---
title: Retractions and tangent charts
summary: A retraction moves a point of a manifold along a tangent vector; it makes each tangent space a chart around its point, and gives the operations ⊕ and ⊖.
---

# Retractions and tangent charts

Estimation moves points. On $\R^\dimension$ a step is an addition, $\point + \tangent$; on a curved manifold, such
as the rotations or the unit sphere, there is no addition, and a *retraction* takes its place. This
note defines it, shows how it turns every tangent space into a chart, and introduces the notation
$\oplus$ and $\ominus$ used throughout these notes.

## The definition

Let $\manifold$ be a smooth manifold of dimension $\dimension$. In computation, a tangent space
$\T_\point\manifold$ is never handled abstractly: each is identified with $\R^\dimension$ by a basis chosen
smoothly in $\point$, and a tangent vector is a column of $\dimension$ numbers. We assume such an identification
throughout. Some manifolds have none that is smooth everywhere: by the hairy ball theorem, no
smooth choice of tangent bases covers the whole sphere $S^2$. There the identification is smooth on
an open part of the manifold, and a computation stays inside it.

> **Definition (Retraction).** A _retraction_ on a manifold $\manifold$ of dimension $\dimension$ is a smooth map
> $R : \T\manifold \to \manifold$, written $(\point, \tangent) \mapsto R_\point(\tangent)$, such that for every
> $\point \in \manifold$, with $\T_\point\manifold$ identified with $\R^\dimension$,
>
> 1. $R_\point(0) = \point$, and
> 2. the derivative of $R_\point$ at $0$ is the identity: $\D R_\point(0) = I_\dimension$.

The definition is the one used in optimization on manifolds, where retractions replace the
exponential map of a Riemannian manifold. The exponential map is a retraction, but
so is any map that agrees with it to first order, which is usually cheaper. Nothing here needs a
metric.

Two examples. On $\R^\dimension$, $R_\point(\tangent) = \point + \tangent$. On the positive reals, with
$\T_\point(0, \infty)$ identified with $\R$ by measuring relative change,
$R_\point(\tangent) = \point\,e^{\tangent}$: it is defined for every $\tangent$, never leaves the positive reals,
and satisfies $\frac{\dd}{\dd\tangent} \point e^\tangent \big|_{0} = \point$, which is the identity
in the chosen identification (a tangent at $\point$ of coordinate $1$ is the velocity $\point$).

## Tangent charts

Because $\D R_\point(0)$ is invertible, the inverse function theorem makes $R_\point$ a diffeomorphism
from a neighbourhood of $0 \in \R^\dimension$ onto a neighbourhood of $\point$. So $R_\point$ is a *chart* of
$\manifold$ around $\point$, the tangent chart at $\point$, and every point near $\point$ has coordinates in it.
As is common in robotics, we write

$$
	\point \oplus \tangent \defeq R_\point(\tangent), \qquad \otherPoint \ominus \point \defeq R_\point^{-1}(\otherPoint), @tag(oplus)
$$

the second for $\otherPoint$ in the neighbourhood where $R_\point$ is invertible. In words, $\otherPoint \ominus \point$ is where
$\otherPoint$ sits in the chart at $\point$. By construction,

$$
	(\point \oplus \tangent) \ominus \point = \tangent @tag(roundtrip)
$$

for every $\tangent$ in the chart's domain, and $\point \oplus (\otherPoint \ominus \point) = \otherPoint$ for every $\otherPoint$ in its
image. On $\R^\dimension$, $\oplus$ and $\ominus$ are addition and subtraction; on the positive reals above,
$\otherPoint \ominus \point = \log(\otherPoint / \point)$.

## The mirror identity

On $\R^\dimension$, $\point \ominus (\point \oplus \tangent) = -\tangent$ as well: the step back from $\point \oplus \tangent$ to
$\point$ is the reverse of the step forward. On a curved manifold it generally is not, because the left
side is measured in the chart at the *moved* point $\point \oplus \tangent$, not at $\point$.

> **Lemma (The mirror identity holds to first order).** For $\tangent$ small enough, $\point \ominus (\point \oplus \tangent) = -\tangent + O(\lVert\tangent\rVert^2)$.

<Proof>

Write $\otherPoint = \point \oplus \tangent$ and $\varphi(\tangent, c) = R_\point^{-1}(R_{\point \oplus \tangent}(c))$, the chart
at $\otherPoint$ read in the chart at $\point$. It is smooth near $(0, 0)$, with $\varphi(\tangent, 0) = \tangent$
and, since $R_{\point \oplus 0} = R_\point$, $\varphi(0, c) = c$. Hence
$\varphi(\tangent, c) = \tangent + c + O(\lVert\tangent\rVert\,\lVert c\rVert)$. The coordinate
$c = \point \ominus \otherPoint$ is the one with $\varphi(\tangent, c) = R_\point^{-1}(\point) = 0$, so
$c = -\tangent + O(\lVert\tangent\rVert\,\lVert c\rVert)$, and $\lVert c \rVert = O(\lVert\tangent\rVert)$
gives the claim.

</Proof>

The identity holds exactly in some cases: on $\R^\dimension$, on the positive reals above, and on any Lie
group whose retraction is $R_\point(\tangent) = \point \operatorname{Exp}(\tangent)$, since then
$\point \ominus (\point \oplus \tangent) = \operatorname{Log}(\operatorname{Exp}(\tangent)^{-1}) = -\tangent$. It
fails at second order for other retractions, and a computation that needs "the point whose
coordinate is $-\tangent$" must not assume it.

A manifold whose retraction is $\point + \tangent$, so that both identities hold exactly and every chart is
the same affine chart, is called *flat* in these notes.