We denote :

• Pressure : $\nu \in \lbrack 0,1\rbrack$

• time : $t \geq 0$

• point : defined by position $x$ and $y$ (or $p = (x,y) \in {\mathbb{R}}^{2}$)

• Raw inputs are denoted by a tuple $i\lbrack k\rbrack = \left( \nu\lbrack k\rbrack,t\lbrack k\rbrack,x\lbrack k\rbrack,y\lbrack k\rbrack \right)$ with $k \in {\mathbb{N}}$.

An input stream is a sequence of raw inputs $i = (\nu,t,x,y)$

• with time $t\lbrack k\rbrack$ $\nearrow \nearrow$ strictly increasing

• starts with a Down event

• contains Move for $k \geq 1$

• ends either with a Move or a Up . If it is a Up we say the input stream is complete

We addition define

• $v_{x},v_{y},a_{x},a_{y}$ as the velocity and acceleration.

With the vector shorthand $v = \left( v_{x},v_{y} \right) \in {\mathbb{R}}^{2}$ and $a = \left( a_{x},a_{y} \right) \in {\mathbb{R}}^{2}$