# A quantitative central limit theorem for
Poisson horospheres in high dimensions

Zakhar Kabluchko111University of Münster, Germany. Email:
<EMAIL_ADDRESS>Daniel Rosen222Ruhr University Bochum,
Germany. Email<EMAIL_ADDRESS>Christoph Thäle333Ruhr University Bochum,
Germany. Email<EMAIL_ADDRESS>
###### Abstract

Consider a stationary Poisson process of horospheres in a $d$-dimensional
hyperbolic space. In the focus of this note is the total surface area these
random horospheres induce in a sequence of balls of growing radius $R$. The
main result is a quantitative, non-standard central limit theorem for these
random variables as the radius $R$ of the balls and the space dimension $d$
tend to infinity simultaneously.

Keywords: central limit theorem, horospheres, hyperbolic stochastic geometry,
Poisson processes
MSC: 52A55, 60D05

##  1 Introduction and main result

The study of random geometric systems in non-Euclidean geometries is a recent
and fast growing branch of stochastic geometry. We refer to [1, 2, 3, 4, 5, 6,
7, 9, 10, 12, 15] for selected works on hyperbolic random geometric graphs,
random tessellations and random polytopes.

In this note we address an interesting generalization of the Poisson
hyperplane process to hyperbolic geometry. The study of Euclidean Poisson
hyperplanes is by now classical [11, 14, 16, 19] and was extended in [12] to
hyperbolic space, where Poisson processes of totally geodesic hypersurfaces
are studied, see also [18] for mean values in the planar case. Even more
recently, in [13] it was observed that this model fits into a one-parameter
family of so-called _Poisson $\lambda$-geodesic hyperplanes_, and the
fluctuations of the total hyperbolic surface area of such a process within a
sequence of growing balls were examined in detail. The special case we
consider here (corresponding to the choice $\lambda=1$ in [13]) is the
_Poisson horosphere process_.

Let us recall some definitions; for more details we refer the reader to [13]
and the references cited therein. A _horosphere_ in a $d$-dimensional
hyperbolic space $\mathbb{H}^{d}$ is, intuitively speaking, a sphere of
infinite radius. More formally, it is a complete totally umbilic hypersurface
of constant normal curvature $1$. For concreteness, in the Poincaré ball model
of hyperbolic space, horospheres are realized as Euclidean spheres tangent to
the boundary, see Figure 1. We denote by $\mathcal{H}$ the space of all
horospheres in $\mathbb{H}^{d}$. This space admits a transitive action by the
group of hyperbolic isometries and an invariant measure for this action, which
is unique up to a multiplicative constant and will be denoted by $\Lambda$,
see [8, 20].

Figure 1: Simulation of a Poisson process of horospheres in the Poincaré disc
model for the hyperbolic plane.

Now, let $\eta_{d}$ be a Poisson process on $\mathcal{H}$ with intensity
measure $\Lambda$, see Figure 1 for a simulation in the case $d=2$. For $R>0$,
we consider the total surface area

$S_{R,d}:=\sum_{H\in\eta_{d}}\mathcal{H}^{d-1}(H\cap B^{d}_{R})$

of $\eta_{d}$ within a hyperbolic ball $B^{d}_{R}$ around an arbitrary but
fixed point in $\mathbb{H}^{d}$ and hyperbolic radius $R>0$. Here,
$\mathcal{H}^{d-1}$ stands for the $(d-1)$-dimensional Hausdorff measure with
respect to the hyperbolic metric. In [13] it was proven that, for a fixed
space dimension $d$, the centred and normalized surface area satisfies a non-
standard central limit theorem. Namely, it converges in distribution, as
$R\to\infty$ and after centring and normalization by the standard deviation,
to a Gaussian random variable of variance ${1\over 2}$. The main result of the
present note extends this in two directions: first, we provide estimates on
the rate on convergence. Second, our bounds depend explicitly on the
dimension, providing central limit theorems for Poisson horospheres in
increasing space dimensions. To measure the distance between two random
variables $X$ and $Y$ we use the Wasserstein metric, which is defined by

$d_{\operatorname{Wass}}(X,Y):=\sup\big{|}\mathbb{E}[h(X)]-\mathbb{E}[h(Y)]\big{|},$

where the supremum is taken over all Lipschitz functions
$h:\mathbb{R}\to\mathbb{R}$ with Lipschitz constant at most one.

Our first result provides a quantitative non-standard limit theorem in a fixed
spacial dimension.

######  Theorem 1 (Central limit theorem for fixed $d$).

Let $N_{\scalebox{0.65}{$\frac{1}{2}$}}$ be a centred Gaussian random variable
of variance $\frac{1}{2}$. Fix $d\in\mathbb{N}$ and consider the surface
functional $S_{R,d}$. Then there exists a constant $C>0$ only depending on $d$
such that

$\displaystyle
d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)$
$\displaystyle\leq C\,R^{-1/2}.$

######  Remark 2.

We note that the rate of convergence $R^{-1/2}$ is the same in all space
dimensions. The same convergence rate, in all dimensions, is observed in the
central limit theorem for the total surface area of Poisson hyperplanes in
Euclidean space (see [14], but the result is also a special case of (1)
below). Let us remark that this limiting behaviour is in sharp contrast with
the cases of $\lambda$-geodesic hyperplanes with $\lambda<1$ (we recall that
horospheres correspond to $\lambda=1$). As described in [13], in those cases
the fluctuations of the surface functional are non-Gaussian in every fixed
dimension $\geq 4$. The geometric distinction between the two cases is that
horospheres are _intrinsically Euclidean_ , while the intrinsic geometry of
other $\lambda$-geodesic hyperplanes is hyperbolic.

Our second result concerns the surface functional in high dimensions. We
consider an arbitrary sequence $R=R_{d}$ which satisfies
$R_{d}\xrightarrow[d\to\infty]{}\infty$. We denote in this case by

$S_{d}:=S_{R_{d},d}=\sum_{H\in\eta_{d}}\mathcal{H}^{d-1}(H\cap B^{d}_{R_{d}})$

the high-dimensional surface area functional. Here we prove the following
quantitative non-standard limit theorem.

######  Theorem 3 (Central limit theorem for $d\to\infty$).

Let $N_{\scalebox{0.65}{$\frac{1}{2}$}}$ be a centred Gaussian random variable
of variance $\frac{1}{2}$. Let $R=R_{d}$ be a sequence satisfying
$R_{d}\to\infty$ as $d\to\infty$.

  * (a)

Suppose that $\limsup\limits_{d\to\infty}(R_{d}-\log d)<+\infty$. Then there
exists a constant $C>0$ such that

$d_{\operatorname{Wass}}\left(\frac{S_{d}-\mathbb{E}S_{d}}{\sqrt{\operatorname{Var}S_{d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq
Ce^{-R_{d}/2}.$

  * (b)

Suppose that $\limsup\limits_{d\to\infty}(R_{d}-\log d)=+\infty$. Then there
exists a constant $C>0$ such that

$d_{\operatorname{Wass}}\left(\frac{S_{d}-\mathbb{E}S_{d}}{\sqrt{\operatorname{Var}S_{d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq
C\left({1\over\sqrt{d}\,(R-\log d)}+{1\over d\,\sqrt{R-\log d}}\right).$

In particular, the surface functional satisfies a non-standard central limit
theorem as soon as $R_{d}\to\infty$.

For example, taking the radius as $R_{d}=\alpha\log d$ for $\alpha>0$, Theorem
3 gives

$d_{\operatorname{Wass}}\left(\frac{S_{d}-\mathbb{E}S_{d}}{\sqrt{\operatorname{Var}S_{d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq
C\,\begin{cases}d^{-\alpha/2}&:\alpha\leq 1\\\ d^{-1/2}(\log
d)^{-1}&:\alpha>1.\end{cases}$

######  Remark 4.

  * (i)

We note that the convergence rate in the first case of Theorem 3 is always
worse than $d^{-1/2}$ (and in particular, worse than in the second case).
Indeed, by assumption $e^{R_{d}}\leq M\,d$ for some $M>0$ and hence
$e^{-R_{d}/2}$ converges to zero slower than $d^{-1/2}$.

  * (ii)

It is natural to ask whether similar bounds holds when the Wasserstein metric
is replaced by the Kolmogorov metric $d_{\rm Kol}$. For two random variables
$X$ and $Y$ the latter is defined as $d_{\rm
Kol}(X,Y):=\sup_{s\in\mathbb{R}}|\mathbb{P}(X\leq s)-\mathbb{P}(Y\leq s)|$.
For any random variable $X$ one has the inequality $d_{\rm
Kol}(X,N_{\scalebox{0.65}{$\frac{1}{2}$}})\leq\left[{2\over\sqrt{\pi}}d_{\operatorname{Wass}}\big{(}X,N_{\scalebox{0.65}{$\frac{1}{2}$}}\big{)}\right]^{1/2}$,
see e.g. [17, Proposition 1.2.(2)]. In conjunction with Theorems 1 and 3 this
provides Kolmogorov bounds the for the normalized random variables $S_{R,d}$,
but these are likely to be not optimal. So far, we were unable to prove bounds
on the Kolmogorov distance of the same order as for the Wasserstein distance
and leave this as an open problem for future research.

  * (iii)

In the high-dimensional regime, that is if $d\to\infty$ and $R=R_{d}$, it is
also natural to ask for sharp conditions on $R$ which ensure that the centred
and normalized total surface area is asymptotically Gaussian. Theorem 3 shows
that $R_{d}\to\infty$ is sufficient, but for fixed $R$ our bounds do not yield
a central limit theorem for the surface functional. We have to leave this as
an open problem as well.

  * (iv)

The reader might be interested in a comparison with the Euclidean case, where
one considers the total surface area $S_{R,d,e}$ of a stationary and isotropic
Poisson process on the space of hyperplanes in $\mathbb{R}^{d}$ within a
centred ball of radius $R$. In this situation it holds that

$d_{\rm
Wass}\Big{(}{S_{R,d,e}-\mathbb{E}S_{R,d,e}\over\sqrt{\operatorname{Var}S_{R,d,e}}},N\Big{)}\leq
C\,d^{1/4}R^{-1/2}$ (1)

for some absolute constant $C>0$ and where $N$ denotes a standard Gaussian
random variable. Since we could not locate this result in the literature, we
provide an argument in Section 4. In particular, the bound shows that if
$d\to\infty$ we need that $R$ grows faster than $\sqrt{d}$ in order to deduce
a central limit theorem.

