I tried the following last night, and was almost done, but mixed up the radius and diameter, so will try again later.
Probably would work for diameters by multiplying the equation by 2 ?
Symmetric Spheroform Tetrahedron
The symmetric spheroform tetrahedron with uniform width begins with a tetrahedron of side one, easily built in MoI.
The Reuleaux tetrahedron is created using Trim on intersecting spheres of radius one, centered at the corners of the tetrahedron. Circular array 3 is helpful.
The Reuleaux tetrahedron is not quite of uniform width.
To achieve uniform width, areas near the 4 edges must be rounded over by
replacing each of four "wedge" areas near each edge, by trimming in a particular solid formed of a swept SPHERE (NOT vertical circle).
To form the swept circle solid, in MoI Top View, sweep a vertical circle placed at the center of two rail curves, along the two rails.
Rail one is the line between x = -0.5 and x = +0.5.
The "bottom" of the SPHERE (NOT circle) sweeps along rail_1.
The SPHERE (NOT circle) shrinks down to zero radius at the ends.
The vertical SPHERE (NOT circle) to be swept is of radius "r". It's center is located at (x,y) = (0,r), which is the center of rail_2.
The center of the SPHERE sweeps along rail_2.
Using Max Smirnov's MoI script FxGraph, rail_2 may be easily plotted.
The formula for the rail_2 NURBS curve is given in the pdf, by the equation:
y = f(x) = ((4 * sqrt(2) - 8) * x*x + 2 - sqrt(2)) / 8 ;
min(x) = -0.5;
max(x) = +0.5;
The number of points plotted was 100.
Tab pasting the following to the command window accomplishes the same radius curve: (If _FxGraph is in command folder, or use FxGraph without _)
_FxGraph ((4*sqrt(2)-8)*x*x+2-sqrt(2))/8;-0.5;0.5;100
MoI does not have SweepSphere command, so some other method must be found and used...
The Sphere to "fake-sweep) can be placed in the middle of the wedge to be.
Place a point at the top of the SPHERE to assist with future alignment of the swept_wedge.
Then perform the fake_sweep with some new method..
Draw appropriate alignment lines, (bothsides setting), from edge center to opposite edge center to place one swept_wedge.
Trim the swept_wedge and the tetrahedron edge area. The trimmed swept_wedge should be tangent to two adjoining faces of the tetrahedron. Perform circular array (3) of the trimmed swept_wedge.
Place appropriate CPlane, to permit circular array of one of the trimmed swept_wedge, along bottom face of the tetrahedron.
Join will now form the Symmetric Spheroform Tetrahedron.
- Brian
I'll post a solid if creation is successful.
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