

In [5]:

    
from __future__ import division
import sympy
from sympy import *
import numpy as np
import matplotlib.pyplot as plt
from sympy import pprint as pp

# Enable LaTeX-based rendering
#%load_ext sympy.interactive.ipythonprinting
# New way of setting printing:
from sympy import init_printing
init_printing()

from IPython.display import display
from IPython.display import Image

1 - Design parameters



In [ ]:

    
w, h, b, d, c1, c2, k1, k2, r_sys, r_ref = symbols("w, h, b, d, c_1, c_2, k_1, k_2, r_{sys}, r_{ref}", real=True)
# Constraints for hyperboloids:
k1_constraint = k1 > 2
k2_constraint = k2 > 2
c1_constraint = c1 > 0
c2_constraint = c2 > 0
xw, yw, zw = symbols("x_w, y_w, z_w", real=True)
# Local image:
# Image(filename='../images/geometric_model.png')

System Height

From Geometry Expressions, we obtain:

$h= \left\lvert{\dfrac{c_{1}}{2} + \dfrac{c_{2}}{2} - d + \sqrt{\dfrac{c_{1}^{2} \left(k_{1} - 2\right)}{4 k_{1}} + w^{2} \left(\dfrac{k_{1}}{2} + 1\right)} + \sqrt{\dfrac{c_{2}^{2} \left(k_{2} - 2\right)}{4 k_{2}} + w^{2} \left(\dfrac{k_{2}}{2} + 1\right)}}\right\rvert $



In [322]:

    
h_eqn = abs((c1/2)+(c2/2) - d + sqrt((k1-2)*(c1)**(2)/(4*k1)+ (w**2)*(1+k1/2))+ sqrt((k2-2)*(c2)**(2)/(4*k2)+ (w**2)*(1+k2/2)))
h_eqn
display(simplify(h_eqn))
baseline_eqn = b-(c1 + c2 - d)
pp(baseline_eqn)
#print(sympy.latex(h_eqn))
#h_eqn
#print(sympy.latex(baseline_eqn))
baseline_sln = solve(baseline_eqn, b)
display(baseline_sln)









    





$$\left\lvert{\frac{c_{1}}{2} + \frac{c_{2}}{2} - d + \frac{1}{2} \sqrt{\frac{1}{k_{1}} \left(c_{1}^{2} \left(k_{1} - 2\right) + 2 k_{1} w^{2} \left(k_{1} + 2\right)\right)} + \frac{1}{2} \sqrt{\frac{1}{k_{2}} \left(c_{2}^{2} \left(k_{2} - 2\right) + 2 k_{2} w^{2} \left(k_{2} + 2\right)\right)}}\right\rvert$$






    



b - c₁ - c₂ + d






    





$$\begin{bmatrix}c_{1} + c_{2} - d\end{bmatrix}$$

Lens Hole Radius

From Geometry Expressions, we obtain:

$\beta = arctan\left (\dfrac{r_{sys}}{\dfrac{c_{1}}{2}+\sqrt{\dfrac{c_{1}^{2}\cdot \left (-2+k_{1}\right )}{4\cdot k_{1}}+r_{sys}^{2}\cdot \left (-1+\dfrac{k_{1}}{2}\right )}}\right )$

$r_{\beta} =\begin{array}{l}\dfrac{\sqrt{k_{1}}\cdot r_{sys}\cdot \left (2\cdot d\cdot \sqrt{k_{2}}-c_{2}\cdot \sqrt{k_{2}}-c_{2}\cdot \sqrt{-2+k_{2}}\right )}{\sqrt{k_{2}}\cdot \left (c_{1}\cdot \sqrt{k_{1}}+\sqrt{-2\cdot c_{1}^{2}+c_{1}^{2}\cdot k_{1}-4\cdot k_{1}\cdot r_{sys}^{2}+2\cdot k_{1}^{2}\cdot r_{sys}^{2}}\right )}\\d-\dfrac{c_{2}}{2}-\dfrac{c_{2}\cdot \sqrt{-2+k_{2}}}{2\cdot \sqrt{k_{2}}}>0\end{array}$

2 - Mirror profiles (expressions)

Mirror 1 (Top)

\begin{align} {\left(z_1 - \frac{{{c_1}}}{2} \right)^2} - ({x_1^2} + {y_1^2})\left(\frac{{{k_1}}}{2} - 1\right) &= \frac{{c_1^2}}{4}\left(\frac{{{k_1} - 2}}{{{k_1}}}\right) \end{align}



In [294]:

    
x1, y1, z1 = symbols("x_1, y_1, z_1", real=True)



In [324]:

    
mirror_1_eqn = (z1 - c1/2)**2 - (x1**2 + y1**2)*(k1/2 - 1) - (c1**2/4)*((k1 - 2)/k1) 
display(expand(mirror_1_eqn))
mirror_1_eqn_top = z1 - c1/2 - abs(sqrt((x1**2 + y1**2)*(k1/2 - 1) + (c1**2/4)*((k1 - 2)/k1)))
display(mirror_1_eqn_top)
display(expand(mirror_1_eqn_top))









    





$$\frac{c_{1}^{2}}{2 k_{1}} - c_{1} z_{1} - \frac{k_{1} x_{1}^{2}}{2} - \frac{k_{1} y_{1}^{2}}{2} + x_{1}^{2} + y_{1}^{2} + z_{1}^{2}$$






    





$$- \frac{c_{1}}{2} + z_{1} - \left\lvert{\sqrt{\frac{c_{1}^{2} \left(k_{1} - 2\right)}{4 k_{1}} + \left(\frac{k_{1}}{2} - 1\right) \left(x_{1}^{2} + y_{1}^{2}\right)}}\right\rvert$$






    





$$- \frac{c_{1}}{2} + z_{1} - \left\lvert{\sqrt{\frac{c_{1}^{2}}{4} - \frac{c_{1}^{2}}{2 k_{1}} + \frac{k_{1} x_{1}^{2}}{2} + \frac{k_{1} y_{1}^{2}}{2} - x_{1}^{2} - y_{1}^{2}}}\right\rvert$$

Line passing through points $P_w$ and $F_1$

Line equations:



In [296]:

    
lambda1, s1 = symbols("lambda_1, s_1", real=True)
pos_F1 = (0, 0, c1)

x1_on_line = xw + s1 * (pos_F1[0]-xw)
display(x1_on_line)

y1_on_line = yw + s1 * (pos_F1[1]-yw)
display(y1_on_line)

z1_on_line = zw + s1 * (pos_F1[2]-zw)
display(z1_on_line)









    





$$- s_{1} x_{w} + x_{w}$$






    





$$- s_{1} y_{w} + y_{w}$$






    





$$s_{1} \left(c_{1} - z_{w}\right) + z_{w}$$

Let $\lambda_1 = 1 - s_1$, so that:



In [297]:

    
s1_eqn = 1 - lambda1

xr1_on_line = simplify(x1_on_line.subs({s1: s1_eqn}))
display(xr1_on_line)

yr1_on_line = simplify(y1_on_line.subs({s1: s1_eqn}))
display(yr1_on_line)

zr1_on_line = expand(z1_on_line.subs({s1: s1_eqn}))
display(zr1_on_line)









    





$$\lambda_{1} x_{w}$$






    





$$\lambda_{1} y_{w}$$






    





$$- c_{1} \lambda_{1} + c_{1} + \lambda_{1} z_{w}$$



In [298]:

    
mirror_1_intersection = mirror_1_eqn_top.subs({x1: xr1_on_line, y1: yr1_on_line, z1:zr1_on_line})
display(mirror_1_intersection)









    





$$- c_{1} \lambda_{1} + \frac{c_{1}}{2} + \lambda_{1} z_{w} - \left\lvert{\sqrt{\frac{c_{1}^{2} \left(k_{1} - 2\right)}{4 k_{1}} + \left(\frac{k_{1}}{2} - 1\right) \left(\lambda_{1}^{2} x_{w}^{2} + \lambda_{1}^{2} y_{w}^{2}\right)}}\right\rvert$$

Solving for intersection point $P_1$



In [299]:

    
lambda1_sln = solve([mirror_1_intersection], lambda1)
lambda1_sln_minus = simplify(lambda1_sln[0])
display(lambda1_sln_minus)
lambda1_sln_plus = together(lambda1_sln[1])
display(lambda1_sln_plus)









