2 Configuration spaces

The compactified configuration space of $k+1$ points in $\mathbb {R}^k$ is the natural setting for finding inscribed simplices in embedded spheres. In this section, we give a very brief overview of compactified configuration spaces. There are many versions of this classical material (see, e.g., [Reference Axelrod and Singer4, Reference Fulton and MacPherson13]), but we follow Sinha [Reference Sinha40], as this approach is more appropriate to our work. A discussion similar to the one found below is found in [Reference Cantarella, Denne and McCleary10].

A reader familiar with configuration spaces may skip much of this section. However, we recommend paying attention to the notation we have used for the spaces. Definitions 2.2, 2.3, 2.9, Remark 2.5 and Theorem 2.7 are particularly useful.

The space $C_n(M)$ is an open submanifold of $M^{\times n}$. We next compactify $C_n(M)$ to a closed manifold-with-boundary and corners, which we will denote $C_n[M]$, without changing its homotopy type. The resulting manifold will be homeomorphic to $M^{\times n}$ with an open neighborhood of the fat diagonal removed. Recall that the fat diagonal is the subset of $M^{\times n}$ of n-tuples for which (at least) two entries are equal, that is, where some collection of points comes together at a single point. The construction of $C_n[M]$ preserves information about the directions and relative rates of approach of each group of collapsing points.

We then compactify $C_n(\mathbb {R}^k)$ as follows:

We now summarise some of the important features of this construction, including the fact that $C_n[M]$ does not depend on the choice of embedding of M in $\mathbb {R}^k$.

The space $C_n[M]$ may be viewed as a polytope with a combinatorial structure based on the different ways groups of points in M can come together. This structure defines a stratification of $C_n[M]$ into a collection of closed faces of various dimensions whose intersections are members of the collection. Full details can be found in [Reference Budney, Conant, Scannell and Sinha8, Reference Sinha40]. The main structure we will consider is the $(0,1,\dots ,k)$-face of $\partial C_n[M]$. This is the boundary component where all the points come together at the same time.

Any pair $\mathbf {p}$, $\mathbf {q}$ of disjoint points in $\mathbb {R}^k$ has a direction $(\mathbf {p}-\mathbf {q})/\left | \mathbf {p}-\mathbf {q} \right |$ associated to it, while every triple of disjoint points $\mathbf {p}$, $\mathbf {q}$, $\mathbf {r}$ has a corresponding distance ratio $\left | \mathbf {p}-\mathbf {q} \right |/\left | \mathbf {p}-\mathbf {r} \right |$. One way to think of the coordinates of $C_n[M]$ is that they extend the definition of these directions and ratios to the boundary.

It turns out that for connected manifolds of dimension at least $2$, the combinatorial structure of the strata of $C_n[M]$ depends only on the number of points. Regardless of dimension, this construction and division of $\partial C_n[M]$ into strata is functorial in the following sense.

For an embedding $f \colon M \hookrightarrow N$, the image of the induced embedding $C_n[f]\colon C_n[M] \hookrightarrow C_n[N]$ will be denoted by $C_n[f(M)]$.

Finally, we will need a metric on the set of evaluation maps $C_n[f] \colon \! C_n[M]\hookrightarrow C_n[N]$. The definition of compactified configuration spaces allows us to view $C_n[N] \subset (\mathbb {R}^k)^{n} \times (S^{k-1})^{n(n-1)} \times [0,\infty ]^{n(n-1)(n-2)}$ as a metric space with the sup norm. If we define the mapping ${\mathrm {pr}}_i$ to be the projection onto the ith space of the product, then this naturally leads to a metric on the set of continuous functions $C^0(C_n[M], C_n[N])$.

Definition 2.9. With the above assumptions, the metric on the set $C^0(C_n[M], C_n[N])$ is given by

$$ \begin{align*}\|F - G\|_0 = \sup_{\overrightarrow{\mathbf{p}} \in C_n[M]} \{\| {\mathrm{ pr}}_i(F(\kern1.5pt\overrightarrow{\mathbf{p}} )) - \mathrm{pr}_i(G(\kern1.5pt\overrightarrow{\mathbf{p}} ))\| \mid \mathrm{for\ all\ }i\}.\end{align*} $$

Thus, given the embeddings $f, g\colon \! M\hookrightarrow N$, we say that the corresponding maps on configuration spaces $C_n[f],C_n[g]\colon \! C_n[M]\hookrightarrow C_n[N]$ are $C^0$-close if for all $\epsilon>0$, we have $\|C_n[f]-C_n[g]\|_0<\epsilon $.

3 Finding special configurations with multijet transversality

In this section, we set up a general method for tackling problems where we seek a special configuration of n points on a compact manifold M that is smoothly embedded in $\mathbb {R}^k$ (cf. [Reference Cantarella, Denne and McCleary10]). In both the square-peg problem and the inscribed simplex problem, M is a sphere (either $S^1$ or $S^{k-1}$). We thus denote the $C^\infty $ smooth embedding by $\gamma \colon \! S^l \hookrightarrow \mathbb {R}^k$, and the corresponding compactified configuration space of n points on $\gamma (S^l)$ is $C_{n}[\gamma (S^l)]$. We let Z denote the subspace of tuples of n points in $\mathbb {R}^k$ that satisfy the conditions of a special configuration. For example, in [Reference Cantarella, Denne and McCleary10], Z is the set of all square-like quadrilaterals in $\mathbb {R}^k$. In this paper, we are interested in $Z=\operatorname {\mathrm {Sim}}(\Delta )$ (see Section 4), which is the set of all $(k+1)$-simplices in $\mathbb {R}^k$ which are similar to a given nondegenerate simplex $\Delta $.

The central idea is as follows: suppose there is a different, well-understood, smooth embedding of $S^l$ in $\mathbb {R}^k$ (via $i\colon \! S^l\hookrightarrow \mathbb {R}^k$), and assume that the corresponding configuration space $C_n[i(S^l)]$ is transverse to Z in $C_n[\mathbb {R}^k]$. Also assume that $i(S^l)$ is smoothly homotopy equivalent to $\gamma (S^l)$ in $\mathbb {R}^k$. Standard transversality arguments should allow us to vary $C_n[i(S^l)]$ to $C_n[\gamma (S^l)]$ while maintaining the transversality of the intersection with Z. There are various technical obstacles to overcome, all of which are handled in detail in [Reference Cantarella, Denne and McCleary10]. Here are the key steps.

Step 1: It is possible that special configurations on $\gamma (S^l)$ shrink away to the boundary of $C_n[\mathbb {R}^k]$ during the isotopy. To prevent this, we first prove

1. (a) that n-tuples of points in $\mathbb {R}^k$ satisfying the geometric condition, Z, are a submanifold in $C_n(\mathbb {R}^k)$, and that $\partial Z\subset \partial C_n[\mathbb {R}^k]$;

2. (b) that $C_n[\gamma (S^l)]$ and Z are boundary-disjoint.

