Understanding Snape

[potioncat]

No: HPFGUIDX 153899

 
> Lunasa:
...Snape then comes to Lupin's office, sets
> the Potion down plans on waiting for Lupin to make sure he drinks 
it,
> spots the map, looks at it, sees Potter and his gang in the Shack 
with Black and a dead man! And then he spots Lupin on his way to the
> Shack, probably looks out the window and sees Lupin on his way
> there! 

Potioncat (using chp 19 of PoA):
No, Snape saw Lupin on the map, disappearing down a passageway. The 
passage runs off the map. While Snape knew it went to the Shack, the 
Shack wasn't on the map. Nor was Black, Pettigrew or the Trio.

He assumed Lupin would be meeting Black (I think) and went to see. 
Once he saw Harry's cloak at the Whomping Willow, Snape knew that 
Lupin had Harry and possibly the other two.

The rest is history--or possibly interpretation.