# TPMS2STEP Supplements: Constraints Matrices and Convergence Proof of
TPMS2STEP

Yaonaiming Zhao  Qiang Zou

###### Abstract

Triply periodic minimal surface (TPMS) is emerging as an important way of
designing microstructures. However, there has been limited use of commercial
CAD/CAM/CAE software packages for TPMS design and manufacturing. This is
mainly because TPMS is consistently described in the functional representation
(F-rep) format, while modern CAD/CAM/CAE tools are built upon the boundary
representation (B-rep) format. One possible solution to this gap is
translating TPMS to STEP, which is the standard data exchange format of
CAD/CAM/CAE. Following this direction, this paper proposes a new translation
method with error-controlling and $C^{2}$ continuity-preserving features. It
is based on an approximation error-driven TPMS sampling algorithm and a
constrained-PIA algorithm. The sampling algorithm controls the deviation
between the original and translated models. With it, an error bound of
$2\epsilon$ on the deviation can be ensured if two conditions called
$\epsilon$-density and $\epsilon$-approximation are satisfied. The
constrained-PIA algorithm enforces $C^{2}$ continuity constraints during TPMS
approximation, and meanwhile attaining high efficiency. A theoretical
convergence proof of this algorithm is also given. The effectiveness of the
translation method has been demonstrated by a series of examples and
comparisons. All relevant Refs.: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]

##  1 Part of constraint matrices for Gyroid, Diamond, and Schwarz_P in CPIA

The constraint matrices for Gyroid, Diamond, and Schwarz_P that are not given
in the article are given here.

For Gyroid, matrices $\mathbf{T}_{1g}$-$\mathbf{T}_{7g}$ and
$\mathbf{M}_{ijg}(i=1,2,3,4,5$ and $j=1,2,3,4)$ are given.

$\displaystyle\mathbf{T}_{1g}=\begin{bmatrix}0&0&-1&1\\\ 1&0&0&0\\\ 0&1&0&0\\\
0&0&0&1\\\ \end{bmatrix}$
$\displaystyle\mathbf{T}_{2g}=\begin{bmatrix}0&0&1&1\\\ 1&0&0&-1\\\
0&-1&0&1\\\ 0&0&0&1\\\ \end{bmatrix}$
$\displaystyle\mathbf{T}_{3g}=\begin{bmatrix}0&-1&0&1.5\\\ 1&0&0&-0.5\\\
0&0&-1&0.5\\\ 0&0&0&1\\\ \end{bmatrix}$
$\displaystyle\mathbf{T}_{4g}=\mathbf{T}_{1g}\mathbf{T}_{3g}$
$\displaystyle\mathbf{T}_{5g}=\mathbf{T}_{4g}^{-1}$
$\displaystyle\mathbf{T}_{6g}=\mathbf{T}_{2g}\mathbf{T}_{3g}$
$\displaystyle\mathbf{T}_{7g}=\mathbf{T}_{6g}^{-1}$

The matrices $\mathbf{M}_{ijg}(i=1,2,3,4,5$ and $j=1,2,3,4)$ are derived from
the method mentioned below. To extract the control points and calculate the
first-order and second-order derivatives, three steps are conducted: (1)
number the control points; (2) map a specific part of the control points to
the corresponding control points (e.g., to calculate the first-order
derivatives, the second row of control points are mapped to the first row of
control points); and (3) utilize the coordinates of these points to calculate
the local first-order and second-order derivatives. In Step 2, the mapping is
constructed using a matrix, denoted as $\mathbf{M}$. $\mathbf{M}$ is actually
one of $\mathbf{M}_{ijg}(i=1,2,3,4,5$ and $j=1,2,3,4)$. Specifically, if the
matrix $\mathbf{P}$ is the control points, then $\mathbf{MP}$ sets some rows
in $\mathbf{P}$ to 0 and changes the position of other rows of elements. In
the matrix $\mathbf{MP}$, a non-zero element at Row i and Column j in
$\mathbf{M}$ repositions the jth line of $\mathbf{P}$ to the ith line of
$\mathbf{P}$. That is, this element maps the jth control point to the ith
control point. After the mapping, calculations can be done to the control
points that are mapped together.

$\mathbf{M}_{ijg}(i=1,2,3,4,5$ and $j=1,2,3,4)$ are given as:

$\displaystyle\mathbf{M}_{11g}=\sum_{i=0}^{n-5}\mathbf{J}_{n^{2}-n+2+i,n^{2}-2n+2+i}$
$\displaystyle\mathbf{M}_{21g}=\sum_{i=0}^{n-5}\mathbf{J}_{n^{2}-2n+2+i,n^{2}-2n+2+i}$
$\displaystyle\mathbf{M}_{31g}=\sum_{i=0}^{n-7}\mathbf{J}_{n^{2}-n+3+i,n^{2}-3n+3+i}$
$\displaystyle\mathbf{M}_{41g}=\sum_{i=0}^{n-7}\mathbf{J}_{n^{2}-2n+3+i,n^{2}-3n+3+i}$
$\displaystyle\mathbf{M}_{51g}=\sum_{i=0}^{n-7}\mathbf{J}_{n^{2}-3n+3+i,n^{2}-3n+3+i}$

where $\mathbf{J}_{i,j}$ is the matrix with a unique non-zero element $1$ at
the position $(i,j)$ and n is the number of control points of the
approximation surface in the u and v direction.

$\displaystyle\mathbf{M}_{12g}=\sum_{i=0}^{n-5}\mathbf{J}_{2n+in,2n+1+in}$
$\displaystyle\mathbf{M}_{22g}=\sum_{i=0}^{n-5}\mathbf{J}_{2n+1+in,2n+1+in}$
$\displaystyle\mathbf{M}_{32g}=\sum_{i=0}^{n-7}\mathbf{J}_{3n+in,3n+2+in}$
$\displaystyle\mathbf{M}_{42g}=\sum_{i=0}^{n-7}\mathbf{J}_{3n+1+in,3n+2+in}$
$\displaystyle\mathbf{M}_{52g}=\sum_{i=0}^{n-7}\mathbf{J}_{3n+2+in,3n+2+in}$
$\displaystyle\mathbf{M}_{13g}=\sum_{i=0}^{n-3}\mathbf{J}_{2+i,n+2+i}$
$\displaystyle\mathbf{M}_{23g}=\sum_{i=0}^{n-5}\mathbf{J}_{n+2+i,n+2+i}$
$\displaystyle\mathbf{M}_{33g}=\sum_{i=0}^{n-7}\mathbf{J}_{3+i,2n+3+i}$
$\displaystyle\mathbf{M}_{43g}=\sum_{i=0}^{n-7}\mathbf{J}_{n+3+i,2n+3+i}$
$\displaystyle\mathbf{M}_{53g}=\sum_{i=0}^{n-7}\mathbf{J}_{2n+3+i,2n+3+i}$
$\displaystyle\mathbf{M}_{14g}=\sum_{i=0}^{n-5}\mathbf{J}_{3n-1+in,3n-2+in}$
$\displaystyle\mathbf{M}_{24g}=\sum_{i=0}^{n-5}\mathbf{J}_{3n-2+in,3n-2+in}$
$\displaystyle\mathbf{M}_{34g}=\sum_{i=0}^{n-7}\mathbf{J}_{4n-1+in,4n-3+in}$
$\displaystyle\mathbf{M}_{44g}=\sum_{i=0}^{n-7}\mathbf{J}_{4n-2+in,4n-3+in}$
$\displaystyle\mathbf{M}_{54g}=\sum_{i=0}^{n-7}\mathbf{J}_{4n-3+in,4n-3+in}$