For $Z=\operatorname {\mathrm {Sim}}(\Delta )$, we prove 1(a) in Theorems 4.8 and 4.9. We prove 1(b) in Proposition 5.1.

Step 2: For a standard embedding $i\colon \! S^l\hookrightarrow \mathbb {R}^k$, we need to do two things

1. (a) prove that the intersection between $C_n[i(S^l)]$ and Z is nonempty and transverse (in other words, $C_n[i]\pitchfork Z$).

2. (b) compute the homology class of the intersection $C_n[i(S^l)] \cap Z$ in Z.

For our inscribed simplex problem, we use the standard embedding $\operatorname {\mathrm {id}} \colon \! S^{k-1}\hookrightarrow \mathbb {R}^k$ of $S^{k-1}$ in $\mathbb {R}^k$. We prove 2(a) in Proposition 5.2 and 2(b) in Proposition 5.3.

Step 3: Note that in order to apply our transversality arguments, we need to be able to perturb $C_n[\gamma (S^l)]$ so the intersection of Z remains transverse. However, there is no guarantee that the perturbed submanifold consists of configurations on a perturbed smooth embedding of $S^l$ in $\mathbb {R}^k$. We deal with this issue by applying the multijet transversality theorem [Reference Golubitsky and Guillemin14, Theorem II.4.13]. This allows us to conclude (see [Reference Cantarella, Denne and McCleary10] Theorem 17) that for any $\epsilon>0$, there is a $C^\infty $-open neighborhood of $\gamma $ in which there is, for all m, a $C^m$-dense set of smooth embeddings $\gamma '\colon \! S^l \hookrightarrow \mathbb {R}^k$, such that $\|C_n[\gamma '] - C_n[\gamma ]\|_0< \epsilon $, and $C_n[\gamma '] \pitchfork Z$, and for which $\partial Z$ and $\partial C_n[\gamma '(S^l)]$ are disjoint in $\partial C_n[\mathbb {R}^k]$.

In order to fully appreciate Step 3, first recall that the metric on the set $C^0(C_n[S^l], C_n[\mathbb {R}^k])$ was given in Definition 2.9. Second, to understand the statement about density, we need to give the topology of the spaces we are working in. In general, for manifolds $M(=S^l)$ and $N(=\mathbb {R}^k)$, the space $C^\infty (M,N)$ has the Whitney $C^\infty $-topology (see, e.g., [Reference Hirsch20]). The sets of the form

$$ \begin{align*}\mathcal{N}^t(f; (U,\phi), (V,\psi), \delta)\end{align*} $$

give a subbasis for the Whitney $C^t$-topology on $C^t(M,N)$ (where t is finite). This subbasis consists of subsets of functions $g\colon M\to N$ that are smooth, and for coordinate charts $\phi \colon (U' \subset M) \to (U\subset \mathbb {R}^m)$ and $\psi \colon (V' \subset N) \to (V\subset \mathbb {R}^k)$ and $K \subset U$ compact with $g(\phi (K)) \subset V'$, then we have, for all $s \leq r$, and all $x \in \phi (K)$,

$$ \begin{align*}\| D^s (\psi g \phi^{-1})(x) - D^s(\psi f \phi^{-1})(x)\| < \delta.\end{align*} $$

Here, $D^s F$ for a function $F \colon (U\subset \mathbb {R}^m) \to (V \subset \mathbb {R}^k)$ is the k-tuple of the sth homogeneous parts of the Taylor series representations of the projections of F. Finally, the subspace $C^\infty (M,N)$ has the Whitney $C^\infty $-topology generated by taking the union of all subbases for all $t\geq 0$.

Note that Step 3 can be applied immediately to the inscribed simplex problem.

Step 4: We need to deform standard spheres into spheres of interest and then consider what happens on the level of configuration spaces. We know precisely when such a deformation of spheres exists due to the following result of Haefliger (which we have stated in a form useful to us).

We use Theorem 3.1 to find a smooth map $E\colon \! S^l \times I \rightarrow \mathbb {R}^K$ with $E(-,0)=i$ our standard embedding and $E(-,1) = \gamma '$ (where K may be greater than our original k). Recalling that both $C_n[i]$ and $C_n[\gamma ']$ are transverse to Z allows us to conclude that using functorality, we get a homotopy $H\colon \! C_n(S^l) \times I\rightarrow C_n(\mathbb {R}^K)$ with $H(-,0)=C_n[i]$ and $H(-,1)=C_n[\gamma ']$. In [Reference Cantarella, Denne and McCleary10], we then use the Transversality Homotopy Extension Theorem (see [Reference Guillemin and Pollack17]), to find a map $H'$ homotopic to H and with $H'(-,0)=H(-,0)$, $H'(-,1)=H(-,1)$ and $H'$ is transverse to Z. This then implies that in Z, the intersections $C_n[i(S^l)] \cap Z$ and $C_n[\gamma '(S^l)] \cap Z$ represent the same homology class. In other words, we have sketched a proof of the following:

In this paper, we will not be making full use of Haefliger’s Theorem in Step 4. Instead, we will restrict our attention to smooth embeddings of $S^{k-1}$ in $\mathbb {R}^k$ which are differentiably isotopic to the identity through a differentiable isotopy in $\mathbb {R}^k$. The conclusion of Theorem 3.2 still holds for such embeddings. Putting all of our steps together allows us to conclude the following theorem.

Intuitively, this theorem shows that any smooth embedding $\gamma $ of $S^l$ in $\mathbb {R}^k$ has a neighborhood in which there is a dense set of smooth embeddings $\gamma '$ for which $C_n[\gamma '(S^l)]$ is guaranteed to have certain intersections with various submanifolds of $C_n[\mathbb {R}^k]$ defined by geometric conditions. Stated in a different way, we have the idea that a dense set of embeddings of $S^l$ always contain certain inscribed configurations of points.

4 Simplices

4.1 Nondegenerate simplices

Definition 4.1. By a simplex $\Delta $ in $\mathbb {R}^k$, we mean a set of $k+1$ distinct points $\{\mathbf {p}_0, \dots , \mathbf {p}_k\}$ in $\mathbb {R}^k$ in general position.

By general position, we mean that no hyperplane in $\mathbb {R}^k$ contains more than k points of $\Delta $. As a first consequence of this definition, we note that the volume of the simplex $\Delta $ is nonzero. This means that for us, a simplex is nondegenerate. In addition, to each simplex $\Delta $, we can associate several sets:

• • the nonzero distances $\{d_{ij}(\Delta )\} =\{\|\mathbf {p}_i-\mathbf {p}_j \|\}$, and observe that $d_{ij}=d_{ji}$;

• • the unit vectors $\{\pi _{ij} (\Delta )\} = \{\frac {\mathbf {p}_i-\mathbf {p}_j}{\|\mathbf {p}_i-\mathbf {p}_j\|}\}$;

• • the ratios $\{r_{ijl} (\Delta )\}= \{\frac {\|\mathbf {p}_i-\mathbf {p}_j\|}{\|\mathbf {p}_i-\mathbf {p}_l\|}\}= \{\frac {d_{ij}}{d_{il}}\}$.