##  2 Proof of the main results

Before proving Theorems 1 and 3, we need to recall some preliminaries. First
we need an explicit description of the invariant measure $\Lambda$ on the
space $\mathcal{H}$ of horospheres. We fix an origin
$\mathbf{o}\in\mathbb{H}^{d}$ and parametrize an element $H\in\mathcal{H}$ by
the pair $(s,u)\in\mathbb{R}\times\SS^{d-1}$, where $s\in\mathbb{R}$ is the
signed distance from $H$ to $\mathbf{o}$ (with $s>0$ if $\mathbf{o}$ lies on
the convex side of $H$, and negative otherwise), and $u\in\SS^{d-1}$ is the
unit vector (in the tangent space $T_{\mathbf{o}}\mathbb{H}^{d}$) along the
geodesic passing through $\mathbf{o}$ and intersecting $H$ orthogonally, while
pointing outside of the convex side. The invariant measure is then defined by
the relation

$\int_{\mathcal{H}}f(H)\,\Lambda(\mathrm{d}H)=\int_{\mathbb{R}}\int_{\SS^{d-1}}f(H(s,u))\,e^{-(d-1)s}\,\mathrm{d}u\,\mathrm{d}s,$
(2)

where $f:\mathcal{H}\to\mathbb{R}$ is a non-negative measurable function and
$H(s,u)$ stands for the unique element of $\mathcal{H}$ parametrized by
$(s,u)$ as just described. Here $\mathrm{d}s$ and $\mathrm{d}u$ stand for the
Lebesgue measure on $\mathbb{R}$ and the normalized spherical Lebesgue measure
on $\SS^{d-1}$, respectively.

We will also need the following geometric computation of the volume of the
intersection $H(s)\cap B_{R}^{d}$, where $H(s)\subset\mathbb{H}^{d}$ is a
horosphere of signed distance $s\in\mathbb{R}$ from the origin $\mathbf{o}$.
Observe that this notation is justified, by rotational symmetry around
$\mathbf{o}$. In [13, Proposition 4.1] it is proven that this intersection is
empty for $|s|\geq R$, and otherwise satisfies

$\mathcal{H}^{d-1}(H(s)\cap B_{R}^{d})=\kappa_{d-1}\bigl{[}2e^{s}(\cosh
R-\cosh s)\bigr{]}^{\frac{d-1}{2}},$ (3)

where for an integer $\ell\geq 1$ we write $\kappa_{\ell}$ for the volume of
the $\ell$-dimensional Euclidean unit ball.

We also mention two elementary properties of the Wasserstein metric that will
be useful for us. For any three integrable random variables $X,Y$ and $Z$ one
has

$d_{\operatorname{Wass}}(X+Y,Z)\leq
d_{\operatorname{Wass}}(X,Z)+\mathbb{E}|Y|.$ (4)

Moreover for any $\alpha>0$ one has

$d_{\operatorname{Wass}}(\alpha X,\alpha
Y)=\alpha\,d_{\operatorname{Wass}}(X,Y).$ (5)

First we reduce the normal approximation bound to the following integral
estimate. Define

$J_{R,d}:=\int_{0}^{R}\left(1-{\cosh s-1\over\cosh
R-1}\right)^{d-1}\,\mathrm{d}s.$

######  Proposition 5.

The following bound holds true for all $d\geq 1$ and $R>0$:

$d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq\sqrt{2}\left({1\over\sqrt{d-1}\,J_{R,d}}+{2\over(d-1)\,\sqrt{J_{R,d}}}\right).$

###### Proof.

In view of the representation (2) of the invariant measure $\Lambda$ and the
expression (3) for the intersection volume, we have that

$S_{R,d}=\sum_{s\in\xi}f_{R}(s),$ (6)

where $\xi$ is an inhomogeneous Poisson process on $\mathbb{R}$ with density
$s\mapsto e^{-(d-1)s}$, and the function $f_{R}$ is defined by

$f_{R}(s)=\begin{cases}\kappa_{d-1}\bigl{[}2e^{s}(\cosh R-\cosh
s)\bigr{]}^{\frac{d-1}{2}}&:|s|\leq R,\\\ 0&:\text{else.}\end{cases}$ (7)

We decompose the random variable $S_{R,d}$ into a ‘positive’ and ‘negative’
part as follows:

$S_{R,d}=S_{R,d}^{+}+S_{R,d}^{-},$

where

$S_{R,d}^{+}:=\sum_{\begin{subarray}{c}s\in\xi\\\
s>0\end{subarray}}f_{R}(s)\qquad\text{and}\qquad
S_{R,d}^{-}:=\sum_{\begin{subarray}{c}s\in\xi\\\ s<0\end{subarray}}f_{R}(s).$

We then have

$\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}}=\frac{S_{R,d}^{+}-\mathbb{E}S_{R,d}^{+}}{\sqrt{\operatorname{Var}S_{R,d}}}+\frac{S_{R,d}^{-}-\mathbb{E}S_{R,d}^{-}}{\sqrt{\operatorname{Var}S_{R,d}}}.$

Observe that

$\operatorname{Var}S_{R,d}^{+}=\operatorname{Var}S_{R,d}^{-}={1\over
2}\,\operatorname{Var}S_{R,d},$ (8)

which follows from the evenness of the integrand in the variance
representation

$\operatorname{Var}S_{R,d}=\int_{\mathbb{R}}f_{R}^{2}(s)e^{-(d-1)s}\,\mathrm{d}s=2^{d-1}\kappa_{d-1}^{2}\int_{-R}^{R}(\cosh
R-\cosh s)^{d-1}\,\mathrm{d}s,$

where we used (2) and (3). We deduce that

$d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq
2^{-{1\over
2}}\,d_{\operatorname{Wass}}\left(\frac{S_{R,d}^{-}-\mathbb{E}S_{R,d}^{-}}{\sqrt{\operatorname{Var}S_{R,d}^{-}}},N\right)+2^{-{1\over
2}}\,\mathbb{E}\left|\frac{S_{R,d}^{+}-\mathbb{E}S_{R,d}^{+}}{\sqrt{\operatorname{Var}S_{R,d}^{+}}}\right|,$
(9)

where $N$ is a standard Gaussian random variable, and where we have used (4),
(5) together with the fact that $2^{-{1\over 2}}N$ has the same distribution
as our target random variable $N_{\scalebox{0.65}{$\frac{1}{2}$}}$.

To control the first summand in (9), we apply the following normal
approximation bound, which is a special case of a general bound for so-called
Poisson $U$-statistics [16, Theorem 4.7]. In our case it states that

$d_{\operatorname{Wass}}\left(\frac{S_{R,d}^{-}-\mathbb{E}S_{R,d}^{-}}{\sqrt{\operatorname{Var}S_{R,d}^{-}}},N\right)\leq
2{{\sqrt{\operatorname{cum}_{4}(S_{R,d}^{-})}}\over\operatorname{Var}S_{R,d}^{-}},$

where $\operatorname{cum}_{4}(S_{R,d}^{-})$ denotes the fourth cumulant of
$S_{R,d}^{-}$. The second summand in (9) is easily bounded by (noting that
$S_{R,d}^{+}\geq 0$)

$\displaystyle\frac{\mathbb{E}|S_{R,d}^{+}-\mathbb{E}S_{R,d}^{+}|}{\sqrt{\operatorname{Var}S_{R,d}^{+}}}$
$\displaystyle\leq{2\,\mathbb{E}S_{R,d}^{+}\over\sqrt{\operatorname{Var}S_{R,d}^{+}}}.$

This gives

$\displaystyle
d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)$
$\displaystyle\leq
2^{1/2}{\sqrt{\operatorname{cum}_{4}(S_{{R,d}}^{-})}\over\operatorname{Var}S_{{R,d}}^{-}}+2^{1/2}{\mathbb{E}S_{{R,d}}^{+}\over\operatorname{Var}S_{{R,d}}^{+}}.$
(10)

If we denote further $C_{d}:=2^{(d-1)/2}\kappa_{d-1}$ and define

$\displaystyle I_{1}$ $\displaystyle:=\int_{0}^{R}(\cosh R-\cosh s)^{d-1\over
2}e^{-{d-1\over 2}s}\,\mathrm{d}s,$ $\displaystyle I_{2}$
$\displaystyle:=\int_{0}^{R}(\cosh R-\cosh s)^{d-1}\,\mathrm{d}s,$
$\displaystyle I_{4}$ $\displaystyle:=\int_{0}^{R}(\cosh R-\cosh
s)^{2(d-1)}e^{-{(d-1)}s}\,\mathrm{d}s,$

then we compute, using (6) and (7), that

$\displaystyle\mathbb{E}S_{R,d}^{+}=C_{d}I_{1},\qquad\operatorname{Var}(S_{R,d}^{\pm})={C_{d}^{2}}I_{2},\qquad\operatorname{cum}_{4}(S_{R,d}^{-})={C_{d}^{4}}I_{4}.$

Here the expectation and variance are computed with the help of the
(multivariate) Mecke equation for Poisson processes, and the fourth cumulant
using [14, Corollary 1]. Plugging this into (10) finally gives

$d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq\sqrt{2}\left[{\sqrt{I_{4}}\over
I_{2}}+{I_{1}\over\sqrt{I_{2}}}\right].$ (11)

Now we use the following trivial estimates for $I_{1}$ and $I_{4}$:

$\displaystyle I_{1}$ $\displaystyle\leq(\cosh R_{d}-1)^{d-1\over
2}\cdot{2\over d-1},$ $\displaystyle I_{4}$ $\displaystyle\leq(\cosh
R_{d}-1)^{2(d-1)}\cdot{1\over d-1}.$

Moreover, for $I_{2}$ we write:

$\displaystyle I_{2}$ $\displaystyle=\int_{0}^{R}(\cosh R-\cosh
s)^{d-1}\,\mathrm{d}s$ $\displaystyle=(\cosh
R-1)^{d-1}\int_{0}^{R}\left(1-{\cosh s-1\over\cosh
R-1}\right)^{d-1}\,\mathrm{d}s$ $\displaystyle=(\cosh R-1)^{d-1}J_{R,d}.$

Now plugging all this back into (11) leads to the desired estimate

$d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq\sqrt{2}\left({1\over\sqrt{d-1}\,J_{d}}+{2\over(d-1)\,\sqrt{J_{d}}}\right)$

and completes the proof. ∎

Our next task therefore is to estimate $J_{R,d}$. This is achieved by the
following result.

