java - Quantify a change of direction given an array of 3d points -

i working on piece of software written in java uses processing.core library classes , simpleopenni track user's hand xbox kinect.

what trying figure out how determine when user's hand movement changes direction abruptly.

what have @ disposal array of pvectors (essentially vector of x,y,and z coordinates: a point in 3d space) record user's hand position past 30 frames or so.

i imagine there must way obtain value represents amount of directional change in real-time given recent few points recorded. maybe fit curve , take derivatives?

ideally solution should not computationally expensive, trying implement real-time worthy solution.

any direction can offer appreciated!

let position (that is, 0 frames ago) (x0, y0, z0)
let position n frames ago (xn, yn, zn).
let position 2n frames ago (xm, ym, zm).

then change in position between frames 2n , n (xn-xm, yn-ym, zn-zm).
can think of average velocity during n frames.

, change in position between n , (x0-xn, y0-yn, z0-zn).
represents average velocity during next n frames.

now have velocity n frames ended n frames ago, , have velocity n frames ended now.

the change in velocity during last n frames must difference between 2 vectors. each coordinate:

ax = (x0-xn) - (xn-xm) = x0 -2xn + xm
ay = (y0-yn) - (yn-ym) = y0 -2yn + ym
az = (z0-xn) - (zn-zm) = z0 -2zn + zm

the magnitude of acceleration |a| = sqrt( (ax)^2 + (ay)^2 + (az)^2 )
if you're concerned "large" changes , prefer speed on accuracy, may able away a' = (ax)^2 + (ay)^2 + (az)^2
or a" = abs(ax) + abs(ay) + abs(az)

the idea want each component contribute whole regardless of whether it's positive or negative, "force" positive either squaring or taking absolute value.

hope helps!