spacing
Using
$\begin{array}{ccc}
x & := & \phantom{ a + } b + c\\
y & := & a \phantom{ + b } + c\\
z & := & a + b \phantom{ + c }
\end{array}$
I get
which is not quite what I hoped for as my "phantoms" don't take as much space as expected.
How can I correct this?
Thanks!
TeX ignores spaces after control sequences (formed by letters) and performs complete expansion when doing a \write.
The tokens it sees when you do \writefile{\Insert stuff.} are (separated by • just for clarity)
\Insert•s•t•u•f•f•.
and there's no space. The expansion gives
Insertstuff
When you put a pair of braces, the space after them is not ignored (it doesn't follow a control sequence):
\Insert•{•}• •s•t•u•f•f•.
With \Insert\ stuff it's the same: the token "control space" is unexpandable, so you get
Insert\ stuff.
With \Insert~stuff you lose: the complete expansion of ~ is not what you'd expect: it's the set of instructions necessary to "print a non breaking space". Indeed LaTeX has \protected@write to cope with this kind of commands that should not be expanded during a write.
As zeroth explained, the expansion of \space is a space; but it's not ignored after \Insert, because when TeX is reading tokens it doesn't yet see the expansion of \space, but that token (which undoubtedly isn't a space token).
Calling these spacing commands is a bit misleading they are all commands to set boxes of specified dimensions.
\rlap, \llap \clap are essentially \makebox[0pt][r], \makebox[0pt][l] and \makebox[0pt][c] except they can avoid the complication of looking for optional arguments etc. Also they differ in the way that \hbox differs from \mbox in that following the plain TeX rather than LaTeX tradition they lack a \leavevmode at the start of their definition so they do not start a paragraph if used in vertical mode.
In text mode \smash is in the same way essentially \raisebox{0pt}[0pt][0pt]
In text mode, the phantom commands are all equivalent to using an empty box with dimensions given by another box, so \phantom is essentially
\def\Phantom#1{\savebox{0}{#1}\savebox{2}{}%
\ht2=\ht0 \dp2=\dp0 \wd2=\wd0
\usebox{2}}
\vphantom is the same except that the width is forced to 0pt rather than the original width of the text in the first box.
The math mode versions are all essentially the same except for two complications, math mode has to be re-entered inside the box, and a \mathchoice construct has to be used so they work correctly at display and script sizes. \phantom and \smash have built in math mode tests and then switch between the text and math definitions automatically. For \rlap (just for historical reasons) it is text mode only, so for math mode you need to switch batc to math explictly or use \mathrlap from a suitable package.
Note that in \mathmode always makes a mathord atom which gets no special spacing. In the examples in the question you compared the spacing of - with the mathord spacing and the mahtop spacing but - is a mathbin atom so you need \mathbin{\phantom{-}}
\documentclass{article}
\usepackage{amsmath,amssymb}
\pagestyle{empty}
\begin{document}
\begin{align*}
y &= x - z\raisebox{0ex}[0pt][0pt]{\rule[-4ex]{0.1pt}{4ex}} \\
&= x \mathbin{\phantom{-}} z\raisebox{0ex}[0pt][0pt]{\rule{0.1pt}{4ex}} \\
&= x \mathbin{\phantom{ - }} z \\
\end{align*}
\hspace*{\fill}Or even worse:\hspace*{\fill}
\begin{align*}
y &= x - z\raisebox{0ex}[0pt][0pt]{\rule[-4ex]{0.1pt}{4ex}} \\
&\phantom{{}= x -{}} z\raisebox{0ex}[0pt][0pt]{\rule{0.1pt}{4ex}} \\
\end{align*}
\end{document}
Even then spacing of subscripts may be different
\phantom{P}_x
is like
{P{}}_x
in which the subscript is not the same as
P_x
It's pretty hard to avoid that as the boxing implied by phantom obscures the kerning information in the font metrics, but fortunately it is rare to want a visible subscript on an invisible base.
Best Answer
In the final part of the first row, try
\phantom{a+{}}b+cto make the first plus sign act like a "real" math-binary element. Similarly, you'll also need to adjust the second row toa\phantom{{}+b}+c.