For Diamond, matrices $\mathbf{T}_{1d}$, $\mathbf{T}_{2d}$, $\mathbf{N}_{1}$,
and $\mathbf{N}_{2}$ are given. $\mathbf{M}_{ijd}$ $(i=1,2,3,4,5$ and
$j=1,2,3,4)$ are the same as $\mathbf{M}_{ijg}$ $(i=1,2,3,4,5$ and
$j=1,2,3,4)$ mentioned before, respectively.

$\displaystyle\mathbf{T}_{1d}=\begin{bmatrix}0&-1&0&0\\\ 1&0&0&0\\\
0&0&-1&0\\\ 0&0&0&1\end{bmatrix}$
$\displaystyle\mathbf{T}_{2d}=\begin{bmatrix}-\frac{1}{2}&-\frac{1}{2}&\frac{\sqrt[]{2}}{2}&\frac{\sqrt[]{2}}{2}\\\
-\frac{1}{2}&-\frac{1}{2}&-\frac{\sqrt[]{2}}{2}&\frac{\sqrt[]{2}}{2}\\\
\frac{\sqrt[]{2}}{2}&-\frac{\sqrt[]{2}}{2}&0&0\\\ 0&0&0&1\end{bmatrix}$
$\displaystyle\mathbf{N}_{1}$
$\displaystyle=\begin{bmatrix}\mathbf{I}_{n^{2}\times
n^{2}}&\mathbf{0}_{n^{2}\times n^{2}}\\\ \mathbf{0}_{n^{2}\times
n^{2}}&\mathbf{0}_{n^{2}\times n^{2}}\end{bmatrix}_{2n^{2}\times 2n^{2}}$
$\displaystyle\mathbf{N}_{2}$
$\displaystyle=\begin{bmatrix}\mathbf{0}_{n^{2}\times
n^{2}}&\mathbf{I}_{n^{2}\times n^{2}}\\\ \mathbf{0}_{n^{2}\times
n^{2}}&\mathbf{0}_{n^{2}\times n^{2}}\end{bmatrix}_{2n^{2}\times 2n^{2}}$

where n is the number of control points of the approximation surface in the u
and v direction. $\mathbf{I}$ is the identity matrix. $\mathbf{N}_{1}$ and
$\mathbf{N}_{2}$ are designed to satisfy the equations
$\mathbf{N}_{1}\mathbf{P}=\begin{pmatrix}\mathbf{P}_{1}\\\
\mathbf{0}\end{pmatrix}$ and
$\mathbf{N}_{2}\mathbf{P}=\begin{pmatrix}\mathbf{P}_{2}\\\
\mathbf{0}\end{pmatrix}$ where $\mathbf{P}=\begin{pmatrix}\mathbf{P}_{1}\\\
\mathbf{P}_{2}\end{pmatrix}$.

For Schwarz_P, matrices $\mathbf{T}_{1p}$-$\mathbf{T}_{4p}$ are given.
$\mathbf{M}_{ijp}$ $(i=1,2,3,4,5$ and $j=1,2,3,4)$ are the same as
$\mathbf{M}_{ijg}$ $(i=1,2,3,4,5$ and $j=1,2,3,4)$ mentioned above.

$\displaystyle\mathbf{T}_{1p}=\begin{bmatrix}\frac{1}{2}&-\frac{1}{2}&-\frac{\sqrt[]{2}}{2}&-\frac{\sqrt[]{2}}{4}\\\
-\frac{1}{2}&\frac{1}{2}&-\frac{\sqrt[]{2}}{2}&-\frac{\sqrt[]{2}}{4}\\\
-\frac{\sqrt[]{2}}{2}&-\frac{\sqrt[]{2}}{2}&0&-\frac{1}{2}\\\
0&0&0&1\end{bmatrix}$
$\displaystyle\mathbf{T}_{2p}=\begin{bmatrix}\frac{1}{2}&\frac{1}{2}&-\frac{\sqrt[]{2}}{2}&\frac{\sqrt[]{2}}{4}\\\
\frac{1}{2}&\frac{1}{2}&\frac{\sqrt[]{2}}{2}&-\frac{\sqrt[]{2}}{4}\\\
-\frac{\sqrt[]{2}}{2}&\frac{\sqrt[]{2}}{2}&0&\frac{1}{2}\\\
0&0&0&1\end{bmatrix}$ $\displaystyle\mathbf{T}_{3p}=\begin{bmatrix}0&-1&0&0\\\
-1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{bmatrix}$
$\displaystyle\mathbf{T}_{4p}=\begin{bmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix}$

##  2 Convergence proofs for Diamond and Schwarz_P TPMS

The following sections include the convergence proofs for Diamond and
Schwarz_P TPMS.

###### Proposition 1

The CPIA iterative method for Diamond is convergent and the limit surface is
the least-square fitting outcome of the initial data
$\mathrm{\\{\mathbf{Q}_{ij}\\}^{m_{1},m_{2}}_{i=0,j=0}}$.

