# Optimal triangulation algorithm for 2D concave hull

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I was avoiding posting here because there's plenty of information on triangulation out there, but I'm beginning to find it more of a curse than a blessing due to how specified my case is and how many triangulation algorithms there are.

I have 2D potentially-concave polygons that are defined as an ordered list of vertices in clockwise order. These vertices define the hull or outline of the shape - no vertices are inside the polygon. I was looking at ear-clipping, but the order of complexity seems like it could be improved with a better algorithm.

Anyone have any good resources or personal knowledge specific to my case?

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If you can get an ear-clipping algorithm to work, try it. If that's not fast enough for your purpose, try something fancier. For instance, CGAL has functions for partitioning a polygon into convex polygons, and then triangulating those is trivial: http://doc.cgal.org/latest/Partition_2/index.html

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I've implemented ear clipping. It can be done quite efficiently. My favorite resource was from David Eberly on his website. He makes annoying use of C++ templates, but the algorithm and PDF + drawings he made are really nice. https://www.geometrictools.com/Documentation/TriangulationByEarClipping.pdf

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Thanks for the input. I was thinking partitioning into convex would be faster, so maybe I'll give that a shot.

My assumptions come from this paper: https://www.geometrictools.com/Documentation/TriangulationByEarClipping.pdf

Particularly, in the first section:

Quote

The simplest algorithm, called ear clipping, is the algorithm described in this document. The order is O(n 2 ). Algorithms with better asymptotic order exist, but are more difficult to implement. Horizontal decomposition into trapezoids followed by identification of monotone polygons that are themselves triangulated is an O(n log n) algorithm [1, 3]. An improvement using an incremental randomized algorithm produces an O(n log∗ n) where log∗ n is the iterated logarithm function [5]. This function is effectively a constant for very large n that you would see in practice, so for all practical purposes the randomized method is linear time. An O(n) algorithm exists in theory [2], but is quite complicated. It appears that no implementation is publicly available

To provide a bit of context, I need this for a simple Unity plugin. Originally I was making it to aid with development of one of my game projects, but after seeing there weren't any good, simple and cheap 2D polygon tools in the asset store, I figured I'd try to optimize it and put it up for sale for a low price. For that reason, I'd like usage of the tool to be as seamless as possible. Ideally the polygons would re-triangulate their meshes every update if a vertex was moved, but if it causes any lag at all I'd rather avoid it. Additionally, I'm personally going for a 2D Sonic-esque physics simulation using polygons/edges rather than heightmaps (old school optimization for the 16-bit games), so some ramps and curves would involve a lot of vertices. Considering that, O(n^2) complexity seems a bit risky for constant triangulation updates. I was curious if anyone had any leads on the O(n) theoretical algorithm, but it sounds like a Unicorn so I'll leave it at that.

I'll go ahead with convex-partitioning, but if anyone has anything new to add please feel free to do so.

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• Here is the original blog post.
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Source files are on GitHub.
Shortcut to sterp implementation.
Shortcut to code used to generate animations in this post.
An Alternative to Slerp
Slerp, spherical linear interpolation, is an operation that interpolates from one orientation to another, using a rotational axis paired with the smallest angle possible.
Quick note: Jonathan Blow explains here how you should avoid using slerp, if normalized quaternion linear interpolation (nlerp) suffices. Long store short, nlerp is faster but does not maintain constant angular velocity, while slerp is slower but maintains constant angular velocity; use nlerp if you’re interpolating across small angles or you don’t care about constant angular velocity; use slerp if you’re interpolating across large angles and you care about constant angular velocity. But for the sake of using a more commonly known and used building block, the remaining post will only mention slerp. Replacing all following occurrences of slerp with nlerp would not change the validity of this post.
In general, slerp is considered superior over interpolating individual components of Euler angles, as the latter method usually yields orientational sways.
But, sometimes slerp might not be ideal. Look at the image below showing two different orientations of a rod. On the left is one orientation, and on the right is the resulting orientation of rotating around the axis shown as a cyan arrow, where the pivot is at one end of the rod.

If we slerp between the two orientations, this is what we get:

Mathematically, slerp takes the “shortest rotational path”. The quaternion representing the rod’s orientation travels along the shortest arc on a 4D hyper sphere. But, given the rod’s elongated appearance, the rod’s moving end seems to be deviating from the shortest arc on a 3D sphere.
My intended effect here is for the rod’s moving end to travel along the shortest arc in 3D, like this:

The difference is more obvious if we compare them side-by-side:

This is where swing-twist decomposition comes in.

Swing-Twist Decomposition
Swing-Twist decomposition is an operation that splits a rotation into two concatenated rotations, swing and twist. Given a twist axis, we would like to separate out the portion of a rotation that contributes to the twist around this axis, and what’s left behind is the remaining swing portion.
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Swing-Twist Interpolation
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And as we concatenate these two components together, we get a swing-twist interpolation that rotates the rod such that its moving end travels in the shortest arc in 3D. Again, here is a side-by-side comparison of slerp (left) and swing-twist interpolation (right):

I decided to name my swing-twist interpolation function sterp. I think it’s cool because it sounds like it belongs to the function family of lerp and slerp. Here’s to hoping that this name catches on.
And here’s my code implementation:
public static Quaternion Sterp ( Quaternion a, Quaternion b, Vector3 twistAxis, float t ) { Quaternion deltaRotation = b * Quaternion.Inverse(a); Quaternion swingFull; Quaternion twistFull; QuaternionUtil.DecomposeSwingTwist ( deltaRotation, twistAxis, out swingFull, out twistFull ); Quaternion swing = Quaternion.Slerp(Quaternion.identity, swingFull, t); Quaternion twist = Quaternion.Slerp(Quaternion.identity, twistFull, t); return twist * swing; } Proof
Lastly, let’s look at the proof for the swing-twist decomposition formulas. All that needs to be proven is that the swing component S does not contribute to any rotation around the twist axis, i.e. the rotational axis of S is orthogonal to the twist axis. Let vec{V_{R_para}} denote the parallel component of vec{V_R} to vec{V_T}, which can be obtained by projecting vec{V_R} onto vec{V_T}: vec{V_{R_para}} = proj_{vec{V_T}}(vec{V_R}) Let vec{V_{R_perp}} denote the orthogonal component of vec{V_R} to vec{V_T}: vec{V_{R_perp}} = vec{V_R} - vec{V_{R_para}} So the scalar-vector form of T becomes: T = [W_R, proj_{vec{V_T}}(vec{V_R})] = [W_R, vec{V_{R_para}}] Using the quaternion multiplication formula, here is the scalar-vector form of the swing quaternion: S = R T^{-1} = [W_R, vec{V_R}] [W_R, -vec{V_{R_para}}] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_R} + W_R (-vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R (vec{V_R} -vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}}] Take notice of the vector part of the result: vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}} This is a vector parallel to the rotational axis of S. Both vec{V_R} X(-vec{V_{R_para}}) and vec{V_{R_perp}} are orthogonal to the twist axis vec{V_T}, so we have shown that the rotational axis of S is orthogonal to the twist axis. Hence, we have proven that the formulas for S and T are valid for swing-twist decomposition. Conclusion
That’s all.
Given a twist axis, I have shown how to decompose a rotation into a swing component and a twist component.
Such decomposition can be used for swing-twist interpolation, an alternative to slerp that interpolates between two orientations, which can be useful if you’d like some point on a rotating object to travel along the shortest arc.
I like to call such interpolation sterp.
Sterp is merely an alternative to slerp, not a replacement. Also, slerp is definitely more efficient than sterp. Most of the time slerp should work just fine, but if you find unwanted orientational sway on an object’s moving end, you might want to give sterp a try.

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