TITLE: 3D software like GeoGebra
QUESTION [31 upvotes]: Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on.

REPLY [2 votes]: I have developed a free 3d drawing program that runs in modern browsers (HTML5, based on Three.js). The English version is called Geoservant 3D, the original German version is called Geoknecht 3D.

I would be able to draw spheres, great circles, and so on.

You can easily draw: 

cube, cuboid, cylinder, line, line segment, plane, point, polygon, quadrangle (square), sphere, text, triangle, vector.

Rotation is available for: Cuboid, cube, text, cylinder as well.
After you specified some 3d shapes you can also animate their properties by changing the number values. See description on page how to change values: holding down ALT key and using cursor keys. Also have a look at the gallery on the page to see what things you can achieve with this program. It also calculates basic values of the 3d objects. 
Currently I am using it a lot to solve vector and plane equations. By the visualization I can check if the result is correct. That is one use case, and it is super helpful for me.
Hope that helps you too. 

Example: 

If you search for a program for 2D drawings, you could use the new Geodrafter 2D.
Update: Geoservant is now available in German, English, Chinese, and Swedish.