    





$$\begin{pmatrix}\frac{c_{1} k_{1} \left(c_{1} - z_{w}\right) - \sqrt{k_{1} \left(k_{1} - 2\right) \left(c_{1}^{2} - 2 c_{1} z_{w} + x_{w}^{2} + y_{w}^{2} + z_{w}^{2}\right)} \left\lvert{c_{1}}\right\rvert}{k_{1} \left(2 c_{1}^{2} - 4 c_{1} z_{w} - k_{1} x_{w}^{2} - k_{1} y_{w}^{2} + 2 x_{w}^{2} + 2 y_{w}^{2} + 2 z_{w}^{2}\right)}\end{pmatrix}$$






    





$$\begin{pmatrix}\frac{c_{1} k_{1} \left(c_{1} - z_{w}\right) + \sqrt{k_{1} \left(k_{1} - 2\right) \left(c_{1}^{2} - 2 c_{1} z_{w} + x_{w}^{2} + y_{w}^{2} + z_{w}^{2}\right)} \left\lvert{c_{1}}\right\rvert}{k_{1} \left(2 c_{1}^{2} - 4 c_{1} z_{w} - k_{1} x_{w}^{2} - k_{1} y_{w}^{2} + 2 x_{w}^{2} + 2 y_{w}^{2} + 2 z_{w}^{2}\right)}\end{pmatrix}$$



In [300]:

    
lambda1_func_minus = lambdify([c1, k1, xw, yw, zw], lambda1_sln_minus, modules=['numpy'])
lambda1_func_plus = lambdify([c1, k1, xw, yw, zw], lambda1_sln_plus, modules=['numpy'])

$\lambda_1$, can be simplified to be

\begin{align} \lambda_1=\frac{c_1}{L_{v_1} \sqrt{k_1\cdot(k_1 - 2)} + k_1 \left(c_1 - z_w \right)} \end{align}

where $L_{v_1} = ||{\vec{f_1} - \vec{p_w}}|| =|\sqrt{x_w^2+y_w^2+(c_1 - z_w)^2} |$, which is the distance between $P_w$ and mirror 1's focus point $F_1$.



In [301]:

    
L1w = sqrt(xw**2 + yw**2 + (c1-zw)**2)
lambda1_simple = c1/(L1w*sqrt(k1*(k1-2)) + k1*(c1-zw))
display(lambda1_simple)
lambda1_simple_den = (abs(sqrt(k1*(k1-2)*xw**2 + yw**2 + (c1-zw)**2)) + k1*(c1-zw))
den1_simple = lambda1_simple_den*lambda1_simple_den









    





$$\frac{c_{1}}{k_{1} \left(c_{1} - z_{w}\right) + \sqrt{k_{1} \left(k_{1} - 2\right)} \sqrt{x_{w}^{2} + y_{w}^{2} + \left(c_{1} - z_{w}\right)^{2}}}$$

Mirror 2 (Bottom)

\begin{align} {\left(z_2 - \left(d - \frac{c_2}{2}\right) \right)^2} - ({x_2^2} + {y_2^2})\left(\frac{{{k_2}}}{2} - 1\right) &= \frac{{c_2^2}}{4}\left(\frac{{{k_2} - 2}}{{{k_2}}}\right) \end{align}



In [325]:

    
x2, y2, z2, = symbols("x_2, y_2, z_2", real=True)
mirror_2_eqn = (z2 - (d-c2/2))**2 - (x2**2 + y2**2)*(k2/2 - 1) - (c2**2/4)*((k2 - 2)/k2)
display(expand(mirror_2_eqn))
mirror_2_eqn_bottom = z2 - d + c2/2 + abs(sqrt((x2**2 + y2**2)*(k2/2 - 1) + (c2**2/4)*((k2 - 2)/k2)))
display(mirror_2_eqn_bottom)









    





$$\frac{c_{2}^{2}}{2 k_{2}} - c_{2} d + c_{2} z_{2} + d^{2} - 2 d z_{2} - \frac{k_{2} x_{2}^{2}}{2} - \frac{k_{2} y_{2}^{2}}{2} + x_{2}^{2} + y_{2}^{2} + z_{2}^{2}$$






    





$$\frac{c_{2}}{2} - d + z_{2} + \left\lvert{\sqrt{\frac{c_{2}^{2} \left(k_{2} - 2\right)}{4 k_{2}} + \left(\frac{k_{2}}{2} - 1\right) \left(x_{2}^{2} + y_{2}^{2}\right)}}\right\rvert$$

