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Also, just a side note, since Axe is pretty fast and everything is either monochrome or grayscale you could possibly just let him fall through. Since there aren't multiple colors it might go unnoticed. Just a thought. What ever you do that I'm sure it'll look and feel good game wise
HL2 updates the tilemap every frame (to accommodate for smooth scrolling later), and every sprite is updated each frame, and I haven't noticed slowdown. Also, I made the decision to allow the player to stand on the edge, because I would hate to be playing and find myself forced off of an edge. It is harder to implement, but I think it is worth it.
If I saw a collision, I would stop the physics engine from updating, take manual control of the character, animate it sliding off the edge, and give the physics engine control again. I would probably play around with it to find a suitable method. That is, assuming that I was willing to do that to the player in the first place.