Here, we assume that $i\neq j$ (and $j\neq l, i\neq l$), so the set of $k(k+1) / 2$ distances consists of nonzero values, and the set of ratios has values in $(0,\infty )$.

A natural question to ask is, given a set of nonzero distances $\mathcal {D}=\{d_{ij}\}$, is there a simplex that can be constructed with those distances? The theory of distance geometry allows us to decide which sets of distances $\mathcal {D}$ are constructible. We start be defining the Cayley-Menger determinant for $\{\mathbf {p}_0,\mathbf {p}_1,\dots ,\mathbf {p}_k\}$, or equivalently for $\mathcal {D}=\{d_{ij}\} =\{\|\mathbf {p}_i-\mathbf {p}_j \|\}$ (see, e.g., [Reference Blumenthal and Gillam7, Reference Sippl and Scheraga41]):

$$ \begin{align*} \operatorname{\mathrm{CM}}(\{\mathbf{p}_0,\mathbf{p}_1,\dots ,\mathbf{p}_k\}) = \operatorname{\mathrm{CM}}(\mathcal{D}) = \left| \begin{matrix} 0 & 1 & 1 & \dots & 1 \\ 1 & 0 & d_{01}^2 & \dots & d_{0k}^2 \\ 1 & d_{10}^2 & 0 & \dots & d_{1k}^2 \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & d_{k0}^2 & d_{k1}^2 & \dots & 0 \\ \end{matrix} \right|. \end{align*} $$

Theorem 4.2 ([Reference Berger5] Theorems 9.7.3.4 and 9.14.23).

Given a set of nonzero distances $\mathcal {D}=\{d_{ij}\}$ for $i,j = 0,1,\dots , k$, a necessary and sufficient condition for the existence of a simplex $\{\mathbf {p}_0,\dots , \mathbf {p}_k\}$ with $d_{ij}=\| \mathbf {p}_i-\mathbf {p}_j\|$ is that for every $h=2,\dots ,k$, and every h-element subset of $\{0,1,\dots , k\}$, the corresponding Cayley-Menger determinant be nonzero and its sign be $(-1)^{h}$.

Moreover, when $\mathcal {D}=\{d_{ij}\} = \{\|\mathbf {p}_i - \mathbf {p}_j\|\}$ for $\mathbf {p}_0,\dots ,\mathbf {p}_{k}\in \mathbb {R}^k$, the volume V of the simplex with vertices $\mathbf {p}_0, \dots , \mathbf {p}_{k}$ obeys

$$ \begin{align*} \operatorname{\mathrm{V}}^2 = \frac{(-1)^{k+1}}{2^k(k!)^2} \operatorname{\mathrm{CM}}(\mathcal{D}). \end{align*} $$

Recall that by definition, a simplex is nondegenerate. This means that both the volume of the simplex and Cayley-Menger determinant are nonzero.

The Cayley-Menger determinant generalises standard facts in triangle geometry: for instance, for a triangle with side lengths a, b and c, we can write this determinant explicitly as

$$ \begin{align*} \operatorname{\mathrm{CM}}(\{a,b,c\}) = a^4-2 a^2 b^2-2 a^2 c^2+b^4-2 b^2 c^2+c^4 = -(a+b+c)(a+b-c)(a-b+c)(-a+b+c), \end{align*} $$

and conclude that

$$ \begin{align*} \operatorname{Area}(\{a,b,c\})^2 = \frac{1}{16} (a+b+c)(a+b-c)(a-b+c)(-a+b+c). \end{align*} $$

This is Heron’s formula for the area of the triangle. We can see the triangle inequality (a criteria for constructability of a triangle), in these formulas: Theorem 4.2 holds if and only if one of the side lengths is greater than the sum of the other two.

4.2 Simplices and configuration spaces

We will now shift our viewpoint and, with an abuse of notation, view the simplex $\Delta $ in $\mathbb {R}^k$ as an ordered $(k+1)$-tuple of points $\Delta =(\mathbf {p}_0, \dots , \mathbf {p}_k)$. Thus, the simplex is a point in the open configuration space $C_{k+1}(\mathbb {R}^k)$. We now wish to understand the set of points $\{(\mathbf {q}_0, \dots , \mathbf {q}_k)\}$ in $C_{k+1}(\mathbb {R}^k)$ that correspond to simplices similar to our given simplex $\Delta $. By similar, we mean there is a translation, rotation and nonzero scaling of $\mathbb {R}^k$ which maps $(\mathbf {q}_0, \dots , \mathbf {q}_k)$ to $\Delta $.

Definition 4.3. For a given nondegenerate simplex $\Delta =(\mathbf {p}_0,\dots , \mathbf {p}_k)$, we let $\operatorname {\mathrm {Sim}}(\Delta )\subset C_{k+1}(\mathbb {R}^k)$ denote the space of simplices similar to $\Delta $. Then $\operatorname {\mathrm {Sim}}(\Delta )$ is the set of all $\overrightarrow {\mathbf {q}} \in C_{k+1}(\mathbb {R}^k)$, such that

$$ \begin{align*}r_{ijl}(\kern1.5pt\overrightarrow{\mathbf{q}} )= r_{ijl}(\Delta) \quad \text{for all}\ \ i\neq j\neq l\neq i.\end{align*} $$

That is, when $\overrightarrow {\mathbf {q}} =(\mathbf {q}_0,\dots , \mathbf {q}_k)$ is extrinsically similar to $\Delta $, then we have

$$ \begin{align*}r_{ijl}(\kern1.5pt\overrightarrow{\mathbf{q}} ) = \frac{\|\mathbf{q}_i - \mathbf{q}_j\|}{\|\mathbf{q}_i - \mathbf{q}_l\|} = \frac{\|\mathbf{p}_i - \mathbf{p}_j\|}{\|\mathbf{p}_i - \mathbf{p}_l\|} = r_{ijl}(\Delta).\end{align*} $$

Our next aim is to show that $\operatorname {\mathrm {Sim}}(\Delta )$ is a submanifold of $C_{k+1}(\mathbb {R}^k)$, by proving that $\operatorname {\mathrm {Sim}}(\Delta )$ is diffeomorphic to $\operatorname {O}(k)\times (0,\infty )\times \mathbb {R}^k$. We also aim to understand how the boundary of $\operatorname {\mathrm {Sim}}(\Delta )$ sits inside of $C_{k+1}[\mathbb {R}^k]$. In order to do this, we will associate a matrix to $\operatorname {\mathrm {Sim}}(\Delta )$ and look at the polar decomposition of that matrix. So we will need to use some results from linear algebra along the way. These are all found in Appendix A.