Line passing through points $P_w$ and $F_2$

Line equations:



In [303]:

    
lambda2, s2 = symbols("lambda_2, s_2", real=True)
pos_F2 = (0, 0, d - c2)

x2_on_line = xw + s2 * (pos_F2[0]-xw)
display(x2_on_line)

y2_on_line = yw + s2 * (pos_F2[1]-yw)
display(y2_on_line)

z2_on_line = zw + s2 * (pos_F2[2] - zw) 
display(z2_on_line)









    





$$- s_{2} x_{w} + x_{w}$$






    





$$- s_{2} y_{w} + y_{w}$$






    





$$s_{2} \left(- c_{2} + d - z_{w}\right) + z_{w}$$

Let $\lambda_2 = 1 - s_2$, so that:



In [304]:

    
s2_eqn = 1 - lambda2

xr2_on_line = simplify(x2_on_line.subs({s2: s2_eqn}))
display(xr2_on_line)

yr2_on_line = simplify(y2_on_line.subs({s2: s2_eqn}))
display(yr2_on_line)

zr2_on_line = simplify(z2_on_line.subs({s2: s2_eqn}))
display(zr2_on_line)
display(expand(zr2_on_line))
display(zw*lambda2 + (d-c2)*(1-lambda2))









    





$$\lambda_{2} x_{w}$$






    





$$\lambda_{2} y_{w}$$






    





$$z_{w} + \left(\lambda_{2} - 1\right) \left(c_{2} - d + z_{w}\right)$$






    





$$c_{2} \lambda_{2} - c_{2} - d \lambda_{2} + d + \lambda_{2} z_{w}$$






    





$$\lambda_{2} z_{w} + \left(- c_{2} + d\right) \left(- \lambda_{2} + 1\right)$$



In [305]:

    
mirror_2_intersection = mirror_2_eqn_bottom.subs({x2: xr2_on_line, y2: yr2_on_line, z2:zr2_on_line})
display(mirror_2_intersection)









    





$$\frac{c_{2}}{2} - d + z_{w} + \left(\lambda_{2} - 1\right) \left(c_{2} - d + z_{w}\right) + \left\lvert{\sqrt{\frac{c_{2}^{2} \left(k_{2} - 2\right)}{4 k_{2}} + \left(\frac{k_{2}}{2} - 1\right) \left(\lambda_{2}^{2} x_{w}^{2} + \lambda_{2}^{2} y_{w}^{2}\right)}}\right\rvert$$

Solving for intersection point $P_2$



In [306]:

    
lambda2_sln = solve([mirror_2_intersection], lambda2)
lambda2_sln_minus = simplify(lambda2_sln[0])
display(lambda2_sln_minus)
lambda2_sln_plus = simplify(lambda2_sln[1])
display(lambda2_sln_plus)
display(lambda1_sln_plus)









    





$$\begin{pmatrix}\frac{c_{2} k_{2} \left(c_{2} - d + z_{w}\right) - \sqrt{k_{2} \left(k_{2} - 2\right) \left(c_{2}^{2} - 2 c_{2} d + 2 c_{2} z_{w} + d^{2} - 2 d z_{w} + x_{w}^{2} + y_{w}^{2} + z_{w}^{2}\right)} \left\lvert{c_{2}}\right\rvert}{k_{2} \left(2 c_{2}^{2} - 4 c_{2} d + 4 c_{2} z_{w} + 2 d^{2} - 4 d z_{w} - k_{2} x_{w}^{2} - k_{2} y_{w}^{2} + 2 x_{w}^{2} + 2 y_{w}^{2} + 2 z_{w}^{2}\right)}\end{pmatrix}$$






    





$$\begin{pmatrix}\frac{c_{2} k_{2} \left(c_{2} - d + z_{w}\right) + \sqrt{k_{2} \left(k_{2} - 2\right) \left(c_{2}^{2} - 2 c_{2} d + 2 c_{2} z_{w} + d^{2} - 2 d z_{w} + x_{w}^{2} + y_{w}^{2} + z_{w}^{2}\right)} \left\lvert{c_{2}}\right\rvert}{k_{2} \left(2 c_{2}^{2} - 4 c_{2} d + 4 c_{2} z_{w} + 2 d^{2} - 4 d z_{w} - k_{2} x_{w}^{2} - k_{2} y_{w}^{2} + 2 x_{w}^{2} + 2 y_{w}^{2} + 2 z_{w}^{2}\right)}\end{pmatrix}$$






