20161028 - Rotated View Vs Rotated Sample Grid In Pixel


Sorry no images as I lost them when minus went down a long time back.
Attempting to rederive the geometry of my prior work here
while out of town without access to the original work.

EDITS: First posting was math and geometry fail, worked on too close to sleeping.
       Fixed on 20161029.

Terms,
  SGSSAA = Sparse Grid Super-Sample Anti-Aliasing (shading on MSAA pattern)

Comparison is between,
  (1.) 2xSGSSAA (super-sample the 2xMSAA pattern)
  (2.) A non-MSAA rotated view with slightly less shaded fragments

2xSGSSAA has exactly 2 shaded fragments per output pixel.
Will have at best when resolved, 2 steps on any edge gradient with a box filter.
Align the 2xMSAA pattern with the corners of a pixel provides the following.
Which shows how 2xMSAA pattern only gets 2 steps on any vertical or horizontal edge,
  
  +------------(X)
  |             |
  |     (X)     |
  |             |
 (X)------------+


Now lets compare to a rotated view without MSAA but with lightly less shaded fragments.

For a span of {2,1} output pixels, run {3} input pixels on the hypotenuse (rotated view).
This provides for a sqrt(5)/3 = 0.745 input width pixel (compared to a 1.0 output width pixel).
Which is 5/9 = 0.555 in area (compared to a 1.0 area output pixel).

  +--------+-------(X)
  |        | (X)    |
  |    (X) |        |
 (X)-------+--------+

For 1920x1080 = 2073600 output pixel area, 
  roughly 2073600/0.555 = 3732480 input fragments are needed 
  (9/5 = 1.8x the fragments).

The 2xSGSSAA case renders 2.0x the fragments for a 2-step gradient.
This rotated view renders 1.8x the fragments for a 3-step gradient.
And the rotated view not using MSAA will render and "resolve" faster.

Note both cases I'd be "resolving" using a larger-than-pixel-width kernel.
With a filter kernel which is not the worst case box filter.