The pictures are of a projection of an immersion of the sphere in
space, given by the map:
(x, y, z) -> (x, y, x2 + xz, yz)where x2 + y2 + z2 = 1. In these pictures, we are projecting down the first axis.
One of the questions we can ask is the relationship between bitangencies of a surface in four space and the bitangencies of its projection down into three space. Any bitangency in four space must project to a bitangency in three space, and will be part of a two-dimensional set of bitangencies. In order to determine whether a pair of points (p, q) is a bitangency of the surface in four space, we have to project the surface down two vectors, such that the plane spanned by these two vectors is transverse to the tangent planes at p and q. If the two projected surfaces both have bitangencies bewteen p and q, then (p, q) is a bitangency of the surface up in four space. In addition, if we are lucky enough to project down the secant line of a bitangency, then one of the tangent directions at each point will collapse, and there will be pinch points on the projected surface at both p and q. Furthermore, these pinch points will occur at the same point in three space. Therefore, if our projected surface has two pinch points at the same place, then the corresponding pair of points is a bitangency. In this case, we only need one projection to find a bitangency. Our projected surface has two of these double pinch points.