Setting The Envelope

Geodesica implements a variety of envelopes ranging from superellipsoids to eggs. Bring up the Envelopes window by clicking the png button on the Toolbar.

png

The sphere has two envelopes – one for the Primal manifold and one for the Dual. The envelope type is selected from the pop-up choice. The default setting is that the Dual uses the same envelope as the Primal with a specified offset, although this can be changed on the Dual tab. Note that the Dual tab is only activated when ‘Generate Dual’ is checked in the ‘Modify’ menu.

Extents

The XYZ extents for an envelope are measured from the origin (system center of the sphere). The extents are measured in any unit desired. Set the desired unit type on the ‘Units’ tab of the Preferences window.

IMPORTANT. When you change the extents, you must click 'Apply Primal' or 'Apply Dual' extents, depending on which tab you are on.

Model extents

The ‘Model extents’ are the internal dimensions used by Geodesica. The ‘Model extents’ should ideally be in the range 0.5 to 2.0. The minimum ‘Model extent’ for any axes is 0.1 units, and the maximum ‘Model extent’ is 10 units. It is advised to leave the Model extents at 1.0 unit, and change the scale of the sphere using the ‘Unit radius scale factor’ instead. This is explained below.

Unit radius scale factor

The ‘Unit radius scale factor’ is the amount by which the ‘Model extents’ are scaled, in order to give the ‘Actual extents’ of the sphere. The ‘Unit radius scale factor’ is also found on the ‘Units’ tab of the Preferences window. When you enter values for the ‘Model extents’ the ‘Actual extents’ are recalculated automatically (and vice-versa). The Log always prints the Actual dimensions, rather than the scaled Model dimensions.

NOTE: It is far more accurate to use a ‘Model extents’ of 1.0, rather than trying to make the dome ‘life size’ in the OpenGL view. The larger the envelope, the greater the probability of errors in projected rays. A ray might only be 0.0000001 degrees off true, but when it intersects with a large envelope, the error becomes appreciable. And this is the reason for the ‘Unit radius scale factor’.

Actual extents

The ‘Actual extents’ are the life size dimensions of the sphere. Thus if the Model extents are 1.0 unit, and the ‘Unit radius scale factor’ is 10, the Actual extents radius will be 10 units. The unit can be any type (e.g. feet, yards, meters) and is set on the 'Units' tab of the 'Preferences' window.

IMPORTANT. Geodesica’s Log is the only intended reference for a dome’s dimensions. By default, Geodesica does NOT scale the model during export as this can incur precision errors in the exported data. However, if you want the exported model scaled to actual size, you must tick the checkbox ‘Scale to actual size’ on the Export Format window. Be aware that many 3D programs truncate floating point numbers when importing models from non-propietry formats; and some programs do not even use double precision.

Envelope Types

png

You may choose various envelopes, ranging from superellipsoids to eggs. To avoid confusion of terms, envelopes created by profiles of revolution are named so in the pop-up choice. The axis of revolution is Y. The terms are explained in the table below.

POP-UP NAME

ENVELOPE

Supercircle of Revolution

Supersphere with a fixed north-south exponent of 2.5 and fixed east-west exponent of 2.0.
Expansion is in Y.

Ellipse of Revolution

Spheroid - oblate or prolate.
Expansion is in Y.

Superellipse of Revolution

Superspheroid - oblate or prolate, with a north-south exponent.
Expansion is in Y.

Ellipsoid

Ellipsoid.
Expansion is XYZ.

Super Ellipsoid

Superellipsoid with a north-south exponent and an east-west exponent.
Expansion is in X and Y (or XYZ with option 2).

Egg of Revolution

Bird’s egg with the top pole narrower than the bottom pole.
Expansion is in Y.

Bell of Revolution

A bell dome for tuncation or otherwise.
Expansion is in Y.

Tear drop of Revolution

A Tear drop dome, designed for tuncation or otherwise.
Expansion is in Y.

Axolotl’s Head of Revolution

A very interesting profile, designed for tuncation or otherwise.
Expansion is in Y.

