The two-way preposition "auf" in the accusative and dative case

Question

I'm struggling to decide as to choose between accusative and dative case while using auf. When I think dative should be used, it's accusative and vice versa. I know that with two-way preposition you use accusative for referring to movement or direction and for referring to location or position, you use the dative. But in the following two sentences, both cases are used even though the meaning and logic is very similar:

Plötzlich fällt der Tourist mit dem Kopf auf den Tisch.

Sein Gesicht landet auf dem Teller.

Also I assumed the dative should've been used in this sentence but it turned to be accusative:

Ich mag den Ausblick auf den Main.

What am I missing?

Answer 1

It would be also grammatically correct to say

Ich mag den Ausblick auf dem Main

but it then would have a different meaning, in case of Main (a river) a one that would make little sense.

Usually you use Ausblick for describing what you are looking at, this is a direction and therefore accusative. So when you say

Ich mag den Ausblick auf den Main

that means that you are at a lookout point and enjoy the view to / over the Main.

The version

Ich mag den Ausblick auf dem Main

would mean that you are literally on the Main (on a boat) and enjoy the view from there. This would make more sense in case the water you are on was a big lake, not a river, though.

The part about landen is already well explained in Nico's answer.

Answer 2

These are two interesting cases which show that you can't explain grammar without including conceptualization. "Landen" automatically triggers the dative. Why? Although the verb implies a path or a trajectory, it is only the final step (dative) that it conceptualizes, that is the state of remaining landed, so to say. The other case is rather the opposite inasmuch as "Ausblick" is conceptualized as involving an imaginary path of the eye movement.

Answer 3

It helps to think of it in terms of location versus direction: auf + dative indicates a place (where something is), but auf + accusative a direction or destination, with the verb determining which of the two is appropriate. E.g.:

Plötzlich fällt der Tourist mit dem Kopf auf den Tisch.

The accusative indicates that the head is moving towards the table (direction). Falling is a movement and requires a direction.

Ich klettere auf den Berg.

The accusative indicates that the mountain is the destination of my climb (a directional activity).

Sein Gesicht landet auf dem Teller.

The dative indicates the location where the head is landing. Landing requires a place.

I sitze auf dem Berg.

The mountain is dative and therefore my current location, as required by "sitzen".

In English, dative and accusative are not readily distinguishable, so the preposition has to be changed to disambiguate:

Suddenly the tourist falls with his head onto the table.

His face is landing on the plate.

The same applies to your "Ausblick auf dem/den Main" example, although there is no verb dictating the distinction in this case. If you are on the river, e.g. in a boat, and enjoying the view from that location, you use the dative ("dem"), but if you are looking at/onto the river (e.g. from the river bank), you use the accusative ("den").