To map coordinates to a 3D coordinate system, you must know the scale factors, whether the origin needs to be translated, and whether the target coordinate system uses the right-hand rule or the left-hand rule.
Units of measurement and scale in 3D graphics tend to be determined arbitrarily, but essentially, you need to decide how a movement in the real world should map to a movement in the virtual world.
The origin of coordinates normalized with an interaction box is the bottom, left, rear corner. You can translate the normalized coordinates to move the origin to a more suitable location.
The Leap Motion coordinate system uses the right-hand rule. If your coordinate system uses the left-hand rule, multiply the z-axis by -1 before normalizing (or swap the x and y axes).
The following example illustrates mapping Leap Motion coordinates to a 3D system with the origin of the interaction area at the bottom center of the world, scales the range of the interaction box to 200 3d world units across: