Is there a software package that would allow visulaizing/rendering some example structures in 3-space? Specifically, I'm thinking of something that would provide a 3-D rendering of, say, 3-vectors (as an arrow), then vector addition (showing the arrows stacked up in addition), point lattices under a basis, convex sets, etc. I imagine a black background with scaled back grid lines in three space, and objects rendered as 3-D objects of different colors. You'd be able to navigate three space to spin, zoom, etc. and develop an intuitive feel for the objects in that space. The UI would be math-savy--enter a matrix and click the "eigen-vector" or "span" radio button to add those to the plot; enter a parametric equation to define a convex set and have it show up; Enter vector definitions and expressions and see the vectors plotted, along with results, in varying colors.
Basically, I'm trying to extend the capability to look at the 2-space examples we get in handouts to 3-space and allow building an intuitive concept of these structures in the highest dimensional space we can easily visualize. This would be mostly a learning tool, but can be useful for some problem analysis (like looking at certain linear programming, for example).
The closest tools I can find are either 2-d graphers or generic 3-d rendering packages (that don't understand abstract mathematical objects). Does anyone know of something in the middle?
(I now see this overlaps with this question, which is tightly focused on matrix operations.)
GeoGebra (which came up in an ad for this question) actually seems to have some useful ability in this area. Particularly, this example shows some 3-D vector ops: http://www.geogebratube.org/student/m29087
While it seems to require some amount of scripting for the model, it's at least something. It seems to have built-in semantics for points, lines, circles, triangles in 3-d geometry. A more specific tool with better semantics (of vector spaces, other mathematical structures, etc.) would still be more useful.