# Parabolic curve with straight lines to create a self-contained shape

Honestly I don't know if this is more of a maths question than a design one, but: I have been asked to redraw this shape:

...so that it can be animated. The animation part is no problem, but I'm a bit stuck on how to actually draw the mesh itself.

Given the way the lines tend to bunch up at the edges of the curve and the way they mesh, I figured it was some kind of parabolic curve drawn with straight lines. I can easily achieve something like this:

...and if I rotate the original image 45 degrees and overlay it, the curve I've come up with is pretty close to at least the initial 'waist' of the mesh; but that's still not a complete self-contained shape, and I'm kind of lost on how to achieve that last much more tapered part (going from right to left).

I've seen lots of art done with parabolic curves like this but it's usually shapes made out of the negative space rather than the intersecting lines, but in this case I need it all to be the one shape.

Any ideas on how to draw something like this - ideally with straight lines?

Try a polygon model of a revolution surface. The next example of that idea has low polygon count to keep the image sparse:

As a wireframe it's this:

It's turned to vertical position, the projection is parallel (=no perspective), the view was saved as PDF, opened in Illustrator and all horizontal and vertical lines are manually deleted:

This has nothing elegant, but at least it resembles the wanted pattern.

The revolution surface above was made in a CAD program. The revolved curve was originally a smooth spline. The surface was converted to STL-file with low resolution to get a polygon mesh.

ADD: There's a comment which suggests mapping a curve pattern along the surface. It needs much more advanced software than revolving a spline and converting the surface to a polygon mesh. I actually tried it in Illustrator with 3D revolve + art mapping, but the curves were expanded without asking to filled areas and became non-uniformly wide. The next method keeps the pattern as curves.

Making it as a 2D shape from scratch in Illustrator:

There's a rectangle and a diagonal line is drawn into it. The line is bent to approximate half-cycle sine curve. The node handles are stretched with the anchor type tool horizontally.

In the next image the sine curve is duplicated and both copies are moved to the corners of the rectangle:

The curves snap perfectly if you have smart guides and snap to point ON.

Make blend between the sine curves. I inserted 20 intermediate copies. The result is quite sparse, but easy to see:

Expand and Ungroup the blend to make the curves free. Select all and delete everything which is outside the rectangle with the shape builder tool (hold Alt for delete). It's done in the left in the next image:

In the right half all sine curves are selected and a flipped copy is made (=Object > Transform > reflect > Vertical > Copy). The rectangle is also colored to red.

Select all. Apply Object > Envelope distortion > Make with Mesh. Have 1 row and 2 columns in the mesh. Drag the mesh nodes and handles with the direct selection tool:

This IS tricky. It may be easier to envelope distort with Warp. Use preset "Bulge" with negative expansion. Adjust the mesh nodes and handles as needed.

You may want more room to adjust. More columns in the envelope distortion mesh give it, but things become very soon unmanageable because every mesh node has several handles which all should be right. One must draw some help lines to be able to adjust handles symmetrically in symmetric shape.

Expand the envelope distortion if you want the curves free for individual colors or other edits.

• You could skin it, inside & out with a regular alpha'd mesh pattern, tiled, then it would transform with the model. Jun 12, 2021 at 10:13
• This is an absolutely incredible answer, thank you! 🥳 I'll give it a go on Monday once I'm back at my desk. Jun 12, 2021 at 19:51