I've been trying to design a dynamic sculpture which I can only define with exactitude through mathematical equations. Now, the equations yield only lines, like you can see in the animation below: 
I want to use these blue lines as extrusion paths in blender. How can I do this?
The Add 3D Function Surface addon allows you to do exactly this. You can download this addon here. It also comes included in the built-in Extra Objects addon.
Once you have the addon installed, when you go to add a new mesh, you will have two additional options: 'Z function surface' and 'XYZ function surface'. If you select either of these, you will then be able to add parametric equations in the Toolbar menu to define your shape.
Example XYZ Function Surface:
Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 with bounds x=[0,1], y=[0,1], z=[0,1]. Is there currently a way to render this in Blender with only those 16 parameters (10 from the quadric and 6 for bounds: x_lo, x_hi, y_lo, y_hi, z_lo, z_hi)?
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Commented
Jul 9, 2018 at 22:15
At some point the geometry will get converted into discrete coordinates - it's something we have to accept.
Perhaps a method as simple as generating a profile 'edge' mesh from 20 points on a curve ( blender has curve interpolation functions in python mathutils.geometry to get neat segments) and then use the Screw modifier to Lathe them around