Proof. As the result of the iterative procedure of CPIA, two sequences of the
control points of the offset surface for Diamond
$\mathrm{\\{\mathbf{P}_{1}^{k}(u,v),k=0,1,\cdots\\}}$ and
$\mathrm{\\{\mathbf{P}_{2}^{k}(u,v),k=0,1,\cdots\\}}$ are generated. To show
their convergence, let
$\mathbf{P}^{k}=\\{\mathbf{P}_{0}^{k},\mathbf{P}_{1}^{k},\cdots,\mathbf{P}_{n}^{k}\\}^{T}$
and
$\mathbf{Q}=\\{\mathbf{Q}_{0},\mathbf{Q}_{1},\cdots,\mathbf{Q}_{m}\\}^{T}$.
Here $\mathbf{P}=\begin{pmatrix}\mathbf{P}_{1}\\\ \mathbf{P}_{2}\end{pmatrix}$
and $\mathbf{Q}=\begin{pmatrix}\mathbf{Q}_{1}\\\ \mathbf{Q}_{2}\end{pmatrix}$.

In the (k+1)th iteration, we can derive the coordinates of the newly adjusted
control points as:

$\displaystyle\begin{aligned}
\mathbf{P}^{k+1}&=\mathbf{P}^{k}+(\mathbf{Q-BwP}^{k})+\frac{1}{2}\sum_{i=1}^{2}((\mathbf{M}_{24d}-\mathbf{M}_{14d})\mathbf{N}_{i}\mathbf{P}^{k}\mathbf{T}_{1d}-(\mathbf{M}_{21d}-\mathbf{M}_{11d})\mathbf{N}_{3-i}\mathbf{P}^{k})\\\
&+\frac{1}{2}\sum_{i=1}^{2}((3\mathbf{M}_{44d}-2\mathbf{M}_{34d}-\mathbf{M}_{54d})\mathbf{N}_{i}\mathbf{P}^{k}\mathbf{T}_{1d}-(3\mathbf{M}_{41d}-2\mathbf{M}_{31d}-\mathbf{M}_{51d})\mathbf{N}_{3-i}\mathbf{P}^{k})\\\
&+\frac{1}{2}\sum_{i=1}^{2}((\mathbf{M}_{23d}-\mathbf{M}_{13d})\mathbf{N}_{i}\mathbf{P}^{k}\mathbf{T}_{2d}-(\mathbf{M}_{22d}-\mathbf{M}_{12d})\mathbf{N}_{3-i}\mathbf{P}^{k})\\\
&+\frac{1}{2}\sum_{i=1}^{2}((3\mathbf{M}_{43d}-2\mathbf{M}_{33d}-\mathbf{M}_{53d})\mathbf{N}_{i}\mathbf{P}^{k}\mathbf{T}_{2d}-(3\mathbf{M}_{42d}-2\mathbf{M}_{32d}-\mathbf{M}_{52d})\mathbf{N}_{3-i}\mathbf{P}^{k})\end{aligned}$
(1)

The matrices $\mathbf{T}_{id}(i=1,2)$ are given in Sec. 1 and they are
invertible matrices where the absolute values of the eigenvalues are all 1,
i.e., $\left|\lambda_{i}(\mathbf{T}_{jd})\right|=1,i=1,2,3,4$ and $j=1,2$.
$\mathbf{B}$ refers to the B-spline basis function matrix and $\mathbf{w}$ is
the weight matrix. The matrices $\mathbf{M}_{ijd}$ $(i=1,2,3,4,5$ and
$j=1,2,3,4)$, $\mathbf{N}_{1}$, and $\mathbf{N}_{2}$ are also given in Sec. 1.
To simplify the form of the equations, let
$\alpha_{i}=3\mathbf{M}_{4id}-2\mathbf{M}_{3id}-\mathbf{M}_{5id}$,
$\beta_{i}=\mathbf{M}_{2id}-\mathbf{M}_{1id}(i=1,2,3,4)$,
$\mathbf{D}=\mathbf{I}-\mathbf{Bw}$, and
$\mathbf{r}^{k}=\mathbf{P}^{k}-\mathbf{w}^{-1}\mathbf{B}^{-1}\mathbf{Q}$. Then
Eq. 1 can be transformed into a new form:

$\displaystyle\begin{aligned}
\mathbf{r}^{k+1}=&(\mathbf{D}-\frac{1}{2}(\alpha_{1}+\beta_{1}+\alpha_{2}+\beta_{2})(\mathbf{N}_{1}+\mathbf{N}_{2}))\mathbf{r}^{k}+\frac{1}{2}(\alpha_{4}+\beta_{4})(\mathbf{N}_{1}+\mathbf{N}_{2})\mathbf{r}^{k}\mathbf{T}_{1d}+\frac{1}{2}(\alpha_{3}+\beta_{3})(\mathbf{N}_{1}+\mathbf{N}_{2})\mathbf{r}^{k}\mathbf{T}_{2d}\\\
&=\cdots\\\
&=\sum_{i_{1}=0}^{k+1}\sum_{i_{2}=0}^{k+1}\sum_{i_{3}=0}^{k+1}{k+1\choose
i_{1},i_{2},i_{3}}(\mathbf{D}-\frac{1}{2}(\alpha_{1}+\beta_{1}+\alpha_{2}+\beta_{2})(\mathbf{N}_{1}+\mathbf{N}_{2}))^{i_{1}}\\\
&\left[\frac{(\alpha_{4}+\beta_{4})(\mathbf{N}_{1}+\mathbf{N}_{2})}{2}\right]^{i_{2}}\left[\frac{(\alpha_{3}+\beta_{3})(\mathbf{N}_{1}+\mathbf{N}_{2})}{2}\right]^{i_{3}}\mathbf{r}^{0}\mathbf{T}_{1d}^{i_{2}}\mathbf{T}_{2d}^{i_{3}}\
\end{aligned}$ (2)

Supposing $\\{\lambda_{k}(\alpha_{i})\\}(k=0,1,\cdots,n-1)$ and
$\\{\lambda_{l}(\beta_{j})\\}(l=0,1,\cdots,n-1)$ are the eigenvalues of
$\alpha_{i}$ and $\beta_{j}$ sorted in non-decreasing order. n is the number
of control points in u and v direction. Since
$\mid\lambda_{k}(\alpha_{i})\mid=0\quad or\quad 1,i=1,2,3,4,k=0,1,\cdots,n-1$,
$\mid\lambda_{l}(\beta_{j})\mid=0\quad or\quad 1,j=1,2,3,4,l=0,1,\cdots,n-1$,
$r(\sum_{i=0}^{4}\alpha_{i})<n$, $r(\sum_{i=0}^{4}\beta_{i})<n$,
$\left|\lambda_{i}(\mathbf{T}_{jd})\right|=1,i=1,2,3,4,j=1,2$, and
$r(\mathbf{N}_{1}+\mathbf{N}_{2})=\frac{n}{2}<n$, their powers have no effect
on convergence. $r(\alpha)$ is the rank of $\alpha$.

