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how can I invent a new symbol/character for TeX. For example, say I want to type the Tengwar letter ungwe (2 IV: http://en.wikipedia.org/wiki/File:Tengwar_alphabet.png), or any other symbol that I can draw- how would I go about doing so. I want to be able to use these diverse symbols as variables in a specific document that I am writing (which happens to be for a constructed language).

Note: I am not specifically concerned with Tengwar necessarily. So, if Tengwar is already supported in TeX (probably in a package), this will not necessarily and immediately answer my question. I am more concerned with being generally able to, basically, create any set of symbols and call them an alphabet/writing-system (or, more generally, a list of mathematical symbols) and then be able to reproduce them in a document using TeX as if they were any other letters/characters/symbols. For the most part, I am not inventing mathematical symbols for new operations or anything- I just want unique variables rather than the typical Latin, Greek, and Cyrillic ones. I am not sure if this is rephrasing the issue properly (they may be technically different), but naively, I would suppose that I would like to invent a font.

I am completely new to TeX.

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You are looking for METAFONT, a program also written by Knuth and accompagnying TeX. This program has a close relative METAPOST that can be used for drawing PostScript or SVG figures. Start here:

http://tug.org/metapost.html

I like the tutorial by André Heck, it will give you a feeling of the program.

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