Definition 4.4. Given a configuration $\overrightarrow {\mathbf {q}} =(\mathbf {q}_0,\dots , \mathbf {q}_k)$ in $\operatorname {\mathrm {Sim}}(\Delta )$, we define the $k\times k$ matrix $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$ by

$$ \begin{align*}\Pi(\kern1.5pt\overrightarrow{\mathbf{q}} ) = \begin{bmatrix} \pi_{10}(\kern1.5pt\overrightarrow{\mathbf{q}} ) & \dots & \pi_{k0}(\kern1.5pt\overrightarrow{\mathbf{q}} ) \end{bmatrix}.\end{align*} $$

Proposition 4.5. Given a configuration $ \overrightarrow {\mathbf {q}} = \{\mathbf {q}_0, \ldots , \mathbf {q}_k\}$ in $\operatorname {\mathrm {Sim}}(\Delta )$, the matrix $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$ has rank k, and the $k\times k$ matrix $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )^T\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$ is given by

$$ \begin{align*}\Pi(\kern1.5pt\overrightarrow{\mathbf{q}} )^T\Pi(\kern1.5pt\overrightarrow{\mathbf{q}} ) = \begin{bmatrix}\cos(\theta_{ij})\end{bmatrix} = \begin{bmatrix}\frac12 (r_{0ij}(\kern1.5pt\overrightarrow{\mathbf{q}} ) + r_{0ji}(\kern1.5pt\overrightarrow{\mathbf{q}} ) - r_{ij0}(\kern1.5pt\overrightarrow{\mathbf{q}} ) r_{ji0})(\kern1.5pt\overrightarrow{\mathbf{q}} )\end{bmatrix} = P^2(\Delta),\end{align*} $$

where $P(\Delta )$ is a uniquely determined symmetric positive-definite matrix.

The matrix $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )^T\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$ is called the Gram matrix, and it consists of the dot products of the $\pi _{ij}(\kern1.5pt \overrightarrow {\mathbf {q}} )$, which are the cosines of the angles between the unit vectors $\pi _{ij}(\kern1.5pt \overrightarrow {\mathbf {q}} )$.

Proof. Recall the law of cosines: $c^2 = a^2 + b^2 - 2ab\cos C$, where c is the length of the side of a triangle $\triangle ABC$ opposite angle C and a and b are the lengths of the sides subtending angle C. In the case $a = \| \mathbf {q}_i - \mathbf {q}_0\|$, $b = \| \mathbf {q}_j - \mathbf {q}_0\|$ and $c = \| \mathbf {q}_i - \mathbf {q}_j\|$, angle C is $\theta _{ij}$, and we have

$$ \begin{align*} \cos \theta_{ij} &= \dfrac{a^2 + b^2 - c^2}{2ab} = \dfrac12 \left( \dfrac{a}{b} + \dfrac{b}{a} - \dfrac{c^2}{ab} \right) \\ &= \dfrac12 (r_{0ij}(\kern1.5pt\overrightarrow{\mathbf{q}} ) + r_{0ji}(\kern1.5pt\overrightarrow{\mathbf{q}} ) - r_{ij0}(\kern1.5pt\overrightarrow{\mathbf{q}} ) r_{ji0}(\kern1.5pt\overrightarrow{\mathbf{q}} )). \end{align*} $$

Because the $r_{ijl}$ are the same for all configurations in $\operatorname {\mathrm {Sim}}(\Delta )$, this matrix is the same as for $\Pi (\Delta )$ associated with $\Delta =(\mathbf {p}_0, \dots , \mathbf {p}_k)\in C_{k+1}(\mathbb {R}^k)$. The following shows $\Pi (\Delta )$ has rank k:

$$ \begin{align*} 0 < {\mathrm{Vol}}_k(\Delta) &= \dfrac1{k!} \det\begin{bmatrix} \mathbf{p}_1 - \mathbf{p}_0 & \dots & \mathbf{p}_k - \mathbf{p}_0\end{bmatrix}\\[3pt] &= \dfrac{\| \mathbf{p}_1 - \mathbf{p}_0\|\times \cdots \times \| \mathbf{p}_k - \mathbf{p}_0\|}{k!} \det \begin{bmatrix}\pi_{10}(\kern1.5pt\overrightarrow{\mathbf{p}} ) &\dots & \pi_{k0}(\kern1.5pt\overrightarrow{\mathbf{p}} )\end{bmatrix} \\[3pt] &= \dfrac{\| \mathbf{p}_1 - \mathbf{p}_0\|\times \cdots \times \| \mathbf{p}_k - \mathbf{p}_0\|}{k!} \det\Pi(\Delta). \end{align*} $$

By construction, $\Pi (\Delta )^T\Pi (\Delta )$ is a symmetric matrix. We have just shown that $\Pi (\Delta )$ has full column rank. Thus, by Theorem A.3, we know $\Pi (\Delta )^T\Pi (\Delta )$ is positive-definite. Since $\Pi (\Delta )^T\Pi (\Delta )=\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )^T\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$, we again use Theorem A.3 to deduce $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$ also has rank k. Now, using other standard results from linear algebra (Theorem A.1 and Remark A.2), we can deduce there is a unique symmetric positive-definite matrix $P(\Delta )$, such that $P(\Delta ) = (\Pi (\Delta )^T \Pi (\Delta ))^{1/2}$.

We now use the polar decomposition theorem for a matrix to get a better understanding of $\operatorname {\mathrm {Sim}}(\Delta )$. Recall, from Theorem A.4, that the polar decomposition of a $k\times k$ real matrix A is a factorisation of the form $A=UP$, where U is orthogonal and P is a positive semidefinite symmetric matrix. This decomposition is unique when A is nonsingular (intuitively, if A is interpreted as a linear transformation of $\mathbb {R}^k$, then the polar decomposition separates it into a rotation or reflection U of $\mathbb {R}^k$, and a scaling of the space along a set of k orthogonal axes).

Proposition 4.6. If $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$, then $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} ) = U(\kern1.5pt \overrightarrow {\mathbf {q}} ) P(\Delta )$, where $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ is a uniquely determined $k\times k$ orthogonal matrix which is a smooth function of $ \overrightarrow {\mathbf {q}} $. The $k\times k$ matrix $P(\Delta )$ depends only on $\Delta $, and is a symmetric positive-definite matrix.

Proof. In our case, since $\operatorname {\mathrm {rank}}\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )=k$, the polar decomposition theorem (Theorem A.4) tells us that for the $k\times k$ matrix $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$, there is a unique decomposition into $k\times k$ matrices: $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} ) = U(\kern1.5pt \overrightarrow {\mathbf {q}} )P$. Here, matrix $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ is orthogonal, and $P=(\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )^T \Pi (\kern1.5pt \overrightarrow {\mathbf {q}} ))^{1/2}$. From Proposition 4.5, we know

$$ \begin{align*}P=(\Pi(\kern1.5pt \overrightarrow{\mathbf{q}} )^T \Pi(\kern1.5pt \overrightarrow{\mathbf{q}} ))^{1/2} = (\Pi(\Delta)^T \Pi(\Delta))^{1/2} = P(\Delta).\end{align*} $$

Thus, for each $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$, we have the same symmetric positive-definite $P(\Delta )$ matrix. The dependence of $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$ on $ \overrightarrow {\mathbf {q}} $ is clearly smooth; that $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ depends smoothly on $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} )$, and, hence, on $ \overrightarrow {\mathbf {q}} $, and $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ is smooth, were shown in [Reference Dieci and Eirola12] (see Remark A.5).