The super quadric exponent n is for the north-south direction (Theta), whilst the super quadric exponent e is for the east-west direction (Phi). Some examples are show below.

png png

n=0.3 e=2.0

n=0.5 e=2.0

png png

n=0.75 e=2.0

n=1.0 e=2.0

png png

n=1.5 e=2.0

n=2.0 e=2.0

png png

n=2.5 e=2.0

n=5.0 e=2.0

Super Circle of Revolution
normal offset 1

Fixed exponents n=2.5, e=2.0. Radius in Y. Profile: ‘super circle’ or ‘Super ellipse’.

The supersphere is attributed to the Danish polymath Piet Hein (1905-1996) who devised a profile that mediates between a circle and a square. The profile is an instance of a Lamé curve (after Gabriel Lam).

The supersphere has been used extensively in the architecture of Stockholm and has found other engineering uses – such as the lofting of aircraft fusalages. [Ref: ‘Superquadratics and Their Geometric Properties’, by Franc Solina].

A supersphere is stable in the upright position because its center of gravity is lower than its center of curavture. [Ref: Martin Gardener’s article in Scientific American, September 1965 –a publication I have yet to find].

Ellipse of Revolution
normal offset 1

Fixed exponents: n=2.0, e=2.0. Radius in Y. Profile: Cartesian ellipse.

Use for for oblate (flatenned) or prolate (stretched) domes.

Super Ellipse of Revolution
normal offset 1

Exponents n=1 (variable), e=2 (fixed.) Radius in Y.

Sometimes this superquadric is called a ‘Superegg’, but I can’t think why, because it looks nothing like an egg, whatever its exponents. (An egg always has one pole smaller than the other). To avoid confusion, the term ‘Superegg’ is omitted because a real egg shape is implemented as a profile of revolution. See below...

Egg of Revolution
egg 1

Without Theta correction

egg 2

With Theta correction

When you select this option, an additional pop-up of Egg Types appears at the top-right of the envelope tab; each choice makes a subtle difference to the egg profile. Increasing the expansion increases the narrowing at the pole. If the expansion is only 1, the result is a sphere.

egg 2p0

E = 2.0

egg 2p0

E = 2.5

egg 2p0

I was inspired to implement egg domes after reading the following web page: http://www.mathematische-basteleien.de/eggcurves.htm

Bell of Revolution
png

This interesting shape is rather like a bell, hence the name. The effect increases with expansion in Y. This shape is ideal for truncation at Theta=90. Or the section below the X axis could be burried below ground for a subterranean space.

Tear drop of Revolution
png

Another interesting shape. The effect increases with expansion in Y.

Axolotl’s Head of Revolution
png

For some reason, this profile reminded me of an axolotl’s head when viewed from above. (An axolotyl is a type is aquatic salamander). I came upon this profile by accident whilst experimenting with egg profiles. The effect increases with expansion in Y.

Ellipsoid
normal offset 1

Fixed exponents: n=2.0, e=2.0. Radius in XYZ.

Super Ellipsoid
jpg

Exponents: n=0.5, e=5. Radius in XYZ.

jpg

Exponents: n=1, e=1. Radius in XYZ.

The superellipsoid can create a plethora of shapes from rounded boxes to octahedral diamonds and oblate spheres. I have avoided limiting the exponents to 2 which means concave shapes are allowed; you might think this odd in a program designed to build domes, but an inner concave space could be tied via space frame struts to an outer convex envelope – not for a dome perhaps, but another unforeseen requirement – like high pressure containment vessels.

By varying the exponents of a Super Ellipsoid it is possible to create all the other shapes. For a sphere n=2, e=2. But if your design only requires a sphere or spheroid, do not choose Super Ellipsoid, because a superellipsoid takes longer to compute and project.

Dual offset

This option allows you to offset the Dual a specified distance from the Primal. Enter a value in the offset field and then click ‘Apply Dual Extents’

jpg
jpg

There is a caveat when moving the dual offset inside non-spherical hulls: at a certain point, the primal normals will begin to intersect and the dual will fold in on itself. This only happens when the offset is very large - much larger than a spacefame requires, so this is not really a problem.