Using both $\theta$ and $\vartheta$ in the same statement

Question

$\newcommand{\textand}{\hspace{5pt}\text{and}\hspace{5pt}}$ In a recent proof of mine, I used the two symbols $\theta$ and $\vartheta$ for different variable angles. My supervisor noted that this can be confusing to read since both are a lowercase theta. However, I read $\vartheta$ as ''vartheta'' and thus I thought that this could be a symbolic analogue of using any of the pairs $$ \theta \textand \theta',\hspace{10pt} \theta \textand \widetilde{\theta},\hspace{10pt} \theta \textand \overline{\theta},\hspace{10pt} \theta \textand \hat{\theta},\hspace{10pt} \theta_{1} \textand \theta_{2}, $$ when one would like exactly two different but related variables. I have changed it to $\theta$ and $\phi$, but I have two questions:

$1.$ Is it improper to use both $\theta$ and $\vartheta$ in the same statement to mean different variables?

$2.$ Is it good practise to stick to just one of the two in a document?

Answer 1

Think how confusing it would be if you used $a$ and $\mathrm{a}$ together as different symbols. That's basically what you've done: a cursive and a non-cursive version of the same letter $\theta$.

And won't someone please think of the handwriting! You've completely destroyed any ability for someone to hand-write things by copying them from your paper.

Answer 2

Asking for our opinion on this is just silly. Because of how an analysis of the pros and cons works out:

Against: Whether you agree it's likely to cause confusion, and regardless of how important you think it is, you certainly must agree that it could cause confusion.

For: Reasons why it's important to use both in the same context: None whatever.

Regardless of how much weight you assign the argument against, it's still infinitely more than the weight assigned to the non-existent argument for. SO worrying about whether it's really officially improper is silly - don't do it!

(Note: the above is assuming that your question is motivated by a desire to write well. It seems possible that the real reason you posted the question was just hoping that someone would say you were right and he was wrong. To the extent that's it: bad attitude.)