Approximation of smooth surfaces with polygonal meshes I'm just getting started with Blender. It seems to be a nice tool, but like most 3D programs, it has a strange limitation: it seems to only render polygon meshes.

A polygon mesh is of course a set of perfectly flat surfaces connected by perfectly straight edges. This is great for moddling man-made objects such as cubes, or for genuinely angular things such as crystals. But how the heck do you draw curved things?

Of course, you can approximate any curve with a sufficient number of straight lines. But it's always a pretty crude approximation. You would need billions of polygons to produce a genuinely smooth-looking surface. Even if it were somehow possible to model that, the memory and render time requirements would be crazy.

To get around this, most 3D programs have a button somewhere to blur out the surface normals, creating the illusion of actual curves. But it's just that — an illusion. The outline of the object still has raggid, sharp edges. The shadow of the object still has sharp edges. If two objects intersect, you can see all the straight lines where they meet, no matter how smooth the surface looks.

And then I sat and watched Spring. And you know what? I don't recall seeing a single straight edge anywhere in the entire thing! So how's it done? Where is the secret button that lets you have curved surfaces?
This is a very good question.
It all comes down to fooling your brain. 

1. “Approximating a curve with polygons is pretty crude”
Well, human eyes have considerably limited resolution.
Screen resolution also contributes - it isn't infinite.

You don't need infinite polygons, just enough to fool the human eye.
It's easier to do, than most people think.

2. “You would need billions of polygons for a smooth-looking surface”
Smooth-looking is the key word.

Surface needs far less geometry to look smooth, than a silhouette.
Most of your focus is on surfaces, not silhouettes.   

3. “I don't recall seeing a straight edge anywhere”
Don't recall is the key phrase.

You don't focus on details, but on the moving imagery.
When you pause the animation, chances are you'll find some jagged edges.  

Try pausing older movies like Toy story 1, Shrek or Ice age.
This is what Pixar understood first - you don't need it perfect. Just non-distracting.



Fig. 02 - Surface of a low-poly object looks perfectly smooth, while it's silhouette is jagged