---
title: "Function `signature()` in the `clifford` package"
author: "Robin K. S. Hankin"
output: html_vignette
bibliography: clifford.bib
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{signature}
  %\usepackage[utf8]{inputenc}
---
 

```{r setup, include=FALSE}
set.seed(0)
library("clifford")
options(rmarkdown.html_vignette.check_title = FALSE)
knitr::opts_chunk$set(echo = TRUE)
knit_print.function <- function(x, ...){dput(x)}
registerS3method(
  "knit_print", "function", knit_print.function,
  envir = asNamespace("knitr")
)
```

```{r out.width='15%', out.extra='style="float:right; padding:10px"',echo=FALSE}
knitr::include_graphics(system.file("help/figures/clifford.png", package = "clifford"))
knitr::include_graphics(system.file("help/figures/lorentz.png", package = "lorentz"))
```

```{r showsigdef}
signature
```

To cite the `clifford` package in publications please use
@hankin2022_clifford.  This short document discusses `signature()` in
the `clifford` R package.  As an example we might wish to work in
$\operatorname{Cl}(1,2)$:

```{r label=sig34}
signature(1,2)
```

Thus $e_1^2=+1$, and $e_2^2=e_3^2=-1$:

```{r showp}
c(drop(e(1)^2),drop(e(2)^2),drop(e(3)^2))
```

We might ask what $e_4$ would evaluate to, and this is assumed to be
zero as is $e_i^2$ for $i\geqslant 4$:

```{r showg4}
c(drop(e(4)^2),drop(e(100)^2))
```

If we wish to set paranoid-level safety measures, we would set option
`maxdim` to prevent accidentally working with too-large values of $i$:

```{r usemaxdim}
options(maxdim = 4)
```

Now we work with a four-dimensional vector space in which
$e_1^2=+1,e_2^2=e_3^2=-1,e_4^2=0$, but now $e_5$ is undefined:

```{r showerror, error=TRUE}
c(drop(e(1)^2),drop(e(2)^2),drop(e(3)^2),drop(e(4)^2))
e(5)
```

The operation of `signature()` is modelled on the `sol()` function in
the `lorentz` package [@hankin2022_lorentz].  Thus, if given no
arguments we return the signature:

```{r showsig}
signature()
```

However, the default value is to use an infinite signature which
corresponds to $e_i^2=1\forall i$:

```{r showinfinitesig}
options(maxdim=NULL)
signature(Inf)
signature()
```

Function `signature()` returns an object of (trivial) class `sigobj`
which has a bespoke print method, `print.sigobj()`.  For technical
reasons an infinite signature is not allowed but is represented
internally by a near-infinite integer, specifically
`.Machine$integer.max`:


```{r dputinfinitesig}
dput(signature())
```


# References