######  Lemma 6.

  * (a)

Suppose that $d$ is fixed. Then there exists a constant $C>0$ such that

$J_{R,d}\geq C\,R.$

  * (b)

Consider the case where $d\to\infty$ and $R=R_{d}$ is some sequence satisfying
$R_{d}\to\infty$ as $d\to\infty$. Denote in this case $J_{d}:=J_{R_{d},d}$.

    1. 1.

Suppose that $\limsup\limits_{d\to\infty}(R_{d}-\log d)<+\infty$. Then there
exists a constant $C>0$ such that

$J_{d}\geq C\,{e^{R/2}\over\sqrt{d}}.$

    2. 2.

Suppose that $\limsup\limits_{d\to\infty}(R_{d}-\log d)=+\infty$. Then there
exists a constant $C>0$ such that

$J_{d}\geq C(R-\log d).$

We postpone the proof of Lemma 6 until Section 3, and first use it to deduce
our main results.

###### Proof of Theorem 1 and Theorem 3.

The theorems follow upon combining Proposition 5 with the integral estimates
given by Lemma 6. First suppose that $d$ is fixed and combine Proposition 5
with Lemma 6 (a). This gives

$d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq\widetilde{C}\Big{(}{1\over
R}+{1\over\sqrt{R}}\Big{)}\leq C\,R^{1/2}$

for some constants $\widetilde{C},C>0$ only depending on $d$. This proves
Theorem 1.

Next suppose that $d\to\infty$ and that $R=R_{d}$. In the case where
$\limsup\limits_{d\to\infty}(R-\log d)<+\infty$, we obtain a bound of the form

$d_{\operatorname{Wass}}\left(\frac{S_{R,d}-\mathbb{E}S_{R,d}}{\sqrt{\operatorname{Var}S_{R,d}}},N_{\scalebox{0.65}{$\frac{1}{2}$}}\right)\leq
C\left(e^{-R_{d}/2}+d^{3/4}e^{-R_{d}/4}\right).$

Since by assumption, $e^{R_{d}}\leq Md$ for some $M>0$, the first term on the
right-hand-side is dominant, leading to the bound appearing in part (b) of the
theorem. In the case where $\limsup\limits_{d\to\infty}(R-\log d)=+\infty$, we
obtain immediately the asserted bound in part (a). ∎

##  3 Bounding $J_{{R,d}}$

Here we bound the integral $J_{R,d}$, which we recall is given by

$J_{R,d}=\int_{0}^{R}\left(1-{\cosh s-1\over\cosh
R-1}\right)^{d-1}\,\mathrm{d}s.$

###### Proof of Lemma 6.

We prove only part $(b)$, regarding the case $d\to\infty$. The proof of part
$(a)$ (the case of fixed $d$) is very similar to case $2$ below, and is
omitted. First we write, with the help of the hyperbolic identity $\cosh
x-1=2\sinh^{2}{x\over 2}$:

$\displaystyle J_{d}$
$\displaystyle=\int_{0}^{R}\left(1-{\sinh^{2}(s/2)\over\sinh^{2}(R/2)}\right)^{d-1}\,\mathrm{d}s,$

where we recall our notational convention $J_{d}=J_{R_{d},d}$. Now we make the
substitution $s\mapsto x$, where

${\sinh(s/2)\over\sinh(R/2)}={x\over\sqrt{d}}.$

Then

${1\over\sqrt{d}}\mathrm{d}x={\cosh(s/2)\over\sinh(R/2)}\,{\mathrm{d}s\over
2}=\sqrt{1+{\sinh^{2}(R/2)\over d}x^{2}}\,{\mathrm{d}s\over 2\sinh(R/2)}.$

We denote also $\rho_{d}:={\sinh(R/2)\over\sqrt{d}}$. Note that since
$R_{d}\to\infty$, one has the asymptotic equivalence

$\rho_{d}\sim{1\over 2}\exp\left({R-\log d}\over 2\right),\qquad d\to\infty,$

meaning that the ratio of the left and the right side tends to $1$ as
$d\to\infty$. This gives

$\displaystyle J_{d}$
$\displaystyle=2\rho_{d}\int_{0}^{\sqrt{d}}\left(1-{x^{2}\over
d}\right)^{d-1}{\mathrm{d}x\over\sqrt{1+\rho_{d}^{2}x^{2}}}.$

We now consider the two cases in the lemma separately.

  1. 1.

First suppose that $\limsup\limits_{d\to\infty}(R_{d}-\log d)<+\infty$. Then
by the above,

$L:=\limsup_{d\to\infty}\rho_{d}<+\infty.$

Fatou’s lemma now gives

$\displaystyle\liminf_{d\to\infty}(\rho_{d}^{-1}J_{d})\geq
2\int_{0}^{\infty}e^{-x^{2}}{\mathrm{d}x\over\sqrt{1+L^{2}x^{2}}}.$

Note that the latter integral converges to a strictly positive limit (in fact,
a computation with the aid of Mathematica gives ${1\over 2L}\cdot{e^{1\over
2L^{2}}K_{0}({1\over 2L^{2}})}$, where $K_{0}$ is the modified Bessel function
of the second kind). Therefore, there is a constant $C>0$ so that

$J_{d}\geq C\exp\left({R-\log d\over 2}\right)=C\,{e^{R/2}\over\sqrt{d}}.$

  2. 2.

Suppose now that $\limsup\limits_{d\to\infty}(R_{d}-\log d)=+\infty$. We then
compute with the help of the substitution ${\rho_{d}x\mapsto y}$:

$\displaystyle J_{d}$ $\displaystyle\geq
2\rho_{d}\int_{0}^{1}\left(1-{x^{2}\over
d}\right)^{d-1}{\mathrm{d}x\over\sqrt{1+\rho_{d}^{2}x^{2}}}$
$\displaystyle\geq\left(1-{1\over
d}\right)^{d-1}\int_{0}^{\rho_{d}}{\mathrm{d}y\over\sqrt{1+y^{2}}}$
$\displaystyle\geq{2\over e}{\rm arcsinh}\,\rho_{d}$ $\displaystyle={1\over
e}\,(R-\log d+O(1)),$

where $O(1)$ stands for a sequence which is bounded in $d$. Therefore, the is
some constant $C>0$ such that

$J_{d}\geq C(R-\log d).$

This completes the argument. ∎

##  4 The Euclidean case

Let $\eta_{d}$ be a stationary and isotropic Poisson process on the space
$\mathbb{A}(d,d-1)$ of affine hyperplanes in $\mathbb{R}^{d}$ with intensity
$1$. Its intensity measure $\Lambda_{e}$ is then given by

$\int_{\mathbb{A}(d,d-1)}f(H)\,\Lambda_{e}(\mathrm{d}H)=\int_{\mathbb{R}}\int_{\SS^{d-1}}f(H_{e}(s,u))\,\mathrm{d}u\,\mathrm{d}s,$

where $f:\mathbb{A}(d,d-1)\to\mathbb{R}$ is a non-negative measurable function
and $H_{e}(s,u)$ stands for the unique hyperplane in $\mathbb{R}^{d}$ with
signed distance $s$ from $\bf o$ and unit normal vector $u$. By

$S_{R,d,e}:=\sum_{H\in\eta_{d}}{\mathcal{H}_{e}^{d-1}}(H\cap B_{R,e}^{d})$

we denote the total surface area induced by the hyperplanes of $\eta_{d}$
within a centred Euclidean ball $B_{R,e}^{d}$ of radius $R>0$, where the
Hausdorff measure $\mathcal{H}_{e}^{d-1}$ is understood with respect to the
Euclidean metric. Using [16, Theorem 4.7] we find that

$\displaystyle d_{\rm
Wass}\Big{(}{S_{R,d,e}-\mathbb{E}S_{R,d,e}\over\sqrt{\operatorname{Var}S_{R,d,e}}},N\Big{)}\leq
2{\sqrt{\operatorname{cum}_{4}(S_{R,d,e})}\over\operatorname{Var}S_{R,d,e}}$
(12)

with a standard Gaussian random variable $N$. The variance and the fourth
cumulant of $S_{R,d,e}$ are given explicitly by

$\displaystyle\operatorname{Var}S_{R,d,e}$
$\displaystyle=\int_{\mathcal{H}_{e}}{\mathcal{H}_{e}^{d-1}}(H\cap
B_{R,e}^{d})^{2}\,\Lambda_{e}(\mathrm{d}H),$
$\displaystyle\operatorname{cum}_{4}(S_{R,d,e})$
$\displaystyle=\int_{\mathcal{H}_{e}}{\mathcal{H}_{e}^{d-1}}(H\cap
B_{R,e}^{d})^{4}\,\Lambda_{e}(\mathrm{d}H).$

Denoting by $\kappa_{d-1}$ the $(d-1)$-volume of the $(d-1)$-dimensional
Euclidean unit ball, we have that

$\displaystyle\operatorname{Var}S_{R,d,e}$
$\displaystyle=2\kappa_{d-1}^{2}\int_{0}^{R}(R^{2}-s^{2})^{d-1}\,\mathrm{d}s$
$\displaystyle=2\kappa_{d-1}^{2}R^{2d-1}\int_{0}^{1}(1-t^{2})^{d-1}\,\mathrm{d}t$
$\displaystyle={\pi^{d-{1\over 2}}\Gamma(d)R^{2d-1}\over\Gamma({d\over
2}+{1\over 2})^{2}\Gamma(d+{1\over 2})},$

where we applied the substitution $s\mapsto Rt$. The same computation also
leads to an explicit expression for $\operatorname{cum}_{4}(S_{R,d,e})$:

$\displaystyle\operatorname{cum}_{4}(S_{R,d,e})$
$\displaystyle=2\kappa_{d-1}^{4}\int_{0}^{R}(R^{2}-s^{2})^{2(d-1)}\,\mathrm{d}s$
$\displaystyle=2\kappa_{d-1}^{4}R^{4d-3}\int_{0}^{1}(1-t^{2})^{2(d-1)}\,\mathrm{d}t$
$\displaystyle={\pi^{2d-{3\over 2}}\Gamma(2d-1)R^{4d-3}\over\Gamma({d\over
2}+{1\over 2})^{4}\Gamma(2d-{1\over 2})}{.}$

In conjunction with (12) this gives

$\displaystyle d_{\rm
Wass}\Big{(}{S_{R,d,e}-\mathbb{E}S_{R,d,e}\over\sqrt{\operatorname{Var}S_{R,d,e}}},N\Big{)}\leq{2\over\pi^{1/4}}{\Gamma(d+{1\over
2})\over\Gamma(d)}\sqrt{{\Gamma(2d-1)\over\Gamma(2d-{1\over 2})}}\,R^{-1/2}.$

Using the well-known asymptotics for quotients of gamma functions, as
$d\to\infty$ we arrive at

$d_{\rm
Wass}\Big{(}{S_{R,d,e}-\mathbb{E}S_{R,d,e}\over\sqrt{\operatorname{Var}S_{R,d,e}}},N\Big{)}\leq
C\,d^{1/4}R^{-1/2}$

for some absolute constant $C>0$.