Let

$\displaystyle\mathbf{s}^{k+1}=\mathbf{D}^{k+1}\mathbf{r}^{0}$ (3)

Similar to Gyroid, we first show that $\mathbf{s}^{k+1}$ is convergent. Then
we show that $\mathbf{r}^{k+1}$ is also convergent. To achieve this goal,
$\mathbf{s}^{k+1}$ is turned into the form

$\displaystyle\mathbf{s}^{k+1}=(\mathbf{I}-\mathbf{Bw})^{k+1}\mathbf{r}^{0}$
(4)

Since B is a non-singular matrix, it is positive definite. From Theorem 2.2 in
[1], we know that $\rho(\mathbf{I}-\mathbf{B})<1$, where
$\rho(\mathbf{I}-\mathbf{B})$ is the spectral radius of
$\mathbf{I}-\mathbf{B}$. With uniform weight assignment,
$\rho(\mathbf{I}-\mathbf{Bw})<1$. Therefore, we get the following equation:

$\lim_{k\to\infty}(\mathbf{I}-\mathbf{Bw})^{k}=\lim_{k\to\infty}\mathbf{s}^{k+1}=(\mathbf{0})_{n+1}$
(5)

Then each term in $\mathbf{r}^{k+1}$ has the same spectral radius as each term
in $\mathbf{s}^{k+1}$, as is proved for Gyroid. So if $\mathbf{s}^{k+1}$ is
convergent, $\mathbf{r}^{k+1}$ is also convergent, which could be expressed
as:

$\lim_{k\to\infty}\mathbf{P}^{k}-\mathbf{w}^{-1}\mathbf{B}^{-1}\mathbf{Q}=\mathbf{0}$
(6)

So { $\mathbf{P}^{k}$ } is convergent, and

$\mathbf{P}^{\infty}=\mathbf{w}^{-1}\mathbf{B}^{-1}\mathbf{Q}$ (7)

###### Proposition 2

The CPIA iterative method for Schwarz_P is convergent and the limit surface is
the least-square fitting outcome of the initial data
$\mathrm{\\{\mathbf{Q}_{ij}\\}^{m_{1},m_{2}}_{i=0,j=0}}$.

Proof. As the result of the iterative procedure of CPIA, a sequence of the
control points of the offset surface for Schwarz_P
$\mathrm{\\{\mathbf{P}^{k}(u,v),k=0,1,\cdots\\}}$ is generated. To show its
convergence, let
$\mathbf{P}^{k}=\mathbf{P}_{0}^{k},\mathbf{P}_{1}^{k},\cdots,\mathbf{P}_{n}^{k}\\}^{T}$
and
$\mathbf{Q}=\\{\mathbf{Q}_{0},\mathbf{Q}_{1},\cdots,\mathbf{Q}_{m}\\}^{T}$. In
the (k+1)th iteration, we have

$\displaystyle\begin{aligned}
\mathbf{P}^{k+1}&=\mathbf{P}^{k}+(\mathbf{Q-BwP}^{k})+\frac{1}{2}\sum_{i=1}^{4}((\mathbf{M}_{2ip}-\mathbf{M}_{1ip})\mathbf{P}^{k}\mathbf{T}_{ip}-(\mathbf{M}_{2ip}-\mathbf{M}_{1ip})\mathbf{P}^{k})\\\
&+\frac{1}{2}\sum_{i=1}^{4}((3\mathbf{M}_{4ip}-2\mathbf{M}_{3ip}-\mathbf{M}_{5ip})\mathbf{P}^{k}\mathbf{T}_{ip}-(3\mathbf{M}_{4ip}-2\mathbf{M}_{3ip}-\mathbf{M}_{5ip})\mathbf{P}^{k})\end{aligned}$
(8)

where $\mathbf{M}_{ijp}(i=1,2,3,4,5,j=1,2,3,4)$ and
$\mathbf{T}_{ip}(i=1,2,3,4)$ are given in Sec. 1. $\mathbf{T}_{ip}(i=1,2,3,4)$
are invertible matrices where the absolute values of the eigenvalues are all
1, i.e., $\left|\lambda_{i}(\mathbf{T}_{jd})\right|=1,i=1,2,3,4$ and
$j=1,2,3,4$. $\mathbf{B}$ and $\mathbf{w}$ are mentioned in the proof for
Gyroid and Diamond.

To simplify the expression of the formulas, let
$\alpha_{i}=3\mathbf{M}_{4ip}-2\mathbf{M}_{3ip}-\mathbf{M}_{5ip}$ and
$\beta_{i}=\mathbf{M}_{2ip}-\mathbf{M}_{1ip}(i=1,2,3,4)$.

Let $\mathbf{D}=\mathbf{I}-\mathbf{Bw}$ and
$\mathbf{r}^{k}=\mathbf{P}^{k}-\mathbf{w}^{-1}\mathbf{B}^{-1}\mathbf{Q}$. Then
a new form of Eq. 8 could be derived as

$\displaystyle\begin{aligned}
\mathbf{r}^{k+1}=&(\mathbf{D}-\frac{1}{2}\sum_{i=0}^{4}(\alpha_{i}+\beta_{i}))\mathbf{r}^{k}+\frac{1}{2}\sum_{i=0}^{4}(\alpha_{i}+\beta_{i})\mathbf{r}^{k}\mathbf{T}_{ip}\\\
&=\cdots\\\ &=\sum_{i=0}^{k+1}{k+1\choose
i}(\mathbf{D}-\frac{1}{2}\sum_{n=0}^{4}(\alpha_{n}+\beta_{n}))^{i}(\frac{1}{2}\sum_{n=0}^{4}(\alpha_{n}+\beta_{n}))^{k+1-i}\mathbf{r}^{0}\mathbf{T}_{ip}^{k+1-i}\end{aligned}$
(9)

Supposing $\\{\lambda_{k}(\alpha_{i})\\}(k=0,1,\cdots,n-1)$ and
$\\{\lambda_{l}(\beta_{j})\\}(l=0,1,\cdots,n-1)$ are the eigenvalues of
$\alpha_{i}$ and $\beta_{i}$ sorted in non-decreasing order. Since
$\mid\lambda_{k}(\alpha_{i})\mid=0\quad or\quad 1,k=0,1,\cdots,n-1$,
$\mid\lambda_{l}(\beta_{j})\mid=0\quad or\quad 1,l=0,1,\cdots,n-1$, and
$r(\sum_{i=0}^{4}\beta_{i})<n$, their powers have no effect on convergence.
($r(\sum_{i=0}^{4}\beta_{i})$ is the rank of $(\sum_{i=0}^{4}\beta_{i})$).

