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I've been using Geogebra to create interactive 2D applets for students to learn with. The good thing about Geogebra is that it requires no extra software or browser plugin but just works (even on mobile).

Now I want to do the same thing but in 3D, simple things like rotating paraboloids to see what it looks like but also more complicated things with "sliders" (for example you can see the function $f(x,y) = ax^2+by^2+c$ and have a slider for $a, b, c$ within a given interval). I know Geogebra has a 3D feature but it renders terribly slow and has all kinds of bugs (for me).

I've been using Mathematica to create pictures for this which works perfectly. However, for students to interact (Manipulate[]), they need to either download Mathematica or the free CDF player and the corresponding .cdf files and view the applets locally. The free CDF player is rather large (for the purpose of something like seeing what a paraboloid looks like) and I also don't like that the students have to download a file for every "animation" or "applet". Another option is the Wolfram Cloud, which however works terribly on my PC and I am not even sure if one needs a subscription to view stuff.

What other options do I have? The rendering should be rather fast, the additionally required software should be as little as possible (optimally none at all like for Geogebra) and it should work on as many of the most popular platforms as possible (mobile would be great!). It also should be embeddable online.

EDIT: I found this question but it's six years old and has no answer that I would be happy with since I need more than just basic objects like points, lines, planes and spheres.

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It's a little more programming intensive, but the most powerful online visualization and 3D graphics tool is webgl. People can (and have) written online in-browser games with full 3D physics engines, so there is little competition in terms of capability and speed :). It works pretty much everywhere (including mobile, most of the time). The most popular library to use with it is three.js, but there are some mathematics-specific libraries (e.g. MathBox). I'd also checkout scene.js if all you need is 3D viewpoint/camera manipulation.

Downsides: it has a fair learning curve. Also, there is no Mathematica integration. For that, I think I think the comment above on JavaView looks really promising. As you noted yourself, you could also make the app in Wolfram CDF, but then people/students would need to do some work on their end to view and play with it.

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