Infinite distance (trigonometric): non-iterative, too?

[fgnievinski]

it seems the solution should not be restricted to the iterative or numerical case -- it'd be possible to obtain a trigonometric solution for infinite distance based on a preliminary solution for finite distance using analytical algorithms, no?

another question: does the infinite distance assumption keep the finite-distance specular point unchanged (position coordinates and geocentric angle and arc_len)?

[fgnievinski]

pedir ajuda para Fujimura, Vuorinen et al.

[fgnievinski]

rascunho:

Dear Dr. Vuorinen,

we'd like to respectfully ask if your research group could help us with the following mathematical problem.

• Would it be possible to simplify the quartic involved in the Ptolemy-Alhazen problem by assuming the source is at infinite distance, i.e., defined only by the direction angles?

In our application, the transmitting source is a GPS satellite at 20,000 km altitude and the receiving antenna is only a few meters above the Earth's surface, assumed spherical with 6,000 km radius.

[vitorhjr]

I would only change the information about Earth radius, that we use the average radius of approximately 6370 km. Then, the email draft is:

Dear Dr. Vuorinen,

We'd like to respectfully ask if your research group could help us with the following mathematical problem.

• Would it be possible to simplify the quartic involved in the Ptolemy-Alhazen problem by assuming the source is at infinite distance, i.e., defined only by the direction angles?

In our application, the transmitting source is a GPS satellite at 20,000 km altitude and the receiving antenna is only a few meters above the Earth's surface, assuming a sphere with a 6,370 km radius.

Regards,

Vitor Hugo de Almeida Junior PhD student at Federal University of Rio Grande do Sul

Do you agree with this draft @fgnievinski? I will send it scheduled for Monday.

[fgnievinski]

parece bom

On Fri., May 5, 2023, 21:13 Vitor Hugo de Almeida Junior, < @.***> wrote:

I would only change the information about Earth radius, that we use the average radius of approximately 6370 km. Then, the email draft is:

Dear Dr. Vuorinen,

We'd like to respectfully ask if your research group could help us with the following mathematical problem.

• Would it be possible to simplify the quartic involved in the Ptolemy-Alhazen problem by assuming the source is at infinite distance, i.e., defined only by the direction angles?

In our application, the transmitting source is a GPS satellite at 20,000 km altitude and the receiving antenna is only a few meters above the Earth's surface, assuming a sphere with a 6,370 km radius.

Regards,

Vitor Hugo de Almeida Junior PhD student at Federal University of Rio Grande do Sul

Do you agree with this draft @fgnievinski https://github.com/fgnievinski? I will send it scheduled for Monday.

— Reply to this email directly, view it on GitHub https://github.com/ufrgs-gnss-lab/geo-alhazen/issues/31#issuecomment-1536928646, or unsubscribe https://github.com/notifications/unsubscribe-auth/ABGJHBAAIMGCOMYVYEO6P2DXEWJR7ANCNFSM6AAAAAATZ46NGY . You are receiving this because you were mentioned.Message ID: @.***>

[vitorhjr]

Response from Dr. Vuorinen:

Dear Mr. Vitor Hugo De Almeida Junior,

Thank you for your email. We will think about this and inform you about our ideas, if any.

Best regards, M Vuorinen

[vitorhjr]

Response from Dr. Vuorinen

Dear Mr. Vitor Hugo De Almeida Junior,

The solution to the "source at infinite-distance" Ptolemy-Alhazen problem that you posed is attached. We are currently working on a manuscript on related questions will perhaps include this result there.

Please keep me informed about your work on this topic and whether you will use this result in your work.

Sincerely, Matti Vuorinen

cal20230511.pdf

[vitorhjr]

Dr. Vuorinen also send the following:

Dear Mr. Vitor Hugo De Almeida Junior,

Good to hear. I have a Mathematica notebook on this and I can send it to you if you are interested.

Good luck for your research!

M Vuorinen

[vitorhjr]

Implementation of Dr. Vuorinen's mathematical model

The mathematical model is based on a complex unit circle. The incident ray reach the sphere from the right side parallel to the real axis.

• $f=r e^{i\theta}$ is the receiver, where r is the normalized radius of the receiver. In our case, we can assume r as $r=R_a/R_0$, which Ra is the radius of the antenna and R0 is the radius of the sphere.
• w is the reflection point. It's solved through the quartic polinomial $r e^{-i\theta}w^4-w^3+w-r e^{i\theta}$.
• $\tilde{w} = w+1$. It lies in the same line of w and is parallel to the real axis
• The polynomial has four roots. We can choose the one that satisfies $e^{i\phi}$
• $\phi$ is the phase angle of the reflection point and $0< \phi <\pi/2$
• As $\phi$ is very close $\theta$, we can define $\phi=\theta$. Thus, we define this as a condition to get the correct root of the polynomial
• The specular point is obtained through the equation: $pos_spec = (w/(e^{-i \pi/2-\theta}))*R_0-[0 R_0i]$

Matlab algorithm

R0 = get_earth_radius (); % sphere radius
Ra = R0 + Ha; % antenna radius
theta = deg2rad(90-e); % phase angle of the antenna
ra = Ra./R0; % normalized antenna radius

% Polynomial coefficients
c4 = ra.*exp(-1i.*theta);
c3 = -1;
c2 = 0;
c1 = 1;
c0 = -ra.*exp(1i.*theta);

ws = roots ([c4 c3 c2 c1 c0]); % roots

phis = angle(ws); % phase angle of the specular point
ind = argmin(abs(phis-theta)); % index of the correct root
w = ws (ind); % correct root
pos_spec_complex_local = w./exp(-1i*(pi/2-theta)).*R0-complex(0,R0); % specular position with local complex coordinates
x_complex_local = real(pos_spec_complex_local);
y_complex_local = imag(pos_spec_complex_local);

[vitorhjr]

I implement Vuorinen's model in the geo-alhazen repository in the following commit (https://github.com/ufrgs-gnss-lab/geo-alhazen/commit/ba83c99824cc614eb5e699c110c536cc822e50f3).

• get_reflection_spherical_vuorinen: function with the mathematical model
• get_spherical_reflection: implemented Vuorinen model in the global function to retrieve the specular reflection on a sphere
• demo_vuorinen: demonstration of the _get_reflection_sphericalvuorinen and a comparison with Fujimura model.

[vitorhjr]

Questions about this model:

• Is the elevation zero the limit for this model?
• What do we have to do in case we have two roots with the same condition? Could it occur?

[vitorhjr]

I started discussions about Vuorinen's algorithm and results at https://github.com/ufrgs-gnss-lab/geo-alhazen/discussions/38.