Let $\operatorname {O}(k)$ be the set of all orthogonal $k\times k$ matrices. Roughly speaking, Proposition 4.6 says that for a (nondegenerate) simplex $\Delta $, we can obtain a different configuration in $\operatorname {\mathrm {Sim}}(\Delta )$ by multiplying $P(\Delta )$ on the left by a matrix in $\operatorname {O}(k)$. This leads us to define the following map.

Definition 4.7. The pose map $ps: \operatorname {\mathrm {Sim}}(\Delta ) \rightarrow \operatorname {O}(k)$ is defined by $ps(\kern1.5pt \overrightarrow {\mathbf {q}} )=U(\kern1.5pt \overrightarrow {\mathbf {q}} )$, where $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ is the unique orthogonal matrix $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$, such that $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} ) = U(\kern1.5pt \overrightarrow {\mathbf {q}} )P(\Delta )$ (as found in Proposition 4.6). For a simplex $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$, we call the corresponding element $U(\kern1.5pt \overrightarrow {\mathbf {q}} ) \in \operatorname {O}(k)$ the pose of $ \overrightarrow {\mathbf {q}} $.

We can now deduce the structure of $\operatorname {\mathrm {Sim}}(\Delta )$ as a submanifold of $C_{k+1}[\mathbb {R}^k]$.

Theorem 4.8. If $\Delta $ is a nondegenerate k-simplex in $\mathbb {R}^k$, then $\operatorname {\mathrm {Sim}}(\Delta )\subset C_{k+1}(\mathbb {R}^k)$ is a submanifold diffeomorphic to $\operatorname {O}(k) \times (0, \infty ) \times \mathbb {R}^k$.

Proof. We will define the following pair of maps:

$$ \begin{align*}\operatorname{O}(k) \times (0,\infty) \times \mathbb{R}^k \mathop{\longrightarrow}\limits^{\scriptstyle i_\Delta}_{\scriptstyle} \operatorname{\mathrm{Sim}}(\Delta) \subset C_{k+1}(\mathbb{R}^k) \mathop{\longrightarrow}\limits^{\scriptstyle ps_\Delta}_{\scriptstyle} \operatorname{O}(k) \times (0,\infty) \times \mathbb{R}^k.\end{align*} $$

We will then prove that both maps are smooth, that the first map $i_\Delta $ is onto $\operatorname {\mathrm {Sim}}(\Delta )$ and the composition is the identity. This will prove that $i_\Delta $ is a diffeomorphism onto $\operatorname {\mathrm {Sim}}(\Delta )$.

We have been given $\Delta =(\mathbf {p}_0,\ldots , \mathbf {p}_k)\in C_{k+1}(\mathbb {R}^k)$. We can then define the set of ratios $\{ r_{ijl}(\Delta )\}$, the set of unit vectors $\{\pi _{ij}(\Delta )\}$ and the matrix $\Pi (\Delta )$. We know that $\Pi (\Delta ) = U(\Delta )P(\Delta )$, from Proposition 4.6. Moreover, without loss of generality, we can assume that $\Delta $ is chosen such that $U(\Delta )=I_k$ (the identity matrix), and so $\Pi (\Delta ) = P(\Delta )$.

Let $A=\begin {bmatrix} \mathbf {v}_1 \ \dots \ \mathbf {v}_k\end {bmatrix}\in \operatorname {O}(k)$, and define a new set of unit vectors by $\pi _{ij} =\begin {bmatrix} \mathbf {v}_1 \ \dots \ \mathbf {v}_k\end {bmatrix} \pi _{ij}(\Delta )$ (intuitively, the new $\pi _{ij}$ are $\pi _{ij}(\Delta )$ rotated/reflected by A). If we also let $r_{ijl} = r_{ijl}(\Delta )$, then we define the map $i_\Delta $ by

$$ \begin{align*} i_\Delta (A, \lambda, \mathbf{q}_0) & = (\mathbf{q}_0, \mathbf{q}_0 + \lambda \pi_{10}, \mathbf{q}_0 + \lambda r_{021} \pi_{20}, \ldots, \mathbf{q}_0 + \lambda r_{0k1} \pi_{k0}) = \overrightarrow{\mathbf{q}}. \end{align*} $$

This map $i_\Delta $ is clearly smooth in A, $\lambda $ and $\mathbf {q}_0$. Furthermore, $i_\Delta (A, \lambda , \mathbf {q}_0)$ has $r_{ijl}(\kern1.5pt \overrightarrow {\mathbf {q}} )=r_{ijl}(\Delta )$ by construction, and so lies in $\operatorname {\mathrm {Sim}}(\Delta )$.

We prove that $i_\Delta $ is onto $\operatorname {\mathrm {Sim}}(\Delta )$. Given $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$, we use Proposition 4.6, and our assumption that $U(\Delta )=I_k$, to write

$$ \begin{align*}\Pi(\kern1.5pt \overrightarrow{\mathbf{q}} ) = \begin{bmatrix} \pi_{10}(\kern1.5pt \overrightarrow{\mathbf{q}} )\ \dots \ \pi_{k0}(\kern1.5pt \overrightarrow{\mathbf{q}} )\end{bmatrix} = U(\kern1.5pt \overrightarrow{\mathbf{q}} ) P(\Delta)=U(\kern1.5pt \overrightarrow{\mathbf{q}} )\Pi(\Delta).\end{align*} $$

This means that $\pi _{j0}(\kern1.5pt \overrightarrow {\mathbf {q}} ) = U(\kern1.5pt \overrightarrow {\mathbf {q}} )\pi _{j0}(\Delta )$ for $j=1,\dots , k$. Remembering that $\pi _{j0}=-\pi _{0j}$, we then deduce the other values of $\pi _{j}(\kern1.5pt \overrightarrow {\mathbf {q}} )$ satisfy the same relationship:

$$ \begin{align*} \pi_{ij}(\kern1.5pt \overrightarrow{\mathbf{q}} ) &= \dfrac{\mathbf{q}_j - \mathbf{q}_i}{\|\mathbf{q}_j - \mathbf{q}_i\|} = \dfrac{\mathbf{q}_j - \mathbf{q}_0}{\|\mathbf{q}_j - \mathbf{q}_i\|} + \dfrac{\mathbf{q}_0 - \mathbf{q}_i}{\|\mathbf{q}_j - \mathbf{q}_i\|} \\[5pt] &= \dfrac{\| \mathbf{q}_j - \mathbf{q}_0\|}{\| \mathbf{q}_j - \mathbf{q}_i\|} \dfrac{\mathbf{q}_j - \mathbf{q}_0}{\|\mathbf{q}_j - \mathbf{q}_0\|} + \dfrac{\|\mathbf{q}_0 - \mathbf{q}_i\|} {\| \mathbf{q}_j - \mathbf{q}_i\|} \dfrac{\mathbf{q}_0 - \mathbf{q}_i}{\|\mathbf{q}_0 - \mathbf{q}_i\|} \\[5pt] & = r_{j0i}(\kern1.5pt \overrightarrow{\mathbf{q}} ) \pi_{j0}(\kern1.5pt \overrightarrow{\mathbf{q}} ) + r_{i0j}(\kern1.5pt \overrightarrow{\mathbf{q}} ) \pi_{0i}(\kern1.5pt \overrightarrow{\mathbf{q}} ) \\[5pt] & = r_{j0i}(\kern1.5pt \overrightarrow{\mathbf{q}} ) U(\kern1.5pt \overrightarrow{\mathbf{q}} )\pi_{j0}(\Delta) + r_{i0j}(\kern1.5pt \overrightarrow{\mathbf{q}} ) U(\kern1.5pt \overrightarrow{\mathbf{q}} )\pi_{0i}(\Delta) = U(\kern1.5pt \overrightarrow{\mathbf{q}} )\pi_{ij}(\Delta). \end{align*} $$

Thus, for $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$, we have $\pi _{ij}(\kern1.5pt \overrightarrow {\mathbf {q}} ) = U(\kern1.5pt \overrightarrow {\mathbf {q}} )\pi _{ij}(\Delta )$. For the map $i_\Delta $, we let $\lambda =\|\mathbf {q}_1-\mathbf {q}_0\|$, then $i_\Delta ( U(\kern1.5pt \overrightarrow {\mathbf {q}} ), \|\mathbf {q}_1-\mathbf {q}_0\|, \mathbf {q}_0) = (\mathbf {q}_0,\mathbf {q}_1,\dots , \mathbf {q}_k)= \overrightarrow {\mathbf {q}} $. To see this last equation, note that

$$ \begin{align*}\mathbf{q}_1=\mathbf{q}_0+\lambda\frac{\mathbf{q}_1-\mathbf{q}_0}{\|\mathbf{q}_1-\mathbf{q}_0\|} \quad \text{and}\quad \mathbf{q}_2= \mathbf{q}_0 + \lambda r_{021}(\kern1.5pt \overrightarrow{\mathbf{q}} ) \pi_{20}(\kern1.5pt \overrightarrow{\mathbf{q}} ) = \mathbf{q}_0+\lambda\dfrac{\| \mathbf{q}_2-\mathbf{q}_0 \|}{\| \mathbf{q}_1-\mathbf{q}_0 \|}\dfrac{\mathbf{q}_2-\mathbf{q}_0}{\| \mathbf{q}_2-\mathbf{q}_0 \|}.\end{align*} $$

We can now define the map $ps_\Delta \colon \! \operatorname {\mathrm {Sim}}(\Delta )\rightarrow \operatorname {O}(k)\times (0,\infty )\times \mathbb {R}^k$ using the pose map from Definition 4.7. We define $ps_\Delta (\kern1.5pt \overrightarrow {\mathbf {q}} ) := (ps(\kern1.5pt \overrightarrow {\mathbf {q}} ),\|\mathbf {q}_1-\mathbf {q}_0\|, \mathbf {q}_0) = (U(\kern1.5pt \overrightarrow {\mathbf {q}} ),\|\mathbf {q}_1-\mathbf {q}_0\|, \mathbf {q}_0) $. From Proposition 4.6, we know $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ depends smoothly on $ \overrightarrow {\mathbf {q}} $, and, hence, $ps_\Delta $ depends smoothly on $ \overrightarrow {\mathbf {q}} $. A moment’s thought shows that, by construction, we have $ps_\Delta \circ i_\Delta =\operatorname {\mathrm {id}}$.

This means that $i_\Delta $ (and, hence, $ps_\Delta $) is a diffeomorphism, and $\operatorname {\mathrm {Sim}}(\Delta )$ is a submanifold of $C_{k+1}(\mathbb {R}^k)$.

We note that this theorem shows that the polar decomposition is a diffeomorphism for matrices of full rank. This is analogous to the smoothness result for the polar decomposition of Dieci and Eirola [Reference Dieci and Eirola12]. This theorem is not true for rank-deficient matrices, which explains why we have restricted our attention to spheres $S^{k-1}$ isotopic to each other in $\mathbb {R}^k$; Haefliger’s theorem might guarantee the existence of an isotopy of the spheres in a higher-dimensional $\mathbb {R}^K$, but $\operatorname {\mathrm {Sim}}(\Delta )$ would still consist of rank $k < K$ matrices and, hence, be harder to control.

The next theorem shows $\operatorname {\mathrm {Sim}}(\Delta )$ has a well-understood structure in the boundary $\partial C_{k+1}[\mathbb {R}^k]$.

Theorem 4.9. If $\Delta $ is a nondegenerate k-simplex in $\mathbb {R}^k$, then the boundary of $\operatorname {\mathrm {Sim}}(\Delta )$ corresponds to configurations in the interior of the $(0, \ldots , k)$ face of $\partial C_{k+1}[\mathbb {R}^k]$, and is diffeomorphic to $\operatorname {O}(k) \times \{0\} \times \mathbb {R}^k$.

Proof. By assumption, the given simplex $\Delta =(\mathbf {p}_0, \dots , \mathbf {p}_k)$ is nondegenerate, so none of the vertices of $\Delta $ coincide, and the ratios $r_{ijl}(\Delta )$ are never 0 nor $\infty $. If we take $(A,\lambda ,\mathbf {p})\in \operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$ and consider the points in $i_\Delta (A,\lambda , \mathbf {p}_0)$, we see these points coincide if and only if $\lambda =0$, in which case, they all coincide. This means $i_\Delta (A,0, \mathbf {p}_0)\subset \partial C_{k+}[\mathbb {R}^k]$ and is in the $(0,1,\dots , k)$ face. Since the boundary of the $(0,1,\dots , k)$ face consists of configurations with $r_{ijl}(\Delta ) = 0$, or $r_{ijl}(\Delta ) = \infty $, it follows immediately that $i_\Delta (A,0, \mathbf {p}_0)$ is in the interior of this face.

Corollary 4.10. If $\Delta $ is a nondegenerate k-simplex in $\mathbb {R}^k$, then $\operatorname {\mathrm {Sim}}(\Delta )$ is a submanifold of $C_{k+1}[\mathbb {R}^k]$ that is diffeomorphic to $\operatorname {O}(k) \times [0,\infty ) \times \mathbb {R}^k$.