### Acknowledgement

We wish to thank Matthias Schulte (Hamburg) for motivating us to study the
problem addressed in this paper.
DR and CT were supported by the German Research Foundation (DFG) via CRC/TRR
191 Symplectic Structures in Geometry, Algebra and Dynamics. ZK and CT were
supported by the German Research Foundation (DFG) via the Priority Program SPP
2265 Random Geometric Systems. ZK was also supported by the German Research
Foundation (DFG) under Germany’s Excellence Strategy EXC 2044 – 390685587
Mathematics Münster: Dynamics – Geometry – Structure.

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# Differential geometry of space curves: forgotten chapters

Dmitry Fuchs, Ivan Izmestiev, Matteo Raffaelli, Gudrun Szewieczek, Serge
Tabachnikov

##  1 Introduction

Evolutes, involutes, and osculating circles of curves belong to the main
notions of planar differential geometry, going back to Christiaan Huygens in
the 17th century. They have many interesting properties, including the
surprising Tait–Kneser theorem: the osculating circles of a curve with
monotonic curvature are nested – see, e.g., [2, 10] or [9, Chapter 10].

In this article we are concerned with evolutes and involutes of space curves.
Although much of the material presented is not new and can be found in classic
books such as [1, 5, 13], we believe that a modern and unified treatment,
complemented with several novel observations and results, may be useful.
Besides, the theory is illustrated with the help of computer graphics, a tool
obviously not available to the classical geometers.

The geometry of space curves remains highly relevant in modern mathematics.
For example, it is closely related to the theory of completely integrable
systems: the filament (a.k.a. binormal, smoke ring, local induction) equation
is a completely integrable evolution of space curves, equivalent to the
nonlinear Schrödinger equation, see, e.g., [11]. Another important application
is the study of curved origami, which can be informally described as folding
paper along curves (as opposed to straight lines); see, e.g., [4, 7].

In this article we consider three kinds of evolutes and involutes. They were
all studied in the early days of differential geometry, however the
terminology is not canonical. Following [6] and [8], we define the evolute as
the locus of centers of osculating spheres or, equivalently, the curve whose
osculating planes are the normal planes of the given curve (in [1, 5] the term
evolute means something else, and Uribe-Vargas, in his detailed study of the
evolute in our sense [14], prefers the term focal curve).

Evolutes and involutes of space curves possess many familiar properties of
evolutes and involutes of plane curves, but also have some unexpected
features. We survey these properties and surprises and discuss a natural
modification of the construction in which the normal planes are replaced by
the rectifying planes. For the resulting curve we use a somewhat awkward term
pseudo-evolute. Properties of pseudo-evolutes offer further surprises.

In the plane, involutes are constructed by wrapping a non-stretchable string
around a curve. The same construction in space provides yet another definition
of involute, and one defines the evolute of a curve as the result of the
converse operation. We call this third version Monge evolutes and Monge
involutes – they were studied by Gaspard Monge in the late 18th century.

One of our goals is to describe the intricate interrelations between these
three kinds of evolutes.

In what follows we present some – but not all – calculations behind the
geometrical statements; they involve only elementary calculus, but in some
cases are cumbersome. Readers are encouraged to perform the missing
calculations on their own.

##  2 Textbook material

The following facts are undoubtedly known to the majority of the readers, but
we prefer to provide a brief survey of them to establish the settings,
terminology, and notations.

### Frenet apparatus.

By a curve we mean an arclength-parametrized, generic (this notion will vary,
but will always include the absence of inflexion points), and smooth
(${\mathcal{C}}^{\infty}$, possibly with cusps) curve $\xi=\xi(t)$, or
$x=x(t),y=y(t),z=z(t)$ in ${\mathbb{R}}^{3}$. If a curve has cusps, then the
arcs between the cusps are unit speed parametrized.

The Frenet frame $({\bf t}={\bf t}(t),{\bf n}={\bf n}(t),{\bf b}={\bf b}(t))$
consists of the tangent vector ${\bf t}=\xi^{\prime}$, the (principal) normal,
which is the unit vector in the direction of $\xi^{\prime\prime}$, and the
binormal $\bf b=t\times n$. The dependence of $\bf t,n,b$ on $t$ is described
by the Frenet formulas

$\begin{array}[]{rlrl}{\bf t}^{\prime}=&&k{\bf n}\\\ {\bf n}^{\prime}=&-k{\bf
t}&&+\tau{\bf b}\\\ {\bf b}^{\prime}=&&-\tau{\bf n}\end{array}$

where $k$ and $\tau$ are, respectively, the curvature and the torsion of the
curve $\xi$. We denote the radius of curvature $1/k$ by $r$.

There are well-known formulas for the curvature and torsion:

$k=\|\xi^{\prime\prime}\|,\quad\tau=\frac{\det(\xi^{\prime},\xi^{\prime\prime},\xi^{\prime\prime\prime})}{\|\xi^{\prime\prime}\|^{2}}.$

If the parametrization is not unit-speed, then the formulas become slightly
more complicated:

$k=\frac{\|\xi^{\prime}\times\xi^{\prime\prime}\|}{\|\xi^{\prime}\|^{3}},\quad\tau=\frac{\det(\xi^{\prime},\xi^{\prime\prime},\xi^{\prime\prime\prime})}{\|\xi^{\prime}\times\xi^{\prime\prime}\|^{2}}.$
(1)

The plane spanned by $\bf t$ and $\bf n$ is called the osculating plane of the
curve $\xi$, the plane spanned by $\bf n$ and $\bf b$ the normal plane of
$\xi$, and the plane spanned by $\bf t$ and $\bf b$ the rectifying plane of
$\xi$.

### Developable surfaces.

A surface in space is called developable if it is locally isometric to a
plane. An informal description of a developable surface is that it is a
surface that can be made by bending, but not folding, a piece of paper.
Certainly, we have in mind the “ideal paper”, which is ideally bendable
(without any bounds on the curvature), non-compressible and non-stretchable,
i.e., the length of any curve drawn on the paper remains unchanged in the
process of bending.

The theory of developable surfaces was developed (pardon the unintended pun)
in the late 1700’s by Euler and Monge. Let us translate the main results into
modern language.

A generic (in particular, nowhere planar) developable surface is ruled: every
point belongs to a unique straight line, which is fully contained in the
surface. Moreover, the tangent plane is constant along each line; this
property distinguishes developable surfaces in the class of all ruled
surfaces.

Generically, these lines, called rulings, are tangent to a certain curve,
called the regression edge of the surface. There the surface is not smooth:
all sections by planes, transverse to the regression edge, have cusps.

The latter property (also a characteristic one) provides a universal method
for constructing developable surfaces: take an arbitrary (generic) curve $\xi$
in space, possibly with cusps, and consider the union of all its tangent
lines. This union is a developable surface with regression edge $\xi$, and all
(generic) developable surfaces can be obtained in this way. This surface is
called the tangent developable of $\xi$.

There are two types of non-generic developable surfaces: cylinders and cones
over an arbitrary curve. When the regression edge has a cusp, the developable
surface becomes approximately conical; when the regression edge escapes to
infinity, the developable becomes approximately cylindrical.

Another universal construction of developable surfaces uses an arbitrary
(generic) one-parameter family of planes: for such a family there exists a
unique developable surface, tangent to all the planes, and called the envelope
of the family. These planes are the osculating planes of the regression edge.
For each plane, tangency occurs along a whole line, which is the tangent of
the regression edge.

The last construction provides a convenient analytic description of everything
mentioned above. Namely, if $F(x,y,z;t)=0$ is the equation of the family of
planes (with parameter $t$), then we can form three systems of equations:

$\left\\{\hskip-6.0pt\begin{array}[]{rl}F(x,y,z;t)&=0,\\\
F^{\prime}(x,y,z;t)&=0;\end{array}\right.\left\\{\hskip-6.0pt\begin{array}[]{rl}F(x,y,z;t)&=0,\\\
F^{\prime}(x,y,z;t)&=0,\\\
F^{\prime\prime}(x,y,z;t)&=0;\end{array}\right.\left\\{\hskip-6.0pt\begin{array}[]{rl}F(x,y,z;t)&=0,\\\
F^{\prime}(x,y,z;t)&=0,\\\ F^{\prime\prime}(x,y,z;t)&=0,\\\
F^{\prime\prime\prime}(x,y,z;t)&=0.\end{array}\right.$ (2)

If we exclude $t$ from the first system, then we get the equation of the
surface. If we solve the second system with respect to $x,y,z$, then we obtain
a parametric equation of the regression edge. The solutions of the last system
provide coordinates of the cuspidal points on the regression edge.

Notice that all this may be repeated for the case when $F(x,y,z;t)=0$ is a
family of surfaces, not necessarily planes, but we will not need that level of
generality.

##  3 Evolutes

Let $\xi=\xi(t)$ be a (generic) curve in space. There arise three families of
planes: the osculating planes, the normal planes, and the rectifying planes.
Each of them has an envelope (the regression edge of the developable surface
tangent to the family). The first of these three cases is not interesting: the
envelope is the curve $\xi$ itself.

The envelope of the family of normal planes is the normal developable, and its
regression edge is what we call the evolute of the curve: this is similar to
the definition of the evolute of a planar curve as the envelope of the family
of its normals.

