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I have been trying to understand "Fast Simulation of Mass-Spring Systems" by Liu et al., which can be found here. There is one part in the methods section that is confusing me. Just after equation 12, it states:

Similarly, Si ∈ Rs is the i-th spring indicator, i.e., Si,j = δi,j

The subscript i is for each spring in the system. But I am not sure what δ is supposed to be - it isn't mentioned anywhere else in the paper. I also don't know what the subscript j is for, or what a spring indicator is. My guess was that S is a diagonal matrix, with the deflection of each spring along the diagonal. But this doesn't work in my code. Having S as just the identity matrix works better.

Does anyone know what the S matrix is supposed to be?

Vermillion
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1 Answers1

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The notation $\delta_{i,j}$ is the Kronecker delta, a notation commonly used in physics. It's defined as: $$\delta_{i,j} \equiv \begin{cases}1,&i=j\\0,&i\neq j\end{cases}$$ So, as you suspected, it's essentially a shorthand for the identity matrix.

The notation $\mathbf{S}_i \in \mathrm{R}^s$ means that each $\mathbf{S}_i$ is an $s$-dimensional vector, so the index $j$ labels the components of this vector. The term "indicator" probably refers to the concept of an indicator function, which is a function that labels some set or point with the value 1 while mapping everything else to 0. So, the "$i$-th spring indicator", in this context, is a vector that has a 1 in the $i$-th component and 0 everywhere else. That's exactly the Kronecker delta, i.e. the identity matrix.

The way this "indicator vector" gets used in equation 12 in the paper is in the form of an outer product with another vector, $\mathbf{A}_i \in \mathrm{R}^m$. The two vectors are multiplied together in the form $\mathbf{A}_i \mathbf{S}_i^{\mathrm{T}}$, which is an outer product. The result is an $m \times s$ matrix with the contents of $\mathbf{A}_i$ in the $i$-th column, and zeroes everywhere else.

Nathan Reed
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