Following our choice of base polyhedron, in this case the icosahedron, the portion of it, in this case the upper half, and the face pattern, in this case a pattern of 16 triangles to each face, we then have a variety of projection variations to choose from. Projection refers to the way in which our choice of now-patterned polyhedron is projected on to a surface, and also to the choice of surface. Here we shall be using projection outwards in all directions away from a chosen point, but we will be looking at projection on to a variety of different surfaces.
Projection on to a Sphere from the centre of the Icosahedron
Here our half icosahedron becomes a hemisphere and all our triangles are projected evenly. The point of projection is the centre of the original icosahedron.
Projection on to a Sphere from above the centre of the Icosahedron
Here our half-icosahedron becomes more than a hemisphere, and the triangles are projected unevenly, because the point of projection is now closer to the top of the original icosahedron
Projection on to a Sphere from above the centre of the Icosahedron, and to one side
Here our dome again becomes more than a hemisphere and the triangles are all biased towards the point of projection.
Projection on to an Ellipsoid
Here the original sphere onto which the Icosahedron has been projected has been
elongated both horizontally and vertically. The projection is central.
Projection on to an Ellipsoid
This is another version of a central projection on to an ellipsoid
Projection onto a Paraboloid
This time the surface onto which we are projecting is a paraboloid.
Projection onto an Elliptical Paraboloid
Now our surface is an elliptical paraboloid. This is again a central projection.
Projection onto a Hyperboic Paraboloid
Now our surface is a hyperbolic paraboloid.
Projection onto a Cone
Here our surface is a cone.
Projection onto an Elliptical Cone
Now our surface is a cone that is stretched both vertically and horizontally.