### Equation of the evolute.

Let $\xi=\xi(t)=(x(t),y(t),z(t))$ be an arclength-parametrized curve, and
$e=e(t)$ be its evolute.

###### Proposition 3.1

$e=\xi+r{\bf n}+\displaystyle\frac{r^{\prime}}{\tau}{\bf b}.$

Proof. The proof is based on formulas (2). We denote the coordinates of points
in ${\mathbb{R}}^{3}$ as $P=(X,Y,Z)$, and we use the dot product to make the
formulas more compact.

The family of normal planes to $\xi$ is described by the formula

$F(P;t)=\xi^{\prime}(t)\cdot(P-\xi(t))=0.$

The derivatives of $F$ are

$F^{\prime}(P;t)=\xi^{\prime\prime}(t)\cdot(P-\xi(t))-1,\quad
F^{\prime\prime}(P;t)=\xi^{\prime\prime\prime}(t)\cdot(P-\xi(t))$

(we used the equalities $\xi^{\prime}(t)\cdot\xi^{\prime}(t)=1$ and
$\xi^{\prime\prime}(t)\cdot\xi^{\prime}(t)=0$).

The resulting (middle) system of equations (2) becomes

$\displaystyle\begin{cases}\xi^{\prime}(t)\cdot(P-\xi(t))&=0,\\\
\xi^{\prime\prime}(t)\cdot(P-\xi(t))&=1,\\\
\xi^{\prime\prime\prime}(t)\cdot(P-\xi(t))&=0.\end{cases}$

Solving this system for $P$ gives the evolute $e(t)$.

The Frenet formulas imply

$\xi^{\prime}={\bf t},\ \xi^{\prime\prime}=k{\bf n},\
\xi^{\prime\prime\prime}=-k^{2}{\bf t}+k^{\prime}{\bf n}+k\tau{\bf b},$

hence

$P-\xi=\frac{1}{k}{\bf n}-\frac{k^{\prime}}{k^{2}}\frac{1}{\tau}{\bf b},$

and finally

$e=\xi+\frac{1}{k}{\bf n}-\frac{k^{\prime}}{k^{2}}\frac{1}{\tau}{\bf
b}=\xi+r{\bf n}+\frac{r^{\prime}}{\tau}{\bf b},$

as required $\Box$

In this calculation, we assume that $\tau\neq 0$, otherwise the evolute
escapes to infinity.

### Osculating circles and osculating spheres.

Let $\xi=\xi(t)$ be a curve, and let $\xi_{0}=\xi(t_{0})$ be a point on this
curve. For $t_{1}<t_{2}<t_{3}$ close to $t_{0}$ we denote by
$C_{t_{1},t_{2},t_{3}}$ a circle passing through
$\xi(t_{1}),\xi(t_{2}),\xi(t_{3})$; for $t_{1}<t_{2}<t_{3}<t_{4}$ close to
$t_{0}$ we denote by $S_{t_{1},t_{2},t_{3},t_{4}}$ a sphere passing through
$\xi(t_{1}),\xi(t_{2}),\xi(t_{3}),\xi(t_{4})$.

In the generic case, both $C_{t_{1},t_{2},t_{3}}$ and
$S_{t_{1},t_{2},t_{3},t_{4}}$ are well defined. Moreover, both have limits
when all $t_{i}\to t_{0}$. This limit circle and limit sphere are called the
osculating circle and the osculating sphere of the curve $\xi$ at the point
$\xi_{0}$.

It is well known (and obvious) that the osculating sphere can be described as
the unique sphere which has a tangency of order $\geq 3$ with $\xi$ at
$\xi_{0}$. The osculating circle is the intersection circle of the osculating
sphere with the osculating plane; it has tangency of order $\geq 2$ with $\xi$
at the point $\xi_{0}$.

For our purposes, a more convenient description of the (center of the)
osculating sphere is the following. We take three parameter values,
$t_{1}<t_{2}<t_{3}$ close to $t_{0}$ and consider the three normal planes to
the curve $\xi$ at points $\xi(t_{1}),\xi(t_{2}),\xi(t_{3})$. These three
planes have a common point, and this point approaches the center of the
osculating sphere at the point $\xi_{0}=\xi(t_{0})$ when $t_{1},t_{2},$ and
$t_{3}$ approach $t_{0}$.

### Evolutes and osculating spheres.

###### Proposition 3.2

The evolute of a curve is the locus of the centers of its osculating spheres.

Proof. Let $F(X,Y,Z;t)=0$ be the equation of the normal plane to the curve
$\xi$ at point $\xi(t)$. Then the center of the osculating sphere at $\xi(t)$
is the limit for $\varepsilon\to 0$ of the solution of the system

$\left\\{\begin{array}[]{ll}F(X,Y,Z;t-\varepsilon)&=0,\\\ F(X,Y,Z;t)&=0,\\\
F(X,Y,Z;t+\varepsilon)&=0,\end{array}\right.$

which is equivalent to the system

$\left\\{\begin{array}[]{ll}F(X,Y,Z;t)&=0,\\\
\displaystyle\frac{F(X,Y,Z;t+\varepsilon)-F(X,Y,Z;t-\varepsilon)}{2\varepsilon}&=0,\\\
\displaystyle\frac{F(X,Y,Z;t+\varepsilon)-2F(X,Y,Z;t)+F(X,Y,Z;t-\varepsilon)}{2\varepsilon^{2}}&=0.\end{array}\right.$

In particular, in the limit $\varepsilon\to 0$, this becomes

$\left\\{\begin{array}[]{ll}F(X,Y,Z;t)&=0,\\\ F^{\prime}(X,Y,Z;t)&=0,\\\
F^{\prime\prime}(X,Y,Z;t)&=0.\end{array}\right.$

The solution of the last system is the point $e(t)$ of the evolute. Thus
$e(t)$ is the center of the osculating sphere of $\xi$ at $\xi(t)$. $\Box$

Propositions 3.1 and 3.2 imply

###### Corollary 3.3

Let $R$ be the radius of the osculating sphere of $\xi$ at $\xi(t)$. Then

$R^{2}=r^{2}+\left(\frac{r^{\prime}}{\tau}\right)^{2}.$

### Singularities of the evolute.

From Proposition 3.1 one has

$e^{\prime}={\bf t}+r^{\prime}{\bf n}-kr{\bf t}+\tau r{\bf
b}+\left(\frac{r^{\prime}}{\tau}\right)^{\prime}{\bf
b}-\frac{r^{\prime}}{\tau}\tau{\bf
n}=\left(r\tau+\left(\frac{r^{\prime}}{\tau}\right)^{\prime}\right){\bf b}.$

Thus, analogously to the two-dimensional case, the tangent to the evolute is
always parallel to the binormal. It follows that the cusps of the evolute
occur when

$\sigma:=r\tau+\displaystyle\left(\frac{r^{\prime}}{\tau}\right)^{\prime}=0.$

In particular, $\sigma\equiv 0$ is the condition for a curve to be spherical;
indeed, the evolute of a spherical curve is a point, the center of the sphere.

On the other hand, from Corollary 3.3,

$(R^{2})^{\prime}=2r^{\prime}r+2\frac{r^{\prime}}{\tau}\left(\frac{r^{\prime}}{\tau}\right)^{\prime},$

and so $(R^{2})^{\prime}=2\frac{r^{\prime}}{\tau}\sigma$. This shows that at
every cusp of the evolute, $R^{2}$ has zero derivative; thus, generically, $R$
achieves a maximum or a minimum.

But there is also the possibility that $r^{\prime}=0$ but $\sigma\neq 0$; so,
although $R$ is maximal or minimal, the evolute has no cusp at this point. By
Corollary 3.3, in this case $r=R$, that is, the osculating circle is a great
circle of the osculating sphere, and the center of the osculating sphere is
contained in the osculating plane of the curve. This is one of the essential
differences between the evolutes of planar and space curves.

Remark. The quantity

$\frac{k^{3}\tau^{2}\sigma}{R^{5/2}}$

is called the conformal torsion and is one of the two invariants of space
curves in conformal geometry (the other one being the conformal curvature);
see, e.g., [3]. In conformal geometry, spheres are “flat”, and the conformal
torsion measures the deviation of the curve from its osculating sphere.

### Examples.

Figure 1: A cusp-free evolute (orange) of a closed space curve (blue). It is a
$(1,5)$-torus knot on a torus with minor radius $m=0.15$ and major radius
$l=1$. It is given by the parametrization $t\mapsto((l+m\cos 5t)\cos
t,(l+m\cos 5t)\sin t,-m\sin 5t)$.

A closed curve whose evolute has no cusps. The evolute of a closed convex
plane curve has cusps, in fact, at least four of them, according to the
4-vertex theorem (see, e.g., [10] or [9, chapter 10]). But the evolute of a
closed space curve with non-vanishing curvature and torsion may be free of
cusps, see Figure 1.

Evolute of a curve with a cusp. It is well known that for a planar curve with
a cusp (like a semi-cubic parabola), its evolute passes through the cusp and
has no cusp at this point. The situation for a space curve is entirely
different. Here is a parametric equation of a curve with a cusp and its
evolute (see Figure 2 left):

${\rm curve:}\quad\begin{array}[]{rl}x=&t^{2},\\\ y=&t^{3},\\\
z=&t^{4};\end{array}\quad{\rm
evolute:}\quad\begin{array}[]{rl}x=&\displaystyle\frac{9}{2}t^{4}+20t^{6},\\\
y=&-8t^{3}-32t^{5},\\\
z=&\displaystyle\frac{1}{2}+\displaystyle\frac{9}{2}t^{2}+15t^{4}.\end{array}$

We see that our curve has a cusp at the point $(0,0,0)$, and the evolute has a
cusp at the point $(1/2,0,0)$.

An elliptical helix. The evolute of the standard (circular) helix is just
another helix. For the elliptical helix $x=a\cos t,\,y=b\sin t,z=ct$, the
evolute has 4 cusps for each turn (see Figure 2 right). The heavy dots on the
evolute mark its non-cuspidal points with $r^{\prime}=0$ and $\sigma\neq 0$,
which correspond to maxima or minima of $R$.

Figure 2: Left: the evolute of a curve with a cusp. Right: the evolute of an
elliptical helix.

