Uneven Profile Curve Thickness with Geometry Nodes Curve to Mesh?

Question

I've been trying to build a Geometry Node group that in part is able to create a mesh of even thickness around a three-dimensional curve that features sharp corners.

After much research I'm aware that Blender is unable to do this with it's own Curve Profile system due to flaws in how it handles three-dimensional curves, but is there a way this can be solved in some basic way through Geometry Nodes and something like a Mesh Extrusion node by providing it a correct Offset for each corner?

I'm not great at this kind of mathematics and I don't want to be stuck in a Pythagoras maths hole for a week to find the missing link here. One Stack Overflow post mentioned something involving calculating the Secant but they never explained why and I wasn't able to replicate their math.

Left is what I want, middle is what I get with my current Geometry Nodes stack with just some sharp corners, right is an approximation of the curve being used.

Answer 1

Unfortunately, the problem cannot be solved without mathematics, but with this example you should still be able to achieve the desired result:

Roughly speaking, you would just have to calculate the scaling factor for the cross section at your points and apply it as offset to the mesh.

1. First you need the direction vector between two points.

   To be able to work on cyclic curves with this setup, the math node Wrap helps you to find the right index of the points.

   Additionally, if you are working on multiple curves at the same time, the index value of the point where the individual curves start is added here. You achieve this with the node Accumulate Field.

   This value is then the index of the next point within a spline, which you feed directly into the node Field at Index.

   To get the direction vector simply subtract the position of the current point from the position of the next point.

2. Then you would have to calculate the angle between two edges/lines.

   With the help of the math node Arccosine you can get the angle from the previously calculated direction vector and the normals of the points which you get directly from the Curve to Points node.

   In fact, this result is then the angle bisector between two edges, which I call here $\theta$.

3. After you have applied your Curve to Mesh, you only have to use the Set Position node and offset the individual points of the mesh with a certain scaling factor.

   You get the factor of scaling with the formula $\vec{a} \circ \vec{b} \times(\frac{1}{\sin(\theta)}-1)$.

   Where $\theta$ is the bisected angle between two edges/lines, $\vec{a}$ is the normal of the curve point and $\vec{b}$ is the direction vector between the points of the generated mesh and the original curve point.

   In order to get a correct result even for curves that are not cyclic, create a selection beforehand that omits the first and last points depending on the value of Is Spline Cyclic.

The result should then look like this:

_{(Blender 3.1+)}