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I would like to know of a good and easy-to-use application to create simple but visually appealing vector figures such as spheres, planes, lines.

It should permit transparencies for example, and 3D coordinate representation to be able to show concepts in the 3D space, such as tangency or crossing.

Thanks!

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In this Wikipedia link you can find a list of different vector graphics softwares available, giving some details on their use and characteristics.

From your question it is hard to say what is it that you want to do and seems similar to this question. Plus, there is probably not a single answer to this question. I personally use Inkscape, which is quite powerful in terms of possibilities and it's Open Source !

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