This is my experiment on distributing vertices around ellipsoidal envelopes when standard ‘Root-E’ correction fails to give the desired results. To begin with then, lets look at the problems of ‘Root-E’ correction, which is the standard theta correction used with most ellipsoidal geodesic domes...
Root E Morphs
To begin, open the Envelope window, which is accessed from the ‘Modify’ menu. You can use the default value for ‘Root-E’ Correction or morph vertices using the mouse. To enter your own value, unlick the button marked ‘D’ ‘for Default’. The distribution field activates. Enter a value or click in the field and drag the mouse left and right. This will move cell vertices either toward or away from the Equator…
The animation shows a ‘Root-E’ correction morph on the vertices of an octahedral superellipsoid with a zenith expansion of 1.25 and an exponent of 2.25.
The idea behind the default ‘Root-E’ correction is to stop crowding at the zenith and shorten struts near the equator, where the load is greatest. But when morphing a ‘Root-E’ correction, it soon becomes apparent that the correction can only be applied so far; the more the zenith opens, the more the equator becomes crushed together. My solution is to apply a Theta distribution curve...
Theta Distribution Curve
To apply a theta distrubution curve, choose ‘Theta curve’ from the ‘Expansion function’ menu on the ‘Distribution’ group of the Envelopes window. Ensure the Theta box is checked, then click the “Set curve...” button. This will open the Theta Distribution Curve window. This is a modal dialogue, so before showing it, it is wise to create four viewports and set each camera to Top, Front, Bottom, Orbiter. This will let you view the effect in both hemispheres.
There is one bezier curve per hemisphere. The input theta, from the zenith to nadir, is shown along the X axis. When ‘North’ is selected, the theta range is 0 to 90 (zenith to equator); when ‘South’ is selected, the theta range is 90 to 180 (equator to nadir).
The gradient of the curve determines the amount of correction applied to the input theta. Thus a steep curve corrects vertices more than a shallow curve. Good distributions are achieved with S-shaped curves, where the gradient diminishes toward the shoulder of the curve. An S-shaped curve will disperse the zenith vertices without squashing struts around the equator. A curve with negative slope will work in reverse - expanding equatorial struts whilst crushing those around the zenith.
The interface is the same as for most bezier drawing programs; simply drag on control points to modify the curve. The strength of the correction can be scaled using the slider. Drag the start anchor point (Yellow) to where you want the theta correction to begin. Drag the end anchor point (Blue) to where you want the theta correction to end. Alternatively, enter values in the appropriate fields. As you change the control points, the dome is reprojected using the curve profile.
If an ellipsoidal dome is symmetrical at the equatorial plane, the curve in the southern hemisphere must mirror that of the north. Do this by checking the “South mirrors North” button. This reflects the northern curve at theta = 90 degrees and preserves the symmetry of the system. When “South mirrors North” is ON, the Strength factor is applied equally to both hemispheres. When “South mirrors North” is OFF, the Strength factor is applied only to the selected hemisphere.
Octahedral superellipsoid. Zenith expansion 1.25, exponent 2.25, Class I, method 1, V10. This shows theta curve correction on particles 1 and 5 of an “elongated” dome. The sequence shows the results of first applying Root E correction and then using a theta distribution curve. The intention was to limit variation in strut length. Note: the distribution curve for the southern hemisphere was disabled by setting its strength to zero.
Octahedral superellipsoid. Zenith expansion 0.666666, exponent 2.5, Class I, method 1, V10. This shows theta curve correction on particles 1 and 5 of a “squashed” dome. The sequence shows the results of first applying Root E correction and then using a theta distribution curve. Note: “South mirrors North” if OFF.
NOTE: the greatest concentration of strut members should generally be at the equator, where the forces are greatest; and these struts should be shorter than those at the zenith where there is little to support.
Icosahedron superellipsoid V8. Zenith expansion 0.85, exponent 2.25, Class I, Method 1, dual space frame. This shows theta curve correction on space frame members; the area around the Zenith is much less crowded in the corrected dome (second image); the intention was to limit weight at the zenith, where struts have little to support but themselves.