Let

$\mathbf{s}^{k+1}=\mathbf{D}^{k+1}\mathbf{r}^{0}$ (10)

Then the convergence of $\mathbf{r}^{k+1}$ is the same as $\mathbf{s}^{k+1}$.
Then we only need to show that $\\{\mathbf{s}^{k}\\}$ is convergent.
$\mathbf{s}^{k+1}$ have another form

$\displaystyle\mathbf{s}^{k+1}=(\mathbf{I}-\mathbf{Bw})^{k+1}\mathbf{r}^{0}$
(11)

Since B is a non-singular matrix, it is positive definite. From Theorem 2.2 in
[1], we know that $\rho(\mathbf{I}-\mathbf{B})<1$, where
$\rho(\mathbf{I}-\mathbf{B})$ is the spectral radius of
$\mathbf{I}-\mathbf{B}$. With uniform weight assignment,
$\rho(\mathbf{I}-\mathbf{Bw})<1$. Therefore, we have the following equation

$\lim_{k\to\infty}(\mathbf{I}-\mathbf{Bw})^{k}=\lim_{k\to\infty}\mathbf{s}^{k+1}=(\mathbf{0})_{n+1}$
(12)

After proving the convergence of $\mathbf{s}^{k+1}$, $\mathbf{r}^{k+1}$ is
also convergent according to the proof for Gyroid. This convergence has the
form:

$\lim_{k\to\infty}\mathbf{P}^{k}-\mathbf{w}^{-1}\mathbf{B}^{-1}\mathbf{Q}=\mathbf{0}$
(13)

So { $\mathbf{P}^{k}$ } is convergent, and

$\mathbf{P}^{\infty}=\mathbf{w}^{-1}\mathbf{B}^{-1}\mathbf{Q}$ (14)

##  3 THE SECOND ORDER DERIVATIVES OF THE OFFSET EQUATION

Given $\phi_{1}$, $\phi_{2}$, and $\phi_{3}$ in Section 3.1 of the paper, the
real and imaginary parts of the three complex variables, denoted as
$\mathbf{p}_{1}\mathbf{p}_{2}\mathbf{p}_{3}$ and
$\mathbf{q}_{1}\mathbf{q}_{2}\mathbf{q}_{3}$, are given as:

$\displaystyle\begin{aligned} &\mathbf{p}_{1}=Re(\phi_{1})\\\
&\mathbf{p}_{2}=Re(\phi_{2})\\\ &\mathbf{p}_{3}=Re(\phi_{3})\\\
&\mathbf{q}_{1}=Im(\phi_{1})\\\ &\mathbf{q}_{2}=Im(\phi_{2})\\\
&\mathbf{q}_{3}=Im(\phi_{3})\\\ \end{aligned}$ (15)

To calculate the second order derivatives, the first order and second order
derivatives of $\phi_{1}$, $\phi_{2}$, and $\phi_{3}$ are first calculated,
given as:

$\displaystyle\begin{aligned}
&\mathbf{p}^{\prime}_{1}=Re\left[-\frac{\left(1-\tau^{2}\right)\left(8\tau^{7}-56\tau^{3}\right)}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}-\frac{2\tau}{\sqrt{\tau^{8}-14\tau^{4}+1}}\right]\\\
&\mathbf{p}^{\prime}_{2}=Re\left[\frac{2i\tau}{\sqrt{\tau^{8}-14\tau^{4}+1}}-\frac{i\left(\tau^{2}+1\right)\left(8\tau^{7}-56\tau^{3}\right)}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right]\\\
&\mathbf{p}^{\prime}_{3}=Re\left[\frac{2}{\sqrt{\tau^{8}-14\tau^{4}+1}}-\frac{\tau\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right]\\\
&\mathbf{q}^{\prime}_{1}=Im\left[-\frac{\left(1-\tau^{2}\right)\left(8\tau^{7}-56\tau^{3}\right)}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}-\frac{2\tau}{\sqrt{\tau^{8}-14\tau^{4}+1}}\right]\\\
&\mathbf{q}^{\prime}_{1}=Im\left[\frac{2i\tau}{\sqrt{\tau^{8}-14\tau^{4}+1}}-\frac{i\left(\tau^{2}+1\right)\left(8\tau^{7}-56\tau^{3}\right)}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right]\\\
&\mathbf{q}^{\prime}_{3}=Im\left[\frac{2}{\sqrt{\tau^{8}-14\tau^{4}+1}}-\frac{\tau\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right]\\\
\end{aligned}$ (16) $\displaystyle\begin{aligned}
&\mathbf{p}^{\prime\prime}_{1}=Re\left[-\frac{2}{\sqrt{\tau^{8}-14\tau^{4}+1}}+\frac{2\tau\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}+\left(1-\tau^{2}\right)\left(\frac{3\left(8\tau^{7}-56\tau^{3}\right)^{2}}{4\left(\tau^{8}-14\tau^{4}+1\right)^{5/2}}-\frac{56\tau^{6}-168\tau^{2}}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right)\right]\\\
&\mathbf{p}^{\prime\prime}_{2}=Re\left[\frac{2i}{\sqrt{\tau^{8}-14\tau^{4}+1}}-\frac{2i\tau\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}+i\left(\tau^{2}+1\right)\left(\frac{3\left(8\tau^{7}-56\tau^{3}\right)^{2}}{4\left(\tau^{8}-14\tau^{4}+1\right)^{5/2}}-\frac{56\tau^{6}-168\tau^{2}}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right)\right]\\\
&\mathbf{p}^{\prime\prime}_{3}=Re\left[2\tau\left(\frac{3\left(8\tau^{7}-56\tau^{3}\right)^{2}}{4\left(\tau^{8}-14\tau^{4}+1\right)^{5/2}}-\frac{56\tau^{6}-168\tau^{2}}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right)-\frac{2\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right]\\\
&\mathbf{q}^{\prime\prime}_{1}=Im\left[-\frac{2}{\sqrt{\tau^{8}-14\tau^{4}+1}}+\frac{2\tau\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}+\left(1-\tau^{2}\right)\left(\frac{3\left(8\tau^{7}-56\tau^{3}\right)^{2}}{4\left(\tau^{8}-14\tau^{4}+1\right)^{5/2}}-\frac{56\tau^{6}-168\tau^{2}}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right)\right]\\\
&\mathbf{q}^{\prime\prime}_{2}=Im\left[\frac{2i}{\sqrt{\tau^{8}-14\tau^{4}+1}}-\frac{2i\tau\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}+i\left(\tau^{2}+1\right)\left(\frac{3\left(8\tau^{7}-56\tau^{3}\right)^{2}}{4\left(\tau^{8}-14\tau^{4}+1\right)^{5/2}}-\frac{56\tau^{6}-168\tau^{2}}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right)\right]\\\
&\mathbf{q}^{\prime\prime}_{3}=Im\left[2\tau\left(\frac{3\left(8\tau^{7}-56\tau^{3}\right)^{2}}{4\left(\tau^{8}-14\tau^{4}+1\right)^{5/2}}-\frac{56\tau^{6}-168\tau^{2}}{2\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right)-\frac{2\left(8\tau^{7}-56\tau^{3}\right)}{\left(\tau^{8}-14\tau^{4}+1\right)^{3/2}}\right]\\\
\end{aligned}$ (17)

