Thesis - Instituto de Telecomunicações

4.1. HUMAN-COMPUTER INTERACTION 65To unwrap the angle vector (θ) removing the discontinuities near π and −π, we used thefollowing equations:( )θ i = arctan ∗ ∆y1+∆x 1{ ( ) }∆θ i = min ∆ arctan ∗ ∆yi+2kπ∆x ii∑j=1∆θ j, (4.3)k ∈ Zwhere the function arctan ∗ is the four quadrant arc-tangent of ∆x and ∆y, with thedomain [−π, π].The curvature is defined as follows:c = ∆θ∆s . (4.4)The curvature is inversely proportional to the radius of the intrinsic circumference thatfits the point where the curvature is being calculated. The rate of change in the curvatureis computed using the following equation:Figure 4.3, left column, presents an example of these vectors.c ′ = ∆c∆s . (4.5)Temporal Information In the temporal domain we defined 9 vectors, calculated fromthe original acquired data points:• px — the vector with the input px i . . . px n values.• py — the vector with the input py i . . . py n values.• t — the input time vector t i . . . t n .• v x — horizontal velocity.• v y — vertical velocity.• v — tangential velocity.