### Interior and exterior points of a curve.

Since at a generic point a curve has an odd degree tangency with the
osculating sphere, a neighborhood of such point is contained either in the
interior or the exterior of this sphere. It remains unclear how to visualize
the difference between “interior” and “exterior” points. Just imagine that you
have a rigid curve, say, a twisted bicycle spoke. Can you tell the interior
points from the exterior ones?

Once again let us use system (2). Assuming that the three middle equations
hold, the type of the point, that is, whether it is interior or exterior, is
determined by the sign of $F^{\prime\prime\prime}(x,y,z;t)$. A calculation
that we do not present yields the following result: a point is interior if and
only if

$\frac{\det({\bf t},{\bf t^{\prime\prime}},{\bf
t}^{\prime\prime\prime})}{k^{2}\tau}>k^{2}.$

The points where a curve has a higher-order contact with its osculating sphere
correspond to the singularities of the evolute, that is, to the points where
$\sigma=0$. This suggests that the above inequality can be rewritten in terms
of $\sigma$. Indeed, it turns out that $\det({\bf t},{\bf
t^{\prime\prime}},{\bf
t}^{\prime\prime\prime})=k^{3}\tau^{2}\sigma+k^{4}\tau$, so that the above
inequality is equivalent to $\sigma\tau>0.$

And here is a geometrical interpretation: a curve of positive torsion locally
lies inside its osculating sphere if the center of the osculating sphere moves
in the direction of the binormal to the curve, and the curve lies outside of
its osculating sphere if the center of the sphere moves in the direction
opposite to the binormal.

### Curvature and torsion of the evolute.

As we know, $e^{\prime}=\sigma{\bf b}$. Using the Frenet formulas, we
calculate the next two derivatives of $e$:

$e^{\prime\prime}=\sigma^{\prime}{\bf b}-\sigma\tau{\bf n},\quad
e^{\prime\prime\prime}=\sigma k\tau{\bf
t}-(2\sigma^{\prime}\tau+\sigma\tau^{\prime}){\bf
n}+(\sigma^{\prime\prime}-\sigma\tau^{2}){\bf b}.$

The formulas for $e^{\prime}$ and $e^{\prime\prime}$ show that the Frenet
frame of the evolute has the form $({\bf t}_{e},{\bf n}_{e},{\bf
b}_{e})=(\pm{\bf b},\pm{\bf n},\pm{\bf t})$.

Formulas (1) imply

###### Proposition 3.4

The curvature and the torsion of the evolute are related to the curvature and
the torsion of the initial curve by the formulas

$k_{e}=\frac{|\tau|}{|\sigma|},\quad\tau_{e}=\frac{k}{\sigma},\quad\text{where
}\sigma=r\tau+\left(\frac{r^{\prime}}{\tau}\right)^{\prime}\text{ and
}r=\frac{1}{k}.$

Assume that the evolute has no cusps, that is, $\sigma\neq 0$. Without loss of
generality, $\sigma>0$. The magic cancellations when integrating the curvature
or the torsion of the evolute

$\displaystyle\int_{a}^{b}k_{e}\|e^{\prime}\|\,dt=\int_{a}^{b}\frac{|\tau|}{|\sigma|}|\sigma|\,dt=\int_{a}^{b}|\tau|\,dt,$
$\displaystyle\int_{a}^{b}\tau_{e}\|e^{\prime}\|\,dt=\int_{a}^{b}\frac{k}{\sigma}|\sigma|\,dt=\int_{a}^{b}\mathrm{sgn}(\sigma)k\,dt$

imply

###### Corollary 3.5

The total curvature of the evolute is equal to the total absolute torsion of
the curve. The total torsion of the evolute is equal to the total curvature of
the curve taken with the sign of $\sigma$.

### Curves congruent to their evolutes.

The planar curves congruent to their evolutes (with the congruence compatible
with the parametrization) are logarithmic spirals. What about space curves?

###### Proposition 3.6

The only space curves congruent to their evolutes (with the congruence
compatible with the parametrization) are helices of equal curvature and
torsion, and the curves whose curvature and torsion depend on an arclength
parameter in the following way:

$k=\tau=ct^{-1/2},\quad c>0,$

as well as their mirror images.

Proof. A necessary condition for a congruence compatible with the
parametrization is $|\sigma|=1$.

Note that the reflection in a plane preserves $k$, but changes the sign of
$\tau$, therefore it changes the sign of $\sigma$. Thus, without loss of
generality, we may assume that $\sigma=1$.

A space curve is determined up to congruence by its curvature and torsion as
functions of an arclength parameter. The formulas in Proposition 3.4 imply
that $k=\tau$ and $\sigma=1$ is a necessary and sufficient condition for a
curve to be congruent to its evolute.

Substituting $k=\tau$ into the formula for $\sigma$, we obtain

$\sigma=1+(r^{\prime}r)^{\prime}=1+\left(\frac{r^{2}}{2}\right)^{\prime\prime}.$

Thus $\sigma=1$ if and only if $r^{2}=at+b$. If $a=0$, then
$k=\tau=\mathrm{const}$, and one obtains helices of equal curvature and
torsion. If $a\neq 0$ then, after a time shift and, if necessary, time
reversal, one has $r^{2}=t/c^{2}$, that is, $k=\tau=ct^{-1/2}$. $\Box$

Figure 3 shows one of the curves whose curvature and torsion satisfy
$k=\tau=ct^{-1/2}$ for some $c>0$, together with its congruent evolute. The
curves have constant slope and are geodesics on generalized cylinders that
have the involute of the circle with radius $\frac{1}{4\sqrt{2}c^{2}}$ as base
curve (see [13] for more details). Thus, if the base curve $\gamma$ is
parametrized by

$\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t))=\frac{1}{4\sqrt{2}c^{2}}(\cos z+z\sin
z,\ \sin z-z\cos z),\ \ \text{where}\ \ z=2c\sqrt{2t},$

then the corresponding curve with congruent evolute is given by

$t\mapsto(\gamma_{1}(t),\gamma_{2}(t),\frac{1}{\sqrt{2}}t).$

Figure 3: A curve whose curvature and torsion are both equal to $ct^{-1/2}$
(blue) together with its congruent evolute (orange). The curves have constant
slope and lie on generalized cylinders over the involutes of appropriate
circles.

It is tempting to apply Corollary 3.5 twice: the curvature density and the
torsion density of the second evolute coincide with those of the initial curve
(under the non-restrictive assumption that the torsion is positive). This is
equivalent to the fact that the Frenet frame of the second evolute is parallel
to the Frenet frame of the initial curve (which also follows from the identity
$(t_{e},n_{e},b_{e})=(\pm b,\pm n,\pm t)$ that we mentioned earlier).

Remark. Two simultaneously parametrized curves with parallel tangents at the
corresponding points are called _Combescure transformations_ of each other;
for a detailed study see [12].

A curve is congruent to its second evolute if and only if an arclength
parameter for the curve is also an arclength parameter for the second evolute.
This condition is equivalent to a complicated differential equation

$\frac{1}{r\tau}\left(\frac{r^{\prime}}{\tau}\right)^{\prime}+\left(\frac{1}{\sigma\tau}\left(\frac{\sigma}{\tau}\right)^{\prime}\right)^{\prime}=0.$
(3)

However, if the curvature $k$ is constant, then $\sigma=\frac{\tau}{k}$, and
equation (3) is satisfied. It follows that a space curve of constant curvature
is congruent to its second evolute. See [8] for curves that are homothetic to
their second evolutes.

### Involutes.

By definition, the curve $\xi$ is an involute of a curve $e$ if $e$ is the
evolute of $\xi$. A generic curve has a two-parameter family of involutes:
they are the curves orthogonal to the family of osculating planes of the given
curve.

Let $S$ be the tangent developable of $e$, and let $H$ be the osculating plane
of $e$ at some point. Let us roll the plane $H$ along the surface $S$ without
slipping and twisting. The instantaneous motion of the plane $H$ is a rotation
about a line in $H$. Therefore the velocity of each point of the plane $H$ is
perpendicular to $H$, and the involutes of $e$ are the trajectories of the
points of $H$ in the process of rolling.

This description shows that the involutes of a curve are equidistant: the
distances between the points of the rolling plane do not change. It also shows
that each involute has a cusp every time it reaches $S$.

Assume that $e$ is a generic smooth closed curve with non-vanishing torsion.
After rolling the osculating plane $H$ all the way around $e$, this plane
returns to the original position, but its points do not necessarily return to
their initial positions: a self-map of $H$ arises, which we call the
monodromy.

If we orient the osculating planes of a curve by its binormals, then the
monodromy is an isometry of $H$ that preserves this orientation. A generic
orientation-preserving isometry of the plane is a rotation about some point,
and this point is its only fixed point. We conclude that a generic smooth
closed curve $e$ has a unique closed involute.

This is in contrast with the planar case: for the involute of a closed plane
curve to be closed, the curve must have zero alternating perimeter (the sign
changes after every cusp), and if this alternating perimeter vanishes, then
all involutes are closed; see [10] or [9, Chapter 10].

Consider the trace $e_{H}$ of the curve $e$ in the plane $H$ as this plane
rolls along $e$. This trace is a planar development of the curve $e$: its
curvature, as a function of the arclength, is the same as that of $e$.
(Imagine that a curve is made of wire which is hard to bend but easy to twist.
Then one can flatten this curve without changing its curvature.)

While the curve $e$ is closed, $e_{H}$ in general is not. The end points of
$e_{H}$ and the tangent directions therein are two contact elements in the
plane $H$.

###### Proposition 3.7

The monodromy is the orientation preserving isometry of $H$ which sends the
initial contact element of the curve $e_{H}$ to its terminal contact element.

Proof. When one surface is rolled, without slipping and twisting, along a
curve on another surface, this curve and its trace on the rolling surface have
the same curvature at the respective points. This describes the relation
between the curves $e$ and $e_{H}$.

An instantaneous displacement of the plane $H$ when it is rolled along the
curve $e$ is the parallel translation distance $dt$ in the tangent direction
$e^{\prime}(t)$, combined with the rotation through the angle $k(t)dt$.
Integrated along $e$, this defines the motion described in the statement.
$\Box$

###### Corollary 3.8

The rotation angle of the monodromy is equal to the total curvature of the
evolute.

Assume that a curve has a closed involute. When are all involutes also closed,
that is, when is the monodromy the identity map?