where $\tau$ is the complex variable mentioned in Section 3.1 of the paper.
$\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}\mathbf{p}^{\prime}_{3}$ and
$\mathbf{q}^{\prime}_{1}\mathbf{q}^{\prime}_{2}\mathbf{q}^{\prime}_{3}$ are
the real and imaginary parts of
$\phi_{1}^{\prime}\phi_{2}^{\prime}\phi_{3}^{\prime}$.
$\mathbf{p}^{\prime\prime}_{1}\mathbf{p}^{\prime\prime}_{2}\mathbf{p}^{\prime\prime}_{3}$
and
$\mathbf{q}^{\prime\prime}_{1}\mathbf{q}^{\prime\prime}_{2}\mathbf{q}^{\prime\prime}_{3}$
are the real and imaginary parts of
$\phi_{1}^{\prime\prime}\phi_{2}^{\prime\prime}\phi_{3}^{\prime\prime}$.

Then the second-order derivatives of the offset equation are given as:

$\displaystyle\begin{aligned}
x^{\prime\prime}=&\frac{8\mathbf{A}_{1}(\mathbf{q}_{3}\mathbf{p}^{\prime}_{2}+\mathbf{p}_{2}\mathbf{q}^{\prime}_{3}-\mathbf{q}_{2}\mathbf{p}^{\prime}_{3}-\mathbf{p}_{3}\mathbf{q}^{\prime}_{2})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})(\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime})\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&+\frac{4\mathbf{A}_{1}^{2}(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{2}^{\prime}-2\mathbf{q}_{3}^{\prime}\mathbf{p}_{2}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime\prime}+\mathbf{q}_{2}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime\prime})}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&+\frac{12(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})(\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime})\right]^{2}}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{4\mathbf{A}_{1}(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})\left[(\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}+\mathbf{p}_{1}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{1}^{\prime})^{2}+(\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}+\mathbf{p}_{1}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}-\mathbf{p}_{3}\mathbf{q}_{1}^{\prime})^{2}+(\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}-\mathbf{p}_{3}\mathbf{q}_{2}^{\prime})^{2}\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{4\mathbf{A}_{1}(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(2\mathbf{p}_{2}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{p}_{1}^{\prime}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{2}^{\prime\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{p}_{1}^{\prime}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime\prime})\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{4\mathbf{A}_{1}(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})^{2}(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{2}^{\prime}-2\mathbf{p}_{2}^{\prime}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime\prime}+\mathbf{q}_{2}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime\prime})}{4\mathbf{A}_{1}^{\frac{5}{2}}}\end{aligned}$
(18) $\displaystyle\begin{aligned}
y^{\prime\prime}=&\frac{-8\mathbf{A}_{1}(\mathbf{q}_{3}\mathbf{p}^{\prime}_{1}+\mathbf{p}_{1}\mathbf{q}^{\prime}_{3}-\mathbf{q}_{1}\mathbf{p}^{\prime}_{3}-\mathbf{p}_{3}\mathbf{q}^{\prime}_{1})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})(\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime})\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{4\mathbf{A}_{1}^{2}(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{q}_{3}^{\prime}\mathbf{p}_{1}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime\prime})}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{12(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})(\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime})\right]^{2}}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&+\frac{4\mathbf{A}_{1}(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})\left[(\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}+\mathbf{p}_{1}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{1}^{\prime})^{2}+(\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}+\mathbf{p}_{1}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}-\mathbf{p}_{3}\mathbf{q}_{1}^{\prime})^{2}+(\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}-\mathbf{p}_{3}\mathbf{q}_{2}^{\prime})^{2}\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&+\frac{4\mathbf{A}_{1}(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(2\mathbf{p}_{2}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{p}_{1}^{\prime}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{2}^{\prime\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{p}_{1}^{\prime}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime\prime})\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&+\frac{4\mathbf{A}_{1}(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})^{2}(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{2}^{\prime}-2\mathbf{p}_{2}^{\prime}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime\prime}+\mathbf{q}_{2}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime\prime})}{4\mathbf{A}_{1}^{\frac{5}{2}}}\end{aligned}$
(19) $\displaystyle\begin{aligned}
z^{\prime\prime}=&\frac{8\mathbf{A}_{1}(\mathbf{q}_{2}\mathbf{p}^{\prime}_{1}+\mathbf{p}_{1}\mathbf{q}^{\prime}_{2}-\mathbf{q}_{1}\mathbf{p}^{\prime}_{2}-\mathbf{p}_{2}\mathbf{q}^{\prime}_{1})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})(\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime})\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&+\frac{4\mathbf{A}_{1}^{2}(2\mathbf{p}_{2}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{q}_{2}^{\prime}\mathbf{p}_{1}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{2}^{\prime\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime\prime})}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&+\frac{12(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime})+(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})(\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime})\right]^{2}}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{4\mathbf{A}_{1}(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})\left[(\mathbf{q}_{2}\mathbf{p}_{1}^{\prime}+\mathbf{p}_{1}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{1}\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2}\mathbf{q}_{1}^{\prime})^{2}+(\mathbf{q}_{3}\mathbf{p}_{1}^{\prime}+\mathbf{p}_{1}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{1}\mathbf{p}_{3}^{\prime}-\mathbf{p}_{3}\mathbf{q}_{1}^{\prime})^{2}+(\mathbf{q}_{3}\mathbf{p}_{2}^{\prime}+\mathbf{p}_{2}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{3}^{\prime}-\mathbf{p}_{3}\mathbf{q}_{2}^{\prime})^{2}\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{4\mathbf{A}_{1}(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})\left[(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})(2\mathbf{p}_{2}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{p}_{1}^{\prime}\mathbf{q}_{2}^{\prime}-\mathbf{q}_{2}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{2}^{\prime\prime}+\mathbf{p}_{2}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{2}^{\prime\prime})+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{1}^{\prime}-2\mathbf{p}_{1}^{\prime}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{1}^{\prime\prime}+\mathbf{q}_{1}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{1}^{\prime\prime}-\mathbf{p}_{1}\mathbf{q}_{3}^{\prime\prime})\right]}{4\mathbf{A}_{1}^{\frac{5}{2}}}\\\
&-\frac{4\mathbf{A}_{1}(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})^{2}(2\mathbf{p}_{3}^{\prime}\mathbf{q}_{2}^{\prime}-2\mathbf{p}_{2}^{\prime}\mathbf{q}_{3}^{\prime}-\mathbf{q}_{3}\mathbf{p}_{2}^{\prime\prime}+\mathbf{q}_{2}\mathbf{p}_{3}^{\prime\prime}+\mathbf{p}_{3}\mathbf{q}_{2}^{\prime\prime}-\mathbf{p}_{2}\mathbf{q}_{3}^{\prime\prime})}{4\mathbf{A}_{1}^{\frac{5}{2}}}\end{aligned}$
(20)

