Corner Orientation

Corner Orientation, abbrevaited CO, the orientation of a cube's corners. There are three possible corner cubie orientations. CO is a sub step in many methods.
For the last layer of the 3x3x3 cube there are three main variations used for the seven cases of CO, one that is named pure CO wich orients the corners without affecting anything else, not even corner permutation, one preserves edge orientation wich is OCLL (OLL 21  27) and the last variation orients the corners ignoring edges (use any OLL that twists the same corners for that).
The separation of the corners to their correct layers on the Square1 is often wrongly named "corner orientation".
Algorithms; at this page you can find algortihms for two diffrent variations of CO;
 Pure CO: orients corners preserving edge orientation and both corner and edge permutation.
 OCLLEPP: orients corners preserving edge orientation and permutation for the edges but not the corners (this method is seldomly used).
See also:
 Orientation
 Orient
 Permutation
 Edge Orientation
 Corner Permutation
 OCLL
 Winter Variation
 CLS
 Partial Corner Control
Pure CO
Usage: Besides that these cases are a group in L4C, L3C, ZBLL, ZZLL;
3OP is a blindfold method that orients the pieces before they are placed into position, the algorithms here solves the corner orientation in at the most 3 steps, first, if needed two pieces, one in the top layer and one in the bottom layer to fix orientation parity and then 24 corners in the top layer and then the same for the bottom layer. Algs for pure orientation of the edges can be found at the ELL page.
Any other BLD method, to fix orientation for corners that are in position but unoriented from the scramble (a few setup moves to get the pieces into the same layer may be needed then).
For FMC to solve the last pieces (but that won't give any WR, better to try some diffrent start of the last layer that gives a easier case in the end).
Algorithms:
All cases here have long algorithms, the H case is actually the worst LL case of them all and the pi case is second worst. The good thing is that all cases are solveable using only two sides (RU 2gen). There are diffrent sections for diffrent types of algorithms, first is optimal 2gen followed by 2gens of any length. All cases can be solved using two or more Sune, the third section is for these combinations (double, anti and mirrors included). Next comes algs optimal in Half Turn Metric, using as many sides that is needed for that (26) and the last section is for any other alg that solves the case.
Note that all of these algorithms are written in the Western notation, where a lowercase letter means a doublelayer turn and rotations are denoted by x, y, and z. (how to add algorithms) Click on an algorithm (not the camera icon) to watch an animation of it. 
Two corners unoriented
Note that the Ttwist is the same case as the Utwist if you reorient the cube (for these images that will be z' y2).
Three corners unoriented
SOptimal 2gen:

S
Optimal 2gen:

Four corners unoriented
HOptimal 2gen:

piOptimal 2gen:

OCLLEPP
Orientation of the corners of the last layer  edges permutation preserved; orients the last layer corners preserving edge permutation but ignoring corner permutation.
Usage; to solve the last four corners in two looks if the edges are solved (3look LLEF, OCLLEPP and CPLL, total 26 algs. 4look EO, EP, OCLLEPP and CPLL, total 16 algs).
Average movecount is 9.63 turns optimally, CPLL is 9.12 turns optimally wich gives 19 moves on average for solving L4C in 2looks like this.
Algorithms:
Note that all of these algorithms are written in the Western notation, where a lowercase letter means a doublelayer turn and rotations are denoted by x, y, and z. (how to add algorithms) Click on an algorithm (not the camera icon) to watch an animation of it. 
Two corners
U (EPP)
Optimal 2gen:

T (EPP)Optimal 2gen:
 
L (EPP)Optimal 2gen:

Three corners
S (EPP)Optimal 2gen:

S (EPP)Optimal 2gen:

Four corners
H (EPP)Optimal 2gen:

pi (EPP)Optimal 2gen:
