I am currently learning a little bit about algebraic groups and quotients of varieties with J. Harris book "Algebraic Geometry" (http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Harris_AlgebraicGeometry.pdf). Here, in example 10.21 (page 125), he shows that the quotient of the affine plane $\mathbb{A}^2$ under the group action $(x,y) \mapsto (-x,-y)$ is a quadric cone in $\mathbb{A}^3$. I understand, why that is the case: The invariant ring $K[X,Y]^G$ is isomorphic to $K[X,Y,Z] / (X^2 + Y^2 - Z^2)$, so the quotient is given by the equation $X^2 + Y^2 = Z^2$ which is precisely the equation of a quadric cone. However, there is also this picture in the book:
Sadly, there is no further explanation to this picture, especially it's not clear for me, what the diagonal lines should indicate. Then a thought came to my mind: What if it's possible to turn the affine plane into the quadric cone in a intuitive way, by cutting the affine plane into pieces, and glueing those pieces together to end up with a cone, where a point on the cone uniquely corresponds to a pair of points $a,b$ (or one point if we are at the origin), s.t. $a=(x,y), b=(-x,-y)$. Maybe these diagonal lines are the lines where we cut the affine plane into pieces. I tried it with a paper: I put the origin into the middle of the paper and cutted along the diagonals, getting 4 triangles. Now we just take any triangle, for example the upper triangle, and glue it onto the triangle on the opposite site, so the lower triangle, s.t. points $(x,y)$ and $(-x,-y)$ get glued together. Now we glue the left triangle onto the right triangle in the same way, ending up with 2 triangles. These two triangles can be glued together at their edges, again we only glue those edges together that represent the same points.
This way, we end up with a a "mono-cone" (a cone with only one nappe), and each point of this mono-cone corresponds uniquely to a pair of points $(x,y), (-x,-y)$, except at the origin. However, we wanted to get a cone, and not just a mono-cone. So my question is: Is it possible to do some kind of similar construction but end up with a cone instead? I tried cutting the plane into 8 pieces, by additionally cutting along the axes. We again glue our 8 triangles together and end up with 4 triangles, now my idea was that we could basically glue two triangles respectively together at their edges and end up with a cone, however this is not really possible because we would have to glue edges together that do not represent the same points, so a point on our cone could then actually have 4 corresponding points on the plane, so this idea does not really work. Is there some intuitive way to do it?
