Real projective plane: Geometry of the double line and the centre pinch point

Consider the cross cap representation of $\mathbb{RP}^2$ where the line AB is the double line corresponding to the Whiteney's umbrella Consider the followin injective mapping which embed $\mathbb{RP}^2$ into $\mathbb{R}^4$ (source of $f$ found here) hence remove all self intersections

$$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$

The following youtube animation which rotates the embedded cross cap presumably in the xw plane suggests line AB is actually a circle Therefore what looks like a pinch point B and double line AB in the 3D projection of the cross cap is actually a circle partially pointing in 4D.

However, because the animation have not explored other rotations, it is hard to deduce the nature of the pinch point A.

Using the answer from this link, line AB corresponds to the Whiteney's umbrella region of the cross cap, which when understood in a 4D perspective, the "unusual line intersection" can be reasoned as a parabola orienting in the yw plane and hence there is no actual intersection in 4D.

However, as the gluing continues, since it is required to glue opposite edges of the whiteney's umbrella together, it seems there is no way to prevent the aforementioend yw parabola to become a yw circle and hence suggesting the two surfaces do touch at the origin (pinch point A). I then suspect the actual geometry of the pinch point A, if the neighbourhood is rotated slightly and the result projected back to 3D will be one of the following: The left scenario will suggest even if the surface don't intersect itself in 4D, it still need to touch at the origin to complete the gluing process, thus suggesting A is indeed some kind of double point. Therefore the region near the origin would trace out an actual figure 8 loop.

The middle scenario is perhaps the most interesting, as having the twist right at the middle, it suggests the origin is possibly a saddle point. The figure 8 loop seen in the cross cap 3D projection can then be explained as an artifact due to the projection process mapping distinct points along the w axis to the same point in xyz.

But with the lack of experience on how to probe this from the embedding equations (because the form of the equations there suggest I am dealing with some kind of hyperboloids, and it is unclear how I can mathematically take a section of the neighbourhood near the origin or even the circle-line AB), I am unsure of the geometry near the origin, hence the actual nature of the pinch point A.

Main: What is the nature of the pinch point A, is it really some kind of double point even in 4D?

Math approach: What direction should I try in order to choose a good section from the embedding equations to mathematically investigate the region near the origin and circle-line AB in order to learn how the surface twists there in 4D?

• Where did the function f come from? As written, it is not even a map of the projective plane. – Moishe Kohan Jan 9 '17 at 12:11
• I have updated the question: It's taken from here: math.stackexchange.com/questions/886616/… – Secret Jan 9 '17 at 12:54
• Briefly, referring to your first diagram: The circle $AB$ looks just like the circle $AC$, and the center pinch point $A$ is a saddle for "height" measured along the axis $CB$, but $A$ is a smooth point since the projective plane is a manifold. – Andrew D. Hwang Jan 9 '17 at 13:16