###### Corollary 3.9

If a curve has at least one closed involute, and the total torsion of this
involute is an integer multiple of $2\pi$, then all involutes are closed.

This follows from Corollary 3.5: the total curvature of the evolute is equal
to the total absolute torsion of the curve. The torsion of the involute does
not change its sign provided the curve has no cusps.

An example of a closed space curve whose planar development is also closed is
a curve of constant curvature 1 and length $2k\pi$ (the development is a $k$
times traversed circle).

### Osculating circles.

As we mentioned earlier, the osculating circles of a plane curve with
monotonic curvature are nested. What about the osculating circles of a space
curve? In particular, can the osculating circles at neighboring points be
linked?

The osculating circles of the curve in Figure 4 seem to be unlinked. The next
proposition confirms that, locally, this is always the case.

Figure 4: Osculating circles of a space curve.

###### Proposition 3.10

Let $\xi$ be a curve with non-vanishing torsion. Then its osculating circle at
every point is disjoint from its osculating planes at sufficiently close
points. In particular, the osculating circles at close points are not linked.

Proof. We view the curve as the envelope of the 1-parameter family of planes
$z=a(t)x+b(t)y+c(t)$. We assume that, for $t=0$, the plane is $z=0$, that
$\xi(0)=(0,0,0)$, and the tangent line to $\xi$ at the origin is the $x$-axis.
This means that

$a(0)=b(0)=c(0)=a^{\prime}(0)=c^{\prime}(0)=c^{\prime\prime}(0)=0.$

Thus

$a(t)=at^{2}+O(t^{3}),\ b(t)=bt+O(t^{2}),\ c(t)=ct^{3}+O(t^{4}).$

We will ignore the “big O” terms in what follows: this will not affect the
result, but will make the formulas less awkward.

We find the equation of the curve using the middle equation (2):

$\xi(t)=\left(-\frac{3ct}{a},\frac{3ct^{2}}{b},ct^{3}\right).$

The curvature of this curve at the origin equals $\frac{2a^{2}}{3bc}$, and the
radius of curvature is $\frac{3bc}{2a^{2}}$.

The intersection of the osculating planes of the curve $\xi(t)$, that is, the
planes $z=a(t)x+b(t)y+c(t)$, with the plane $z=0$ are the lines
$a(t)x+b(t)y+c(t)=0$ (when $t=0$, it is the $x$-axis), that is,
$at^{2}x+bty+ct^{3}=0$.

The envelope of this family of lines is the parabola

$x=\\-\frac{2ct}{a},\ y=\frac{ct^{2}}{b}.$

The curvature of this curve at the origin equals $\frac{a^{2}}{2bc}$, and its
radius of curvature $\frac{2bc}{a^{2}}>\frac{3bc}{2a^{2}}$.

Figure 5: To proof of Proposition 3.10.

It follows that the osculating circle of the curve $\xi$ at the origin lies
inside the parabola. Therefore the tangent lines to the parabola, that is, the
intersections of the osculating planes of $\xi(t)$ with its osculating plane
at $\xi(0)$, are disjoint from the osculating circle at $\xi(0)$. This implies
the result, see Figure 5. $\Box$

##  4 Pseudo-evolutes

### Definition and first properties.

A generic space curve $\xi$ determines a one-parameter family of rectifying
planes. This is the family of tangent planes to a certain developable surface
$S$, called rectifying developable, and this surface has a regression edge. We
call this regression edge the pseudo-evolute of the initial curve. In other
words, the pseudo-evolute of a curve $\xi$ is a curve whose osculating planes
are the rectifying planes of $\xi$.

###### Proposition 4.1

The curve $\xi$ is a geodesic on $S$.

Proof. According to one of many description of geodesics on a surface in
space, it is a curve whose principal normals are normal to the surface. For
$\xi$ on $S$ this clearly holds: the principal normals of $\xi$ are
perpendicular to the rectifying planes of $\xi$, that is, to the tangent
planes of $S$. $\Box$

Informally, this means that a piece of paper with a straight line drawn on it
may be, in a unique way, attached (without crumblings and foldings) to a given
space curve in such a way that the line follows the curve (see [7]). This
surface is unique because the normals to this surface are determined: they are
the normals of $\xi$. This gives another description of the pseudo-evolute of
a (generic) curve $\xi$: it is the regression edge of the unique developable
surface $S$, which contains $\xi$ as a geodesic.

### Equation of the pseudo-evolute.

The equation of the rectifying plane to the curve $\xi$ at the point $\xi(t)$
is

$(P-\xi(t))\cdot{\bf n}(t)=0.$

The first and second derivatives of this equations with respect to $t$ are

$\begin{array}[]{rll}&\hskip-6.0pt(P-\xi(t))\cdot(-k{\bf t}(t)+\tau{\bf
b}(t))&\hskip-6.0pt=0\\\ k\,+&\hskip-6.0pt(P-\xi(t))\cdot(-k^{\prime}{\bf
t}+\tau^{\prime}{\bf b}-(k^{2}+\tau^{2}){\bf n})&\hskip-6.0pt=0.\end{array}$

These three equations form the system that has the solution

$(P-\xi(t))\cdot{\bf t}(t)=\frac{k\tau}{k^{\prime}\tau-k\tau^{\prime}},\
(P-\xi(t))\cdot{\bf b}(t)=\frac{k^{2}}{k^{\prime}\tau-k\tau^{\prime}}.$

We obtain the equation of the pseudo-evolute $\varepsilon(t)$ of the curve
$\xi(t)$:

###### Proposition 4.2

The pseudo-evolute $\varepsilon$ of the curve $\xi$ has the equation

$\varepsilon=\xi+\frac{k}{k^{\prime}\tau-k\tau^{\prime}}(\tau{\bf t}+k{\bf
b}).$

### Escapes to infinity and cusps.

###### Proposition 4.3

Let $\xi$ be a generic curve with curvature $k$ and torsion $\tau$.

$(1)$ The pseudo-evolute of $\xi$ escapes to infinity when the first
derivative of the ratio $\displaystyle\frac{\tau}{k}$ vanishes.

$(2)$ The pseudo-evolute of $\xi$ has cusps when the second derivative of
$\displaystyle\frac{\tau}{k}$ vanishes.

Figure 6: Left: the pseudo-evolute of an elliptical helix. Right: the pseudo-
evolute of a closed smooth curve. It has 12 cusps and escapes 4 times to
infinity.

Proof. Part (1) follows from Proposition 4.2: $\varepsilon$ escapes to
infinity, when $k^{\prime}\tau-k\tau^{\prime}=0$.

Part (2) requires some computation also based on the formula from Proposition
4.2. From this formula:

$\varepsilon^{\prime}={\bf
t}+\left(\frac{k}{k^{\prime}\tau-k\tau^{\prime}}\right)^{\prime}(\tau{\bf
t}+k{\bf b})+\frac{k}{k^{\prime}\tau-k\tau^{\prime}}(\tau{\bf t}+k{\bf
b})^{\prime}.\ $ (4)

By Frenet’s formulas, $(\tau{\bf t}+k{\bf b})^{\prime}=(\tau^{\prime}{\bf
t}+k^{\prime}{\bf b})$, so the right hand side of (4) is a linear combination
of $\bf t$ and $\bf b$. The coefficient of $\bf t$ is

$\begin{array}[]{c}\displaystyle{1+\frac{(k^{\prime}(k^{\prime}\tau-k\tau^{\prime})-k(k^{\prime}\tau-k\tau^{\prime})^{\prime}\tau}{(k^{\prime}\tau-k\tau^{\prime})^{2}}+\frac{k\tau^{\prime}}{k^{\prime}\tau-k\tau^{\prime}}}\\\
=\displaystyle{\frac{(k^{\prime}\tau-k\tau^{\prime})+k^{\prime}\tau+k\tau^{\prime}}{k^{\prime}\tau-k\tau^{\prime}}-\frac{k\tau(k^{\prime}\tau-k\tau^{\prime})^{\prime}}{(k^{\prime}\tau-k\tau^{\prime})^{2}}=\left(\frac{2k^{\prime}}{k^{\prime}\tau-k\tau^{\prime}}-\frac{k(k^{\prime}\tau-k\tau^{\prime})^{\prime}}{(k^{\prime}\tau-k\tau^{\prime})^{2}}\right)\tau},\end{array}$

while the coefficient of $\bf b$ is

$\frac{(k^{\prime}(k^{\prime}\tau-k\tau^{\prime})-k(k^{\prime}\tau-k\tau^{\prime})^{\prime}k}{(k^{\prime}\tau-k\tau^{\prime})^{2}}+\frac{kk^{\prime}}{k^{\prime}\tau-k\tau^{\prime}}=k\left(\frac{2k^{\prime}}{k^{\prime}\tau-k\tau^{\prime}}-\frac{k(k^{\prime}\tau-k\tau^{\prime})^{\prime}}{(k^{\prime}\tau-k\tau^{\prime})^{2}}\right).$

Hence, $\varepsilon^{\prime}=0$ if
$\displaystyle{\frac{2k^{\prime}}{k^{\prime}\tau-k\tau^{\prime}}-\frac{k(k^{\prime}\tau-k\tau^{\prime})^{\prime}}{(k^{\prime}\tau-k\tau^{\prime})^{2}}}=0$,
and the last expression is
$\displaystyle{\frac{k^{3}}{(k^{\prime}\tau-k\tau^{\prime})^{2}}\left(\frac{\tau}{k}\right)^{\prime\prime}}$.
$\Box$

Thus the derivative $\displaystyle{\left(\frac{\tau}{k}\right)^{\prime}}$
plays, for pseudo-evolutes, the role similar to that of the curvature for
evolutes of planar curves.

The function $\displaystyle{\frac{\tau}{k}}$ has the following geometrical
meaning. The tangent indicatrix of a space curve $\xi(t)$ is the curve
$\xi^{\prime}(t)$ on the unit sphere (recall that $\xi$ is arclength-
parametrized). The geodesic curvature of the tangent indicatrix equals
$\displaystyle{\frac{\tau}{k}}$.

Proposition 4.3 has the following consequence ([13, Section 2-4, Problem 5]):

###### Corollary 4.4

The rectifying developable is a cone if and only if
$\displaystyle\frac{\tau}{k}$ is a linear function, and it is a cylinder if
and only if $\displaystyle\frac{\tau}{k}$ is a constant.

