Constant Negative Curvature - Tumblr Posts

I know I have drawn this shape so often already, but the process of drawing it is so soothing.

And for that, I have drawn a kind of step-by-step guide how to draw that shape in the top of this drawing:

(from left to right: ) [sorry in advance if I make it sound more complicated than it actually is. If you want to draw it, I would advice you to focus more on these illustrations rather than on my gibberish-text.]

1. draw a 2-dimensional Cartesian plane - or, in other words: just draw a cross like depicted

1.1. mark 2 points on the y-axis/vertical line with same distance to the coordinate origin, then mark 2 points on the x-axis/horizontal line with the same distance to the coordinate origin. (The markings on the y-axis need to be farther away from the origin than the markings of the x-axis)

2. connect the 4 marked points like depicted above. This is a function plot of a tractrix. (it has two mirror symmetry axes. )

3. draw an ellipse and connect the two markings on the x-axis. This becomes a kind of "belt" for the pseudosphere (4th picture)

4. part the ellipse into whatever-amount-you-want of partings (like you would cut a cake) and slightly mark these.

5. now imagine you cut that shape horizontally on the outer surface. (In the 5th picture I depicted that with red-ish pen across the pseudosphere. ) -

6. then the cut shape needs to be "(shape) shifted". For that we use a set of marked points we did in step 4). Furtherly, we "cut" the ellipse open, and push one end of it to the top, and the other end to the bottom. (depicted in picture 6 )

7. Then we connect the rest to get that shape: