How do I test overlap (or collision) of two arbitrary shapes

Question

I'm looking for a Java library which I can use to compute the collision of two simple (no intersecting lines or holes) 2D polygons.

The specific implementation is a drag-and-drop "sticker" applicator where the user drags various solid shapes around the screen and the shapes may not overlap.

I have a vague idea of how to write this myself but am hoping to avoid reinventing this bicycle, however, after a number of searches I'm unable to find anything useful.

Answer 1

Box2D

Features

Rigid body physics
Stable stacking
Gravity
Fast persistent contact solver
Dynamic tree broadphase
Sliding friction
Boxes, circles, edges and polygons
Several joint types: distance, revolute, prismatic, pulley, gear, mouse
Motors Sleeping (removes motionless bodies from simulation until touched)
Continuous collision detection (accurate solving of fast bodies)
Ray casts
Sensors
Serialization
Dynamic, Kinematic, and Static bodies
Liquid particle simulation from Google's LiquidFun