Linear Blend Skinning

A rigged character is really two things glued together: a skeleton of joints that an animator poses, and a skin — a single smooth mesh of thousands of vertices — that has to follow the bones without tearing at the seams. The question every real-time engine must answer, sixty times a second for every vertex, is deceptively simple: when the elbow bends, where does this particular skin vertex go?

The industry's default answer, running in nearly every game and phone on Earth, is Linear Blend Skinning (LBS) — also called skeletal subspace deformation or just smooth skinning. It is beautifully cheap, it fits a GPU vertex shader perfectly, and it has one famous, ugly flaw that has a nickname: the candy-wrapper collapse. This page builds the method, works its collapse out on paper, and shows you exactly why a linear blend of rigid motions refuses to stay rigid.

One vertex, several bones, a weighted average

In the bind pose the skin sits in its neutral rest shape and each vertex is painted with a few skin weights — numbers w_j saying how much joint j owns this vertex. A vertex deep in the forearm is owned entirely by the forearm bone (w = 1); a vertex right over the elbow crease has its ownership split between the upper-arm and forearm bones (say w = 0.5 each). The weights on any vertex always sum to one.

LBS then does the obvious thing. It transforms the rest vertex by each owning bone as if that bone owned it completely, and takes the weighted average of the results:

The product M_j B_j^{-1} is a single skinning matrix per joint that the engine computes once per frame and stores in a small matrix palette. Per vertex the whole cost is then: fetch two-to-four matrices, scale each by its weight, add them, multiply the vertex once. No trig, no branches — a straight weighted sum of matrices, which is why it flies on hardware.

Cost, and nothing but cost. A modern character mesh has tens of thousands of vertices and the skinning must finish inside a couple of milliseconds. LBS needs only a handful of multiply-adds per vertex, maps directly onto a GPU vertex shader, and the expensive part — building each M_j B_j^{-1} — happens just once per bone per frame, not once per vertex. Cap the influences at four bones per vertex and you can pack the weights into a single vertex attribute. Decades of "good enough" motion have been shipped on exactly this arithmetic; the fancier methods on this course all exist to fix the specific places where this weighted average goes wrong.

Watch the collapse happen

Below is a two-bone arm: a fixed upper arm and a forearm you can bend with the slider. The skin is drawn as an outline of vertices. The highlighted vertices near the elbow are the shared ones, split half-and-half between the two bones; the rest are owned rigidly by a single bone. Drag the bend toward a sharp angle and watch the inner elbow: the shared vertices do not sweep cleanly around the corner — they migrate inward, pinching the surface and losing volume. Push it toward a full fold and the crease nearly vanishes.

The rigidly-owned vertices (unhighlighted) travel perfectly with their bone — no distortion at all. Every bit of the trouble lives in the blended band, and it gets worse the more the two bones disagree about where the vertex should go.

Worked example: the 90° elbow that shrinks

Take a vertex sitting right on the elbow, a distance r = 0.6 straight out from the joint, shared equally (w = 0.5) between the upper-arm bone (which does not move here — its transform is the identity I) and the forearm bone, which rotates by 90^\circ about the joint. Because both bones share the same pivot, the only thing that differs between them is the rotation, so the blended transform acting on the vertex's offset from the joint is

M \;=\; 0.5\,I \;+\; 0.5\,R(90^\circ) \;=\; 0.5\begin{pmatrix}1&0\\0&1\end{pmatrix} + 0.5\begin{pmatrix}0&-1\\1&0\end{pmatrix} = \begin{pmatrix}0.5&-0.5\\0.5&0.5\end{pmatrix}.

Is M a rotation? A rotation has determinant 1. Here \det M = (0.5)(0.5) - (-0.5)(0.5) = 0.5 — not one. In fact M factors cleanly:

M \;=\; \tfrac{1}{\sqrt2}\,R(45^\circ) \;=\; 0.707\times(\text{rotation by }45^\circ).

So the blended transform is a rotation by 45^\circ (half of ninety — sensible) but with a uniform shrink to 0.707 baked in. The vertex that lived 0.6 from the joint now lands only 0.6 \times 0.707 \approx 0.424 from it: the surface has been sucked 29% inward. That inward suck, all around the joint, is the elbow pinch. The general rule falls straight out — the equal-weight blend 0.5\,(I + R(\theta)) equals \cos(\theta/2)\,R(\theta/2), a half-angle rotation scaled by \cos(\theta/2). The scale is 1 only at \theta = 0; it falls the more the bones disagree.

The candy-wrapper twist

The pinch turns catastrophic at a twist. Imagine the wrist rotating about the length of the forearm while the forearm itself stays put — the two bones sharing the wrist vertices are then two rotations almost 180^\circ apart. Plug \theta = 180^\circ into the blend:

0.5\,(I + R(180^\circ)) \;=\; 0.5\,(I - I) \;=\; \mathbf{0}, \qquad \cos(90^\circ) = 0.

The blended matrix is the zero matrix: it maps the entire shared band onto the single joint point. The skin collapses to a thin pinched neck — exactly the twisted-shut look of a sweet wrapper, which is where the nickname comes from. Anywhere between 0^\circ and 180^\circ you get partial shrink; at the full half-turn you get total collapse, because averaging a rotation with its opposite gives nothing at all.

Here is the misconception that causes every one of these bugs. LBS looks like it is averaging two motions, so it feels like the answer should be "half-way between the two poses" — and a half-way pose ought to be a perfectly good rigid pose. But that is not what the formula does. It averages the two matrices, entry by entry, and the set of rotation matrices is curved — it is not closed under straight-line averaging. Take two rotations, blend their matrices linearly, and the result leaves the rotation group entirely: its determinant drops below one, so it necessarily shrinks. That single fact — a linear blend of rigid transforms is not itself rigid — is the whole story behind the elbow pinch and the candy-wrapper. So: