How can I scale a 2D shape with constant thickness inwards

Question

Suppose I have a mesh or a simple curve consisting of four (or some more) points.

Now I want to scale them inward with constant distance to the edges.

This works fine (with a little effort) under some circumstances.

But when the scaling reaches a certain level, the points start to protrude into each other. This is especially noticeable with curves that consist of several points, or that do not contain straight lines, as well as points that are close together.

My goal would be the following (simplified sketched):

On the left is the starting point, in the middle a scaling with constant strength, and on the right the result I would like to have as soon as the distance is a little higher.

Clearly, the inner curve in this example would then have to consist of only three points instead of four, which is absolutely OK.

Only how do I get to this result in the first place?

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The solution here helped me, but causes the points to protrude into each other from a certain curvature or thickness:
Solving Uneven Profile Curve Thickness with Mesh Extrusions?

This solution is also not uninteresting, but much too computational in this case:
How to displace a curve without introducing artifacts at tight turns?

I am looking for a simple mathematical solution for a relatively simple shape on the X/Y plane (Geometry Nodes only!)

Answer 1

You could try breaking this one.. it Mitres a 2D Mesh-Line to a given width:

It:

• Converts one branch to curve, so radius can be set
• Extrudes 2D mesh vertically, to get edge angle
• Wrangles indices, and tranfers edge-angle from extrusion edge to curve point
• Does trig on the edge angle to get mitre-joint length (curve radius)
• Converts the curve back to a mesh

Answer 2

As already mentioned in the comments, I don't consider this solution to be less computational and probably even not less compuationally intensive. But hopefully it makes you happy, as it does not include any Geometry Proximity ;-)

The general idea is, to build up the inset of quads, create the union, triangulate it and then remove all inner vertices.

Depending on the shape of the polygon, the inset may result in geometry outside of the polygon. Thus I implemented the Point-in-Polygon algorithm according to Weiler, Kevin (1994), "An Incremental Angle Point in Polygon Test", in Heckbert, Paul S. (ed.), Graphics Gems IV, San Diego, CA, USA: Academic Press Professional, Inc., pp. 16–23, ISBN 0-12-336155-9. Applying this to all points leads to a complexity of O(n²) where n is the number of points of the polygon. I use this for removing the original polygon from the inset as well.

In some cases I got some artifacts inside the polygon near the outline of the polygon. This is, why I remove points with less than 2 neighbouring vertices at the end. I did not test the solution with many different shapes. So I don't know, if it will work in every case.