The raw input is processed as follow

raw input $\rightarrow$ wobble smoother $\rightarrow$ upsampled $\rightarrow$ position modeling

The position of the pen is modeled as a weight connected by a spring to an anchor.

The anchor moves along the resampled dewobbled inputs, pulling the weight along with it across a surface, with some amount of friction. Euler integration is used to solve for the position of the pen.

The physical model that is used to model the stroke is the following $$\frac{d^{2}s}{dt^{2}} = \frac{\Phi(t) - s(t)}{k_{\text{spring }}} - k_{\text{drag }}\frac{ds}{dt}$$ where

• $t$ is time

• $s(t)$ is the position of the pen

• $\Phi(t)$ is the position of the anchor

• $k_{\text{spring}}$ and $k_{\text{drag}}$ are constants that sets how the spring and drag occurs

$k_{\text{spring}}$ is given by position_modeler_spring_mass_constant and $k_{\text{drag}}$ by position_modeler_drag_constant

We will thus have as input the upsampled dewobbled inputs taking the role of discretized $\Phi(t)$ and $s(t)$ will be the output

Modeling a stroke.

• Input : input stream $\left\{ \left( p\lbrack k\rbrack,t\lbrack k\rbrack \right),0 \leq k \leq n \right\}$

• Output : smoothed stream $\left\{ \left( p_{f}\lbrack k\rbrack,v_{f}\lbrack k\rbrack,a_{f}\lbrack k\rbrack \right),0 \leq k \leq n \right\}$

We define $\Phi\lbrack k\rbrack = p\lbrack k\rbrack$. An euler scheme integration scheme is used with the initial conditions being $v\lbrack 0\rbrack = 0$ and $p_{f}\lbrack 0\rbrack = p\lbrack 0\rbrack$ (same initial conditions)

Update rule is simply $$\begin{array}{r} a_{f}\lbrack j\rbrack = \frac{p\lbrack j\rbrack - p_{f}\lbrack j - 1\rbrack}{k_{\text{spring }}} - k_{\text{drag }}v_{f}\lbrack j - 1\rbrack \\ v_{f}\lbrack j\rbrack = v_{f}\lbrack j - 1\rbrack + \left( t\lbrack j\rbrack - t\lbrack j - 1\rbrack \right)a_{f}\lbrack j\rbrack \\ p_{f}\lbrack j\rbrack = p_{f}\lbrack j - 1\rbrack + \left( t\lbrack j\rbrack - t\lbrack j - 1\rbrack \right)v_{f}\lbrack j\rbrack \end{array}$$ The position $s\lbrack j\rbrack$ is the main thing to export but we can also export speed and acceleration if needed. We denote $$q\lbrack j\rbrack = \left( p_{f}\lbrack j\rbrack,v_{f}\lbrack j\rbrack,a_{f}\lbrack j\rbrack,t\lbrack j\rbrack \right)$$ and this will be our output with $0 \leq j \leq n$