I am currently modelling the dynamic engulfment of a sphere. The sphere is fixed. Here is the model.
The ODE part is very simple. The data are associated to A_2d_psi0 (2D signal for A when there is no rotation, or psi=0). A_3d ODE is the reconstruction of A_2d assuming that k2_3d=k1. The signal increase and decrease for these two ODEs are regulated by events, so up to a certain time point, the signal is only increasing, whereas after that point, the signal is only decreasing. The signal peak corresponds to 1/2 engulfment of the sphere (maximum circumference surrounding the sphere) and this perimeter is 1.
As the maximum signal intensity is 1, I can infer:
- $R$, the radius of the sphere
- $\theta$, the extending angle of this engulfing circumference
- $r1$, the radius of this engulfing circumference
- $r2$, the second radius if the engulfment is projected. This is equal to $r1$ if $\psi=0$, but decreases if we increase $\psi$ to $\pi/2$. So the projection of the engulfing circumference to 2D coordinates is an ellipse.
- $h$, the height of the expanding circumference from the North $N$.
- $E$, the perimeter of the projected ellipse
Now the problem.
What I am interested is the length of the visible curve (green) and the area of the visible surface (red) in 2D coordinates.
I think I correctly extracted the thresholds where the formulas for these should change from this. Here is an image of these thresholds and the graphical representation of the engulfment: Engulfment upon different threshold ranges.
The previous image exactly shows the visible parts that I am interested. This must be computed reflecting changes in $\psi=[0,\pi/2]$.
In the current model I have the computation of the visible green line (C_2d) is an approximation, but it is not exact. I am totally stuck with the computation of the visible red area (SC_2d).
Could you help me, please?