where $\mathbf{A}_{1}$ is derived as:

$\mathbf{A}_{1}=(\mathbf{p}_{2}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{2})^{2}+(\mathbf{p}_{3}\mathbf{q}_{1}-\mathbf{p}_{1}\mathbf{q}_{3})^{2}+(\mathbf{p}_{3}\mathbf{q}_{2}-\mathbf{p}_{2}\mathbf{q}_{3})^{2}$

Numerical methods are applied to calculate the maximum of the second-order
derivatives given above.

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in our context is seen in Algorithm 3.

Algorithm 3 Adapted Floyd-Warshall Algorithm

1:$D\in\mathbb{R}^{K\times K}_{\geq 0}$

2:$D=D^{\top}$

3:Trim entries below 0: $D[D<0]\leftarrow 0$

4:for $k\in\\{1,2,\dots,n\\}$ do

5: for $j\in\\{1,2,\dots,n\\}$ do

6: for $i\in\\{1,2,\dots,n\\}$ do

7: if $D_{ij}>(D_{ik}+D_{kj})$ then

8: $D_{ij}\leftarrow(D_{ik}+D_{kj})$

9: end if

10: end for

11: end for

12:end for

One can draw similarities here to the problem of sparse metric repair. Metric
repair seeks to adjust the fewest entries in a noisy distance matrix so that
it still preserves the properties of being a metric (positivity, triangle
inequality). Gilbert and Jain [2017] illustrated this Floyd-Washall algorithm
to be a solution to a special type known as decrease only metric repair.

##  Appendix C Other link functions:

Another common link function is the and formulation of the model is via the
following:

$\displaystyle\nu_{i}$ $\displaystyle\sim F_{\nu},\quad\nu_{i}\leq 0$
$\displaystyle Z_{i}$ $\displaystyle\sim F_{Z},\quad Z_{i}\in\mathcal{M}^{p}$
$\displaystyle P(G_{ij}=1)$
$\displaystyle=\operatorname*{\text{logit}}(\nu_{i}+\nu_{j}+\varphi-d(Z_{i},Z_{j}))$

We can consistently estimate the node level parameters up to a constant shift
using conditional maximum likelihood as in the semiparametric Rasch model
[Andersen, 1970]. The parameter $\varphi$ controls the global sparsity.
Similar to before, we note that $\nu_{i}$ terms are likely to be very large in
a cliques then we can set the largest parameter in each group to be nearly
zero.

Other link functions may be used but will likely all need specific methods to
estimate $\nu_{i}$ parameters within each clique. However, we have shown how
it can be developed in these two canonical cases.

##  Appendix D Riemannian Geometry Definitions

In this section, we review some definitions of the sectional and scalar
curvature as well as the volume elements. A Riemannian manifold
$\mathcal{M}=(M,g)$ is a smooth manifold $M$ equipped with a Riemannian inner
product $g_{q}$ on the tangent space $T_{q}(\mathcal{M})$ at any point
$q\in\mathcal{M}$, $g_{q}(u,v):T_{q}(\mathcal{M})\times
T_{q}(\mathcal{M})\mapsto\mathbb{R}$.

This inner product can be used to define the Riemann curvature tensor at a
point $q\in\mathcal{M}$ $R_{q}(u,v)w$, which takes $3$ vectors $u,v,w$ and
returns an element of the tangent space

$\displaystyle R_{q}(u,v)w$ $\displaystyle:\quad T_{q}(\mathcal{M})\times
T_{q}(\mathcal{M})\times T_{q}(\mathcal{M})\mapsto T_{q}(\mathcal{M})$
$\displaystyle R_{q}(u,v)w$
$\displaystyle:=[\nabla_{u},\nabla_{v}]w-\nabla_{[u,v]}w$

where $[u,v]$ is the lie bracket of vector fields and
$[\nabla_{u},\nabla_{v}]$ is the commuter of differential operators. The
Riemann curvature tensor can be used to define our main quantity of interest,
the sectional curvature at a point $\kappa_{q}(u,v):\times
T_{q}(\mathcal{M})\times T_{q}(\mathcal{M})\mapsto\mathbb{R}$. The sectional
curvature takes two linearly independent elements of the tangent space and
maps them to the real line.

$\kappa_{q}(u,v):=\frac{g_{q}(R_{q}(u,v)v,u)}{g_{q}(u,u)g_{q}(v,v)-g_{q}(u,v)^{2}}.$

The sectional curvature is independent of the coordinate system used, but
depends only on the linear subspace spanned by $u,v$. Furthermore, in the
canonical manifolds $\kappa_{q}(u,v)=\kappa$ by construction.

From the sectional curvature, we can define the scalar curvature $S(m)$,

$S(q):=\sum_{i\not=j}\kappa(e_{i},e_{j})$

where $\\{e_{i}\\}_{i=1}^{p}$ form an orthonormal frame for
$T_{q}(\mathcal{M})$. We can think of the scalar curvature as an “average of
sectional curvatures” across the manifold.

We next define a volume form (also known as the Levi-Civita Tensor) via the
Riemannian inner product $g$. If $\omega$ is a local oriented coordinate
system near a point $q$ then

$\displaystyle dV$ $\displaystyle:=\sqrt{|\text{det}(g)|}d\omega.$

where $g_{q}$ is the metric tensor evaluated on the basis coordinate system
$\omega$. For further details on these quantities, see Klingenberg [1995].