### Examples.

Figure 6 shows two examples: the pseudo-evolutes of the same elliptical helix
as in Figure 2, and of the curve $x=\cos t,y=\sin t,z=\frac{1}{2}\sin 2t$.

### Pseudo-evolute of a curve with a cusp.

Our definition works in this case, but we are puzzled by the geometry of the
result, see Figure 7.

Figure 7: The pseudo-evolute of a curve with a cusp.

We consider the standard example of the curve with a cusp:
$x=t^{2},y=t^{3},z=t^{4}$. It is true that the principal normal of a curve (at
a regular point) has the directiion of $\bf t\times(t\times\dot{t})$ which,
for our curve, is

$(18t^{4}+64t^{6},48t^{7}-12t^{3},-32t^{4}-36t^{6})=2t^{3}(9t+32t^{3},24t^{4}-6,-16t-18t^{3}).$

Thus, the rectifying plane of our curve at a point with the parameter value
$t\neq 0$ has the equatiion

$\begin{array}[]{rl}F(x,y,z,t)=(9t+32t^{3})(x-t^{2})+(24t^{4}-6)(y-t^{3})-(16t+18t^{3})(z-t^{2})&\\\
=(9t+32t^{3})x+(24t^{4}-6)y-(16t+18t^{3})z-(6t^{7}+16t^{5}+3t^{3})=0.&\end{array}$

This equation makes sense for $t=0$ (it becomes $y=0$), so we have a family of
rectifying planes parametrized by $t\in\mathbb{R}$.

To find the envelope of this family, we consider the system
$F=F_{t}=F_{tt}=0$, and we cancel the equation $F_{tt}=0$ by $t$. After doing
that, we solve the system using the Cramer rule and obtain:

$\begin{array}[]{rl}x=&\displaystyle\frac{4(324t^{10}+1440t^{8}+1335t^{6}+720t^{4}+320t^{2}+18)}{525(1-4t^{4})},\\\
y=&-\displaystyle\frac{8t^{5}(9t^{2}+8)}{9(1-4t^{4})},\\\
z=&\displaystyle\frac{460t^{10}+7880t^{8}+11520t^{6}+8490t^{4}+1440t^{2}+81}{1050(1-4t^{4})}.\end{array}$
(5)

These are parametric equations of a curve that can be considered as the
pseudo-evolute of the curve $x=t^{2},y=t^{3},z=t^{4}$.

The curve (5) has a singularity at the point
$\displaystyle\frac{3}{350}(16,9)$, but this singularity is a “degenerate
cusp”: it is easy to find that

$16z-9x=\displaystyle\frac{4(36t^{10}+16t^{8}+63t^{6}+60t^{4})}{3(1-4t^{4})},$

which shows that the “type of the cusp” is $(t^{2},t^{4},t^{5})$. How to
relate this to our geometric interpretation of pseudo-evolutes remains unclear
to us.

### Pseudo-involutes.

Every curve has a two-parameter family of pseudo-involutes, namely the curves
whose pseudo-evolute is the initial curve; indeed, a curve $\eta$ is the
regression edge of a developable surface, the union of tangent lines of
$\eta$, and the geodesics of this surface are pseudo-involutes of $\eta$.

Pseudo-involutes may have cusps: it happens every time when the involute
reaches $\eta$. (This makes pseudo-involutes similar to involutes of planar
curves, but there is a big difference: pseudo-involutes of $\eta$ do not need
to be perpendicular to $\eta$ at the cuspidal points.)

A pseudo-involute may be a smooth closed curve – see Figure 6, right. Then it
is a closed geodesic on a developable surface, and its equidistant curves on
this surface are also closed geodesics. Hence such a curve is included in a
1-parameter family of smooth closed pseudo-involutes.

### An annoying question.

We see that pseudo-evolutes have many properties similar to those of evolutes.
Still their geometric meaning remains enigmatic. Evolutes are the loci of
centers of osculating spheres. And pseudo-evolutes are the loci – of
what?111Spoiler alert: the answer will be given at the very end of the
article.

##  5 Monge evolutes

### Monge involutes.

Attach one end of a non-stretchable string to a space curve $\eta$, pull the
string tight, and wrap it around $\eta$. The velocity of the free end of the
string is always orthogonal to the string (it is non-stretchable). The
trajectory of this free end is the curve $\xi$, a Monge involute of $\eta$.
Changing the length of the string yields a 1-parameter family of these
involutes.

Differentiating the formula

$\xi(t)=\eta(t)+(\ell-t){\bf t}_{\eta},$ (6)

where $\ell$ is the length of the string attached at point $\eta(0)$, yields

$\xi^{\prime}=(\ell-t)k_{\eta}{\bf n}_{\eta}.$ (7)

It follows that if the curvature of $\eta$ never vanishes, then the cusps of
$\xi$ lie on $\eta$, which happens when the free end of the string lands on
the curve.

### Monge evolutes.

If $\xi$ is a Monge involute of $\eta$ then, by definition, $\eta$ is a Monge
evolute of $\xi$. Does every curve has such an evolute, and if so, how many?

If $\eta$ is an evolute of $\xi$, then one has

$\eta=\xi+y{\bf n}+z{\bf b}.$

Differentiate with respect to the arclength parameter of $\xi$ and apply the
Frenet formulas:

$\eta^{\prime}=(1-ky){\bf t}+(y^{\prime}-\tau z){\bf n}+(z^{\prime}+\tau
y){\bf b}.$

By assumption, $\eta^{\prime}$ must be parallel to $\eta-\xi=y{\bf n}+z{\bf
b}$, which leads to a system of equations

$1-ky=0,\quad\frac{y^{\prime}-\tau z}{y}=\frac{z^{\prime}+\tau y}{z}.$

This system has a one-parameter family of solutions

$y=r,\quad z=-r\,\tan\int\tau\,dt.$

This leads to

###### Proposition 5.1

The Monge evolutes of an arclength-parametrized curve $\xi$ are given by

$\eta=\xi+r{\bf n}-r\,\tan\alpha\,{\bf b},\quad\alpha^{\prime}=\tau.$

In particular, spatial Monge evolutes of a plane curve are geodesics on the
cylinder over the plane evolute of this curve.

###### Corollary 5.2

The corresponding points of two different evolutes are seen from the
corresponding point of the curve under a constant angle.

### Singularities of Monge evolutes.

Proposition 5.1 implies that

$\|\eta-\xi\|=\frac{1}{k|\cos\alpha|}.$

On the other hand, due to (6), one has

$\|\eta-\xi\|=\int_{t_{0}}^{t}\|\eta^{\prime}\|\,dt.$

Therefore

$\|\eta^{\prime}\|=\left(\frac{1}{k|\cos\alpha|}\right)^{\prime}.$

It follows that the cusps of a Monge evolute correspond to the critical points
of the function $k\cos\alpha$. This generalizes a property of plane evolutes:
their cusps correspond to the vertices of the curve.

Proposition 5.1 also allows to determine when an evolute escapes to infinity:
this happens if either $k=0$ or $\alpha=\frac{\pi}{2}+k\pi$.

### Closed Monge evolutes and Monge involutes.

Proposition 5.1 has the following consequence.

###### Proposition 5.3

Monge evolutes of a closed space curve are closed if and only if the total
torsion of the curve is an integer multiple of $\pi$. In particular, Monge
evolutes of a centrally symmetric curve are closed.

The last statement is due to the obvious fact that a central symmetry reverses
the sign of the torsion.

Figure 8: A Monge evolute (green) of a centrally symmetric curve.

As to the closed Monge involutes, the situation is exactly the same as in the
plane:

###### Proposition 5.4

Monge involutes of a closed space curve are closed if and only if the curve
has zero length (the length element changes sign after each cusp).

### Interrelations between the evolutes.

As promised, we describe how the three kinds of evolutes interact with each
other.

The line $\\{\xi+r{\bf n}+\lambda{\bf b}\mid\lambda\in{\mathbb{R}}\\}$ is
called the _polar line_ of the curve $\xi$ at the corresponding point. This is
the line that goes through the center $\xi+r{\bf n}$ of the osculating circle
and is orthogonal to the osculating plane of $\xi$. The center of the
osculating sphere lies on the polar line.

Recall the formula $e^{\prime}=\sigma{\bf b}$. It implies that the polar line
is tangent to the evolute of $\xi$. It follows that the polar lines are the
rulings of the normal developable of the curve $\xi$.

Proposition 5.1 implies that each point of a Monge evolute lies on some polar
line, and every point of a polar line lies on a unique Monge evolute. It
follows that the normal developable surface of the curve $\xi$ is foliated by
its Monge evolutes.

A Monge evolute of $\xi$ meets the evolute of $\xi$ at the points where
$r^{\prime}=r\tau\tan\alpha$. The tangent to the Monge evolute at those points
coincides with the binormal of $\xi$, that is, the Monge evolute is tangent to
the evolute.

As a result we have

###### Proposition 5.5

The evolute is the envelope of Monge evolutes.

Figure 9: The evolute (orange) and some Monge evolutes (green) of a curve
$\xi$ (blue) together with its normal developable (orange surface). Left: The
polar lines are tangent to the evolute and lie on the normal developable of
$\xi$. Middle: The tangents of the Monge evolute intersect $\xi$ orthogonally.
Right: The Monge evolutes also foliate the normal developable of $\xi$.

We conclude with yet another relation between the different types of evolutes
([13, Section 2-4, Exercise 3]).

###### Proposition 5.6

The pseudo-evolute of any space curve is the evolute of any of its Monge
involutes.

Proof. Let $\eta$ be a space curve, and let $\xi$ be a Monge evolute of
$\eta$. One has to show that the normal planes of $\eta$ are the rectifying
planes of $\xi$. These planes are parallel because their normals are parallel;
see Equation (7). And they coincide because $\xi-\eta$ is tangent to $\eta$
and normal to $\xi$, and thus is contained in both planes. $\Box$

In particular, this provides an answer to the above “annoying question”: the
pseudo-evolute of a curve is the locus of the osculating spheres of its Monge
evolute.

### Acknowledgements.

I. Izmestiev and M. Raffaelli were supported by the Austrian Science Fund
(FWF), project F 77 (SFB ”Advanced Computational Design”); S. Tabachnikov was
supported by NSF grant DMS-2005444.

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