At this point is the checking extension that is longest, as extension. But that doesn't mean that it is better than b. The extension to b does more to secure the corner. If Black plays as shown, the right-hand corner will need two more plays to become territory, at least.
On top of these considerations, there is the rest of the board to take into account, as far as direction of play is concerned. $$$$
Bill: After White is a bit cramped. would be better at a. (People do not worry so much about that these days, but it is less than ideal.) So the wedge at a is normally better than .
Why play it, then? The answer lies with the rest of the board. Typically White plays because if she played at a Black's approach from the left would be too good. Usually that means that Black has something on the left side (like a long extension) that would work with that approach. (See nirensei v nirensei for an example.)
From a Tong Yang Securities Cup game, Seo Pong-su-Kataoka Satoshi 1998-01-21. Up to is expected. Now with White pursues a strong group plan. After White still has chances of invading, in either corner. In older games White invaded at c with .
Black has also played this way, to cultivate the corner. White can of course answer at d, but in top games White has also ignored the checking extension and gone elsewhere. holds back, so that White can consider this a proper plan.
When Black plays from the other direction, White has a choice of ways to approach the right-hand star point. White at a is normal thinking, at b was played by Sakata (see wedge - four-space extension), and at c is light and aims at influence, since it is easy for Black to invade in that case.
Since Black may not immediately find a clear plan, it often happens that White plays here once more. In that case is a popular choice in pro games. One can see it as something Black probably answers, to avoid a double kakari, and then White's group on the side shouldn't become very weak.
Against the orthodox fuseki pattern, a wedge is urgent. here is also good, though White at a is the popular choice. White at b doesn't have so much point to it, since Black at would be an excellent checking extension in front of the enclosure.
White's plan, though that may be putting it too strongly, anticipates and then (ignoring the checking extension is known here, having been played in particular by Otake). Now White at a makes for a big swing in power; and it might be premature for Black to attack at once.
For Black to play the other checking extension isn't necessarily wrong. But it's a minor victory for White, since the other wedge play would allow Black one more line in front of the enclosure.
In this sub-orthodox formation, as you could call it, Black's 4-4 point is replaced by a 3-3 point (marked). The side is very open, with a space 12 lines wide. The normal wedge employed by White is shown here. After that is natural, followed by to play lightly and take sente. The other choices of wedge shown are also known from pro games: White at b is common enough because Black's checking extension at a leaves White the chance of playing a shoulderhit in the left corner. White at a was played by the dour Kada Katsuji.
In this case is the popular wedge, at the point Black wants (see Mark II Kobayashi formation page). at a has been played by Kubouchi Shuchi, and White at b, which is also a perverse wedge, by Go Seigen.