How can I manually fix nonplanar geometry?

Question

I'm going through a course on blender. The course teaches blender through the use of several projects. Right now I'm modeling a low-poly chess set, and we've arrived at the knight. Because we made the knight free-form (just extruding faces and edges up from the base), and because we want to stay as quad-based as possible (to preserve loop cuts), there are a lot of nonplanar faces.

I've modeled my knight, and would now like to fix the nonplanar quads. Most of them only need one or two vertices pushed a small distance (typically less than 0.05BU). My problem is that I can't push them the right way.

In all of the transform orientations (global, normal, gimbal, etc.) the axes never point in the exact direction I need to push the vertex. The closest I can come is the view oreintation, which isn't exact for those vertices not perpendicular/parallel to a side (front, top, right, etc.).

So how can I fix my faces? Is there a way to fix nonplanar quads? Do note that this is a low-poly model, meant to look blocky. The subdivision modifier will not be used.

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Note that in the course, the instructor simply uses the edge split modifier. I don't like that, as it is not fixing the geometry, and my model still has triangulation on it (the modifier essentially does nothing). Even if it did work, I would prefer to actually fix the problem instead of pretending it's not there.

You can download the file here. Note that I am using the mirror modifier, which is why there is only half of a knight. You'll have to turn both modifiers off and turn mesh analysis on to see which faces are non-planar.

Answer 1

^{I will compile multiple methods here from various sources.}

The methods for a single face will work for multiple faces if you choose Individual Origins as the Pivot center for rotating/scaling. However, this will only work for faces which aren't adjacent. If they share common edges, the single face options will treat them as one large face.

Scale along normal (single face)

Select the vertices of a face, choose the Normal Transform Orientation, then scale along the Normal Z axis. You can do this with the scale manipulator or by pressing SZZ. You can make a single face planar by scaling to 0 with SZZ0.

LoopTools Flatten (single face)

If not already, activate the preinstalled LoopTools Addon. Select the face and choose the Fatten operation from the LoopTools addon. It is available in the 3DView toolshelf T under tools > LoopTools or in the specials menu W > LoopTools Flatten.

Make Planar Faces (multiple faces)

From this answer, there is a tool to make faces planar.

Shift single Vertex

In this answer is a script to make a face planar by only changing a single vertex. Select the three vertices of a face, which shouldn't change, then Shift select the vertex which should be shifted, marking it as the active vertex.
Then, call the Flatten Vertex operator (see the linked answer for details).

Note, that this will not update the Mesh Analysis display. But that will update after a Zero operation, like G0.