary) are run between the spheres placed at either end of edges formed by adjacent planes. Each plane may then be dealt with individually, with identical planes set parallel on the cylinders, as was done previously with the triangle.
Spaces formed between base planes and their externally translated replicas are equal to the spheres’ radii and can be used to run wires and cables; machinery, mechanisms, and electronics; and to provide storage or recesses for movable conveniences.
All the preceding involves convex surfaces only. As far as I have been able to determine, there is no possible way to retain 100% surface continuity when concave surfaces are introduced – and doing so while adhering to code geometry.
However, there are a few basic ways of dealing with concavity on an external convex surface in a sensible and harmonious manner which include proceeding as usual with spheres, cylinders, and parallel planes, and simply accepting the inevitable concave crease; and slicing along the underlying base planes that form the concavity.
A 3rd way to deal with concave discontinuity is to lay spheres and cylinders (of equal diameters) into concave creases and longitudinally slicing the unnecessary cylindrical cross section away. The same result is achieved by having the sphere/cylinders serve as the cap to a mold filled with a casting. Discontinuity is limited to the ends of any such application where it is made minimal by slicing orthonally into the cross sections. Without such slicing, this technique poses one way of rounding the interior of a shell, the other way simply going with the inside of the external shell and discarding or disregarding the base (planar) shell.
Such are the most basic methods of rounding the templates hard angles. Advanced rounding techniques entail employment of varying sized spheres or conical, parabolic, ellipsoidal, or sinusoidal wave forms. To describe these require other basic principles, and as these are best introduced later, exploration of advanced rounding will have to wait.