5 Simplices inscribed in spheres

We next move to Step 1 of the method described in Section 3. We will be looking at $C^\infty $-smooth embeddings of $S^{k-1}$ in $\mathbb {R}^k$. We will always assume that these embeddings are regular, that is, the embedding induces an injection of tangent spaces everywhere. This is particularly relevant for embeddings of $S^1$ in $\mathbb {R}^2$, where we assume the tangent vector is nowhere zero (otherwise, it is possible to smoothly describe an embedded curve with corners, by allowing the tangent vector to smoothly change to zero at each corner). Now, given any such regular $C^\infty $-smooth embedding $\gamma \colon \! S^{k-1}\hookrightarrow \mathbb {R}^k$, we can view the corresponding configuration space $C_{k+1}[\gamma (S^{k-1})]$ as a submanifold of $C_{k+1}[\mathbb {R}^k]$.

We next move to Step 2. We take the standard embedding of the $(k-1)$ sphere in $\mathbb {R}^k$, that is, $\operatorname {\mathrm {id}} \colon \! S^{k-1} \hookrightarrow \mathbb {R}^k$, and consider the corresponding configuration spaces. In this special case, we use the notation $C_{k+1}[S^{k-1}] = C_{k+1}[\operatorname {\mathrm {id}}(S^{k-1})]$.

We need to show that there is a transverse intersection between $C_{k+1}[S^{k-1}]$ and $\operatorname {\mathrm {Sim}}(\Delta )$ in $C_{k+1}[\mathbb {R}^k]$, in other words, $C_{k+1}(\operatorname {\mathrm {id}})\pitchfork \operatorname {\mathrm {Sim}}(\Delta )$. We also need to compute the homology class of the intersection of $C_{k+1}[S^{k-1}]\cap \operatorname {\mathrm {Sim}}(\Delta )$ in $\operatorname {\mathrm {Sim}}(\Delta )$.

Proof. Since $\operatorname {\mathrm {Sim}}(\Delta )$ and $C_{k+1}[S^{k-1}]$ are boundary disjoint (Proposition 5.1), we just need to consider $\operatorname {\mathrm {Sim}}(\Delta ) \cap C_{k+1}(S^{k-1})$. Now, every simplex $\Delta $ has a unique circumsphere; a $(k-1)$-sphere passing through all of the $k+1$ vertices. The radius of the circumsphere and coordinates of the circumcentre are well known (see, for instance, Proposition 9.7.3.7 [Reference Berger5] or [Reference Coxeter11, Reference Schoenberg38]). Indeed, the circumradius R of the simplex $\Delta $ is given by

$$ \begin{align*} R^2 = - \frac{ \left| \begin{matrix} 0 & d_{01}^2 & \cdots & d_{0k}^2 \\ d_{10}^2 & 0 & \cdots & d_{1k}^2 \\ \vdots & \vdots & & \vdots \\ d_{k0}^2 & d_{k2}^2 & \cdots & 0 \end{matrix} \right| } {2 \operatorname{\mathrm{CM}}(\{\mathbf{p}_0,\dots, \mathbf{p}_k\})}. \end{align*} $$

Given $\Delta $, we can scale and translate it to give a new $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$ with circumradius $R=1$, and circumcentre $\mathbf {0}$. Thus, $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta ) \cap C_{k+1}(S^{k-1})$ and the intersection is nonempty. Intuitively, we obtain all other points of $\operatorname {\mathrm {Sim}}(\Delta ) \cap C_{k+1}(S^{k-1})$ by rotating/reflecting $ \overrightarrow {\mathbf {q}} $ via $A \overrightarrow {\mathbf {q}} $ for $A\in \operatorname {O}(k)$. More formally, we define the diffeomorphism $r\colon \! \operatorname {O}(k)\rightarrow \operatorname {\mathrm {Sim}}(\Delta ) \cap C_{k+1}(S^{k-1})$ by $r(A) =A \overrightarrow {\mathbf {q}} = (A\mathbf {q}_0, \dots , A\mathbf {q}_k)$.

We now want to show that $\operatorname {\mathrm {Sim}}(\Delta )$ intersects $C_{k+1}[S^{k-1}]$ transversally. Specifically, we take any inscribed simplex $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )\cap C_{k+1}[S^{k-1}]$, and we want to show that

$$ \begin{align*}T_{\overrightarrow{\mathbf{q}} } (\operatorname{\mathrm{Sim}}(\Delta)) \oplus T_{\overrightarrow{\mathbf{q}} } (C_{k+1}[S^{k-1}]) = T_{\overrightarrow{\mathbf{q}} }(C_{k+1}[\mathbb{R}^k]).\end{align*} $$

We first note that the orthogonal complement of $T_{ \overrightarrow {\mathbf {q}} } (C_{k+1}[S^{k-1}])$ in $T_{ \overrightarrow {\mathbf {q}} }(C_{k+1}[\mathbb {R}^{k}])$ is the $(k+1)$-dimensional space with orthonormal basis $\mathcal {B} = \{(\mathbf {q}_0, 0, \dots , 0), \dots , (0,\dots ,\mathbf {q}_{k})\}$. Next, observe that $T_{ \overrightarrow {\mathbf {q}} } (\operatorname {\mathrm {Sim}}(\Delta )) \cong T_{ \overrightarrow {\mathbf {q}} }(\operatorname {O}(k))\oplus T_{ \overrightarrow {\mathbf {q}} }[0,\infty ) \oplus T_{ \overrightarrow {\mathbf {q}} }(\mathbb {R}^k)$. The tangent space $T_{ \overrightarrow {\mathbf {q}} }(\operatorname {\mathrm {Sim}}(\Delta ))$ thus contains the vectors $(\mathbf {e}_1, \dots , \mathbf {e}_1), \dots , (\mathbf {e}_k,\dots ,\mathbf {e}_k)$ from the translational component of $\operatorname {\mathrm {Sim}}(\Delta )$, as well as the vector $(\mathbf {q}_0,\dots ,\mathbf {q}_{k})$ from scaling the configuration $ \overrightarrow {\mathbf {q}} $. Writing these vectors in the basis $\mathcal {B}$, we get the matrix:

$$ \begin{align*} M = \left( \begin{matrix} q_{0,0} & q_{1,0} & \cdots & q_{k,0} \\ q_{0,1} & q_{1,1} & \cdots & q_{k,1} \\ \vdots & \vdots & & \vdots \\ q_{0,k} & q_{1,k} & \cdots & q_{k,k} \\ 1 & 1 & \cdots & 1 \\ \end{matrix} \right). \end{align*} $$

Subtracting the last column from the rest, we get

$$ \begin{align*} M' = \left( \begin{matrix} \mathbf{q}_0 - \mathbf{q}_{k} & \mathbf{q}_1 - \mathbf{q}_{k} & \cdots & \mathbf{q}_{k-1} - \mathbf{q}_{k} & \mathbf{q}_{k} \\ 0 & 0 & \cdots & 0 & 1 \\ \end{matrix} \right). \end{align*} $$

The determinant of this matrix is $\pm 1$ multiplied by the determinant of the upper-left $k \times k$ principal minor. But that determinant is positive because $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$ and is nondegenerate. Thus, the $k+1$ tangent vectors are linearly independent, and we have proven transversality.