From this definition of a volume, we can define probability density functions
on the manifold. A density function $f$ corresponding to a measure $F$ with
support on the manifold can be defined as follows. For a set
$\mathcal{X}\subset\mathcal{M}$.

$P(X\in\mathcal{X})=\int_{x\in\mathcal{X}}f(x)dV_{x}$

See Pennec [1999] for further introduction for defining probabilities on the
manifold.

##  Appendix E Assumptions on $\mathcal{M}$

Here we verify that the Algebraic Midpoint properties, as well as locally
Euclidean properties are satisfied for a complete simply connected smooth
Riemannian manifolds.

If the algebraic midpoint property is satisfied for any complete metric space
$\mathfrak{M}$ then by Theorem 1.8 of Gromov [2007], then $\mathfrak{M}$ is a
path metric space. The authors follow up in discussion a list of examples of
path metric spaces, which include Riemannian manifolds with boundary.

Secondly, if $\mathcal{M}^{p}$ has scalar curvature at point $q$, $S(q)$ then

$\displaystyle\frac{\text{Vol}(B_{\mathcal{M}^{p}}(\epsilon,q))}{\text{Vol}(B_{\mathbb{E}^{p}}(\epsilon,q))}$
$\displaystyle=1-\frac{S(q)}{6(p+2)}\epsilon^{2}+o(\epsilon^{3})$

by Theorem 3.98 of Gallot et al. [2004]. Since this holds, then for a latent
metric which is generated by distances on a Riemannian manifold, the locally
Euclidean volume property will hold.

##  Appendix F Graph Statistics From Simulations

Columns denote the scale factor used in the simulations.

Table 4: $\kappa=-2$ graph statistics summary.

Scale ($\rho$) | 0.7 | 1 | 2
---|---|---|---
Edge.fraction | 0.016 (0.001) | 0.012 (0.001) | 0.007 (0.001)
Max.Degree | 341.3 (23.596) | 412.29 (26.364) | 656.63 (33.954)
Mean.Degree | 56.726 (5.279) | 58.538 (5.351) | 73.975 (5.448)
Distinct.Cliques$\geq\ell$ | 78.665 (6.955) | 53.675 (5.427) | 43.515 (4.41)
Max.Clique.Size | 24.215 (4.985) | 28.725 (5.494) | NaN (NA)
Mean.Degree.Centrality | 0.155 (0.009) | 0.134 (0.007) | 0.107 (0.005)

Table 5: $\kappa=-1$ graph statistics summary.

Scale ($\rho$) | 0.7 | 1 | 2
---|---|---|---
Edge.fraction | 0.017 (0.002) | 0.012 (0.001) | 0.008 (0.001)
Max.Degree | 343.69 (25.21) | 416.825 (28.796) | 661.12 (32.257)
Mean.Degree | 58.943 (5.354) | 61.168 (5.636) | 78.239 (5.6)
Distinct.Cliques$\geq\ell$ | 80.14 (6.488) | 54.965 (5.404) | 42.32 (5.027)
Max.Clique.Size | 24.005 (5.456) | 28.735 (6) | NaN (NA)
Mean.Degree.Centrality | 0.159 (0.008) | 0.137 (0.007) | 0.112 (0.005)

Table 6: $\kappa=-0.5$ graph statistics summary.

Scale ($\rho$) | 0.7 | 1 | 2
---|---|---|---
Edge.fraction | 0.017 (0.001) | 0.013 (0.001) | 0.008 (0.001)
Max.Degree | 340.785 (21.648) | 413.115 (24.818) | 656.07 (39.433)
Mean.Degree | 59.658 (4.776) | 62.67 (4.989) | 79.873 (6.566)
Distinct.Cliques$\geq\ell$ | 80.205 (6.493) | 55.155 (5.288) | 39.95 (5.289)
Max.Clique.Size | 23.62 (5.216) | 29.52 (6.073) | NaN (NA)
Mean.Degree.Centrality | 0.161 (0.009) | 0.141 (0.007) | 0.114 (0.005)

Table 7: $\kappa=0$ graph statistics summary.

Scale ($\rho$) | 0.7 | 1 | 2
---|---|---|---
Edge.fraction | 0.012 (0.001) | 0.008 (0) | 0.005 (0)
Max.Degree | 205.38 (16.619) | 242.88 (18.178) | 371.805 (24.336)
Mean.Degree | 40.815 (2.626) | 41.759 (2.415) | 52.465 (2.813)
Distinct.Cliques$\geq\ell$ | 70.2 (13.19) | 46.84 (6.057) | 41.54 (4.576)
Max.Clique.Size | 23.735 (5.244) | 28.49 (5.421) | 41.63 (7.913)
Mean.Degree.Centrality | 0.169 (0.017) | 0.149 (0.013) | 0.125 (0.009)

Table 8: $\kappa=0.5$ graph statistics summary.

Scale ($\rho$) | 0.7 | 1 | 2
---|---|---|---
Edge.fraction | 0.017 (0.001) | 0.013 (0.001) | 0.008 (0)
Max.Degree | 267.08 (17.432) | 318.615 (18.062) | 488.545 (25.046)
Mean.Degree | 61.311 (2.803) | 63.539 (2.96) | 80.373 (3.454)
Distinct.Cliques$\geq\ell$ | 87.945 (12.778) | 55.385 (7.343) | 39.535 (5.591)
Max.Clique.Size | 23.595 (4.844) | 28.45 (6.158) | 39.925 (7.861)
Mean.Degree.Centrality | 0.21 (0.016) | 0.183 (0.012) | 0.153 (0.008)

Table 9: $\kappa=1$ graph statistics summary.

Scale ($\rho$) | 0.7 | 1 | 2
---|---|---|---
Edge.fraction | 0.025 (0.001) | 0.018 (0.001) | 0.012 (0)
Max.Degree | 351.89 (16.965) | 421.36 (18.32) | 652.265 (26.131)
Mean.Degree | 88.331 (3.89) | 91.901 (3.805) | 117.569 (4.089)
Distinct.Cliques$\geq\ell$ | 91.71 (30.245) | 64.23 (17.006) | 41.765 (4.761)
Max.Clique.Size | 24.07 (5.334) | 29.29 (6.259) | 40.52 (7.969)
Mean.Degree.Centrality | 0.241 (0.012) | 0.21 (0.01) | 0.174 (0.007)