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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?

pdp10
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  • Update for the visible green curve in 2D dimensions (C_2d). – pdp10 May 19 '17 at 11:58
  • I used the sunrise equation, cos(omega)=-tan(phi) x tan(delta), to calculate the angle of illumination for an actor at a latitude phi when the Sun (the observer) is at declination delta. The engulfment now works as if this actor were moving from the North pole to the South pole (so it is =pi/2-theta). Delta replaces psi. Doing so, Eomega/(2pi) is the proportion of E that I am interested. – pdp10 May 19 '17 at 12:07

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