Before we move on to determine the homology class corresponding to inscribed simplices, we pause to remember that $\operatorname {O}(k)$ has two connected components. One component, $\operatorname {SO}(k)$, is a subgroup of $\operatorname {O}(k)$, and consists of all orthogonal matrices with determinant $+1$. The other component contains all orthogonal matrices with determinant $-1$.

In order to sensibly discuss homology classes, we restrict our attention to the submanifold of simplices diffeomorphic to $\operatorname {SO}(k)\times [0,\infty )\times \mathbb {R}^k$, which we denote $\operatorname {\mathrm {Sim}}^+(\Delta )$. We could equally restrict our attention to $\operatorname {\mathrm {Sim}}^-(\Delta ) := \operatorname {\mathrm {Sim}}(\Delta )\setminus \operatorname {\mathrm {Sim}}^+(\Delta )$.

Proposition 5.3. Given a nondegenerate simplex $\Delta =(\mathbf {p}_0,\dots , \mathbf {p}_k)\in C_{k+1}(\mathbb {R}^k)$, and given the configuration space $C_{k+1}[S^{k-1}]$ corresponding to the standard embedding of $S^{k-1}$ in $\mathbb {R}^k$, then in $\operatorname {\mathrm {Sim}}(\Delta )$

$$ \begin{align*}H_*(\operatorname{O}(k);\mathbb{Z}) \cong H_*(\operatorname{\mathrm{Sim}}(\Delta) \cap C_{k+1}[S^{k-1}];\mathbb{Z}).\end{align*} $$

Moreover, the pose map is a diffeomorphism $ps\colon \! \operatorname {\mathrm {Sim}}^+(\Delta )\cap C_{k+1}[S^{k-1}] \rightarrow \operatorname {SO}(k)$ which induces an isomorphism on integral homology taking the top class of $\operatorname {\mathrm {Sim}}^+(\Delta )\cap C_{k+1}[S^{k-1}]$ in $\operatorname {\mathrm {Sim}}(\Delta ^+)$ to the top class of $\operatorname {SO}(k)$. A similar result holds for $\operatorname {\mathrm {Sim}}^-(\Delta )\cap C_{k+1}[S^{k-1}]$.

Proof. Without loss of generality, we can normalise $\Delta =(\mathbf {p}_1,\dots , \mathbf {p}_k)$ so that it lies on $S^{k-1}$; that is $\Delta $ has circumcentre $\mathbf {0}$ and circumradius $R=1$. We then have the following maps

$$ \begin{align*}\operatorname{O}(k)\xrightarrow{r} \operatorname{\mathrm{Sim}}(\Delta)\cap C_{k+1}[S^{k-1}] \xrightarrow{ps} \operatorname{O}(k).\end{align*} $$

Here, the map r is the rotation/reflections diffeomorphism previously defined in Proposition 5.2. That is, for $A\in \operatorname {O}(k)$, we define $r(A)= A\Delta =(A\mathbf {p}_0, \dots , A\mathbf {p}_k)$. The map $ps$ is the pose map from Definition 4.7: for $ \overrightarrow {\mathbf {q}} \in \operatorname {\mathrm {Sim}}(\Delta )$, we define $ps(\kern1.5pt \overrightarrow {\mathbf {q}} )=U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ (where $U(\kern1.5pt \overrightarrow {\mathbf {q}} )$ is the unique orthogonal matrix, such that $\Pi (\kern1.5pt \overrightarrow {\mathbf {q}} ) = U(\kern1.5pt \overrightarrow {\mathbf {q}} )\Pi (\Delta )$). From Proposition 5.2, we know that r is a diffeomorphism, and by construction, $ps\circ r = \operatorname {\mathrm {id}}$. Hence, the pose map is also a diffeomorphism.

From Proposition 5.2, we recall that $\operatorname {\mathrm {Sim}}(\Delta )$ intersects $C_{k+1}[S^{k-1}]$ transversally, and, moreover, that the intersection is diffeomorphic to $\operatorname {O}(k)$. Now, since $\operatorname {\mathrm {Sim}}(\Delta )\cong \operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$, we have $\operatorname {\mathrm {Sim}}(\Delta )\cap C_{k+1}[S^{k-1}]$ is a deformation retract of $\operatorname {\mathrm {Sim}}(\Delta )$.

When we put this together and take the homology of the spaces, we see

$$ \begin{align*}H_*(\operatorname{O}(k))\xrightarrow{r_*} H_*( \operatorname{\mathrm{Sim}}(\Delta)\cap C_{k+1}[S^{k-1}]) \cong H_*(\operatorname{\mathrm{Sim}}(\Delta)) \xrightarrow{ps_*} H_*(\operatorname{O}(k)) .\end{align*} $$

Since $ps_* \circ r_* = \operatorname {\mathrm {id}}$, we know $ps_*$ is an isomorphism. Hence,

$$ \begin{align*}H_*(\operatorname{O}(k)) \cong H_*(\operatorname{\mathrm{Sim}}(\Delta) \cap C_{k+1}[S^{k-1}]).\end{align*} $$

We could also choose to restrict r and $ps$ as follows:

$$ \begin{align*}\operatorname{SO}(k)\xrightarrow{r} \operatorname{\mathrm{Sim}}^+(\Delta)\cap C_{k+1}(S^{k-1}) \xrightarrow{ps} \operatorname{SO}(k).\end{align*} $$

Then, repeating the previous argument gives

$$ \begin{align*}ps_*([\operatorname{\mathrm{Sim}}^+(\Delta) \cap C_{k+1}(S^{k-1})] )=[r(\operatorname{SO}(k))] = [\operatorname{\mathrm{Sim}}^+(\Delta)] \in H_{\frac{k(k-1)}{2}}(\operatorname{\mathrm{Sim}}^+(\Delta)).\\[-41pt] \end{align*} $$

We now have all the pieces needed for our main theorem.

We immediately recover Gromov’s result [Reference Gromov16] (but for $C^\infty $-smooth embeddings) as a corollary.

In fact, we have proved more than Gromov.

In [Reference Matschke31], Matschke proves that for generic $C^\infty $-smooth embeddings of the circle in the plane, there are an odd number of loops of inscribed triangles similar to $\Delta $ that wind an odd number of times around the embedded circle. Figure 2 shows such a loop of inscribed equilateral triangles in an irregular embedding of a circle in the plane. We recover Matschke’s result in Corollary 5.6, and strengthen it with the fact that the degree of the map is 1.

Figure 2

In this irregular embedding of a circle in the plane, we see a series of inscribed equilateral triangles. By following the highlighted vertex around the curve, we see there is a loop of such inscribed triangles.

We end by, again, noting that Meyerson [Reference Meyerson34] and Nielsen [Reference Nielsen35] have results about inscribed triangles that are for Jordan curves (just assuming continuity of the embedding). By adding both a generic and smoothness assumption on our embeddings, we are able to provide more information about the structure of the set of inscribed triangles.