Joining Angles

When I first tried to construct a non-spherical object (ShowTent.lig) out of particleboard, I assumed that all face-joining angles could be cut (mitred) using half-interfacial angles and that all faces would fit together perfectly. I soon discovered that this produced minor overlaps at the vertices where a number of faces came together. I solved this problem by cutting small chamfer with a chisel to the inside of interfering face corners. When assembled, the external appearance of the object was unaffected, but the interior revealed small triangular pits at some vertices.

All faces will fit together perfectly at the vertices only if the solid angles used to cut the edges of the faces are allowed to radiate from a single point. In other words, the surfaces represented by the edge cuts of the faces should all pass through a single point -- normally the origin. These ‘origin-centric angles’ were subsequently added to L3DD’s plan as an option that can be selected in the Plan Setup dialog box. The two origin-centric angles of adjoining faces, when added together, equal the full interfacial angle.

The second illustration on the right shows a cross section of a soccer ball made out of thick panel material. The origin-centric angles used to cut the edges of the adjoining faces are not the same (73.53 and 69.09 degrees), but they add up to the full interfacial angle (142.62 degrees).

Although origin-centric angles result in a better fit, they may require more frequent changes to the angle of the saw blade when cutting edges. Also, some angles of less than 45 degrees may be required. Half-interfacial angles allow the saw blade to be set to an angle and then used for twice the number of edge cuts than the origin-centric case. For spherical or highly symmetrical objects, such as geodesic spheres, the half angles are often the same as the origin-centric angles.

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In frameworks generated with the Create Frame command you should always use origin-centric angles for the V-shaped cuts at the ends of frame members – otherwise you may have trouble making snug joints. However, for spherical frameworks (eg., a framework generated from an icosahedron) you will often get the same angles if you use the half-interfacial option.

If you choose the origin-centric angles you should think about moving the whole object so that the origin is at an optimal position for the particular object. This is not necessarily the object’s center of gravity. If you look at ShowTent.lig, it is clear that the origin should be at the center of the base of the dome. In some objects some origin-centric angles will always be too acute to be easily cut with a saw. In this situation consider printing several sets of plans with differing origins. You can then choose the best angles to use in different parts of the object, but don’t combine the origin centric angles between two adjoining faces from more than the one plan. They must add up to the full interfacial angle. You could also use half angles and origin-centric angles in the one object.