How can I highlight part of an equation with TikZ but without extra space around the highlighted part?

Question

I can use TikZ to highlight one or more terms/factors/exponents/etc. in a mathematical expression. But when I have two highlighted entities in a row extra space is introduced that I would like to eliminate, even if that means the highlighting overlaps surrounding entities a bit.

Here is a MWE that illustrates this.

% !TEX TS-program = lualatexmk
\documentclass[10pt]{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{shapes}
\NewDocumentCommand{\hilite}{ O{brown!60} m O{rectangle} }{%
  % See https://texample.net/tikz/examples/beamer-arrows/
  \tikz[baseline]{\node[fill=#1,#3,anchor=base]{\ensuremath{#2}};}% previous working version
}%
\begin{document}
\begin{align}
  (\Delta s)^2 &= -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[rounded rectangle] + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[rectangle] + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[ellipse] + (\Delta y)^2 + (\Delta z)^2 
\end{align}
\begin{align}
  \hilite[orange]{\Delta\mathbf{p}}[circle] &= \mathbf{F}{\textnormal{net}},\Delta t \
  \Delta\mathbf{p} &= \hilite[yellow!50]{\mathbf{F}{\textnormal{net}}}[rounded rectangle],\Delta t \
  \Delta\mathbf{p} &= \mathbf{F}{\textnormal{net}},\hilite[olive!50]{\Delta t}[rectangle] \
  \Delta\mathbf{p} &= \hilite[cyan!50]{\mathbf{F}{\textnormal{net}},\Delta t}[ellipse] 
\end{align}
\end{document}

Here is the result. Note the extra space introduced. I would like to eliminate that.

If I change my hilite command to

\NewDocumentCommand{\hilite}{ O{brown!60} m O{rectangle} }{%
  % See https://tex.stackexchange.com/a/406084/218142
  % See also https://tex.stackexchange.com/a/570789/218142
  % See also https://tex.stackexchange.com/a/79659/218142
  % See also https://tex.stackexchange.com/q/375032/218142
  \tikz[overlay,baseline=(n1.base)]{\node[fill=#1,draw,#3] (n1){\ensuremath{#2}};}%
}%

the result looks like this,

which introduces less space but obviously doesn't work. I've tried to understand the overlay option, and I understand why it causes the collapsed view here. I think the solution may lie with using \tikzmark but I can't seem to understand that concept. Ideally, I would like to be able to hilight any quantity, symbol, operator, etc. while in math mode. The only way I can think of to verbalize how I think the highlighting should look is that is should look like a "spotlight" centered on whatever I pass to the command, and overlapping onto other entities is okay. The shape and color of the "spotlight" can also be specified.

Answer 1

It is not too difficult to literally answer the question. I switch to tikzmark as this library is made for such things. One just need to add a node that has the desired shape underneath using the behind path option.

\documentclass[10pt]{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{fit,shapes,tikzmark}
\NewDocumentCommand{\hilite}{ O{brown!60} m O{rectangle} }{%
  % See https://texample.net/tikz/examples/beamer-arrows/
  \tikzmarknode[alias=n,append after command={%
  node[behind path,fill=#1,#3,overlay,fit=(n)]{}%
  }]{n}{#2}%
}%
\begin{document}
\begin{align}
  (\Delta s)^2 &= -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[rounded rectangle] + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[rectangle] + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[ellipse] + (\Delta y)^2 + (\Delta z)^2 
\end{align}
\begin{align}
  \hilite[orange]{\Delta\mathbf{p}}[circle] &= \mathbf{F}{\textnormal{net}},\Delta t \
  \Delta\mathbf{p} &= \hilite[yellow!50]{\mathbf{F}{\textnormal{net}}}[rounded rectangle],\Delta t \
  \Delta\mathbf{p} &= \mathbf{F}{\textnormal{net}},\hilite[olive!50]{\Delta t}[rectangle] \
  \Delta\mathbf{p} &= \hilite[cyan!50]{\mathbf{F}{\textnormal{net}},\Delta t}[ellipse] 
\end{align}
\end{document}

Almost certainly this is not what you want. It does fulfill the requirements but since the colored areas overlap with earlier elements of the equations, they erase them. This can be cured with some atbegshi trickery. However, I am wondering if this is what you really want. After all, fulfilling your requirements seem to imply that you highlight stuff that is outside of the node that you want to highlight.

On a different note, I'd also like to draw your attention to \tcbhighmath from tcolorbox.

ADDENDUM: This is the version with ship out trickery.

\documentclass[10pt]{article}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{eso-pic}
\usetikzlibrary{fit,shapes,tikzmark}
\newcounter{tikzhighlightnode}
\NewDocumentCommand{\hilite}{ O{brown!60} m O{rectangle} }{%
  % See https://texample.net/tikz/examples/beamer-arrows/
  \stepcounter{tikzhighlightnode}%
  \tikzmarknode{highlighted-node-\number\value{tikzhighlightnode}}{#2}%
  \edef\temp{\noexpand\AddToShipoutPictureBG*{\noexpand\begin{tikzpicture}[overlay,remember picture]
  \noexpand\node[fill=#1,#3,fit=(highlighted-node-\number\value{tikzhighlightnode})]{};%
  \noexpand\end{tikzpicture}}}%
  \temp%
}%
\begin{document}
\begin{align}
  (\Delta s)^2 &= -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[rounded rectangle] + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[rectangle] + (\Delta y)^2 + (\Delta z)^2 \
  (\Delta s)^2 &= \hilite{-(\Delta t)^2 + (\Delta x)^2}[ellipse] + (\Delta y)^2 + (\Delta z)^2 
\end{align}
\begin{align}
  \hilite[orange]{\Delta\mathbf{p}}[circle] &= \mathbf{F}{\textnormal{net}},\Delta t \
  \Delta\mathbf{p} &= \hilite[yellow!50]{\mathbf{F}{\textnormal{net}}}[rounded rectangle],\Delta t \
  \Delta\mathbf{p} &= \mathbf{F}{\textnormal{net}},\hilite[olive!50]{\Delta t}[rectangle] \
  \Delta\mathbf{p} &= \hilite[cyan!50]{\mathbf{F}{\textnormal{net}},\Delta t}[ellipse] 
\end{align}
\end{document}

P.S. I am not a fan of the optional arguments using \NewDocumentCommand, at least not if I have pgf at my disposal. You could just have one argument, which you feed with an arbitrary number of keys.