    





$$\begin{pmatrix}\frac{c_{1} k_{1} \left(c_{1} - z_{w}\right) + \sqrt{k_{1} \left(k_{1} - 2\right) \left(c_{1}^{2} - 2 c_{1} z_{w} + x_{w}^{2} + y_{w}^{2} + z_{w}^{2}\right)} \left\lvert{c_{1}}\right\rvert}{k_{1} \left(2 c_{1}^{2} - 4 c_{1} z_{w} - k_{1} x_{w}^{2} - k_{1} y_{w}^{2} + 2 x_{w}^{2} + 2 y_{w}^{2} + 2 z_{w}^{2}\right)}\end{pmatrix}$$



In [307]:

    
lambda2_func_minus = lambdify([c2, k2, xw, yw, zw, d], lambda2_sln_minus, modules=['numpy'])
lambda2_func_plus = lambdify([c2, k2, xw, yw, zw, d], lambda2_sln_plus, modules=['numpy'])

$\lambda_2$, can be simplified to be

\begin{align} \lambda_2=\frac{c_2}{L_{v_2} \sqrt{k_2\cdot(k_2 - 2)} - k_2 \left(d-c_2-z_w \right)} \end{align}

where $L_{v_2} = ||{\vec{f_2} - \vec{p_w}}|| =|\sqrt{x_w^2+y_w^2+(d-c_2-z_w)^2} |$, which is the distance between $P_w$ and mirror 2's focus point $F_2$.



In [308]:

    
display(expand(d - lambda2*zw - (d - c2)*(1 - lambda2)))
display(expand(c2 + lambda2*(d - c2 - zw)))









    





$$- c_{2} \lambda_{2} + c_{2} + d \lambda_{2} - \lambda_{2} z_{w}$$






    





$$- c_{2} \lambda_{2} + c_{2} + d \lambda_{2} - \lambda_{2} z_{w}$$

Fielf of View Angles

From Geometry Expressions, we obtain:

\begin{align} \theta_{1,max} = -arctan\left (\dfrac{c_{1}\cdot \sqrt{k_{1}}-\sqrt{-2+k_{1}}\cdot \sqrt{c_{1}^{2}+2\cdot k_{1}\cdot r_{sys}^{2}}}{2\cdot \sqrt{k_{1}}\cdot r_{sys}}\right ) \end{align}

or by Mathematica:

\begin{align} \frac{\arctan \left(\sqrt{\frac{(k_1-2) \left(c_1^2+2 k_1 r^2_{sys} \right)}{k_1}}-c_1\right)}{2 r_{sys}} \end{align}



In [309]:

    
theta_1_max_eqn = -atan((c1* sqrt(k1)-sqrt(-2+k1)* sqrt(c1**2 + 2* k1 * r_sys**2))/(2*sqrt(k1) * r_sys))
display(ratsimp(theta_1_max_eqn))









    





$$- \operatorname{atan}{\left (\frac{c_{1}}{2 r_{{sys}}} - \frac{\sqrt{k_{1} - 2}}{2 \sqrt{k_{1}} r_{{sys}}} \sqrt{c_{1}^{2} + 2 k_{1} r_{{sys}}^{2}} \right )}$$

From Geometry Expressions, we obtain:

$$\theta_{1,min} = \arctan\left (\dfrac{\dfrac{-d}{2}+c_{1}}{r_{ref}}\right )$$



In [310]:

    
theta_1_min_eqn = atan((-d/2+c1)/(r_ref))
display(cancel(theta_1_min_eqn))









    





$$\operatorname{atan}{\left (\frac{c_{1}}{r_{{ref}}} - \frac{d}{2 r_{{ref}}} \right )}$$

From Geometry Expressions, we obtain:

$\theta_{2,min} = arctan\left (\dfrac{\dfrac{-c_{2}}{2}+\sqrt{\dfrac{c_{2}^{2}\cdot \left (-2+k_{2}\right )}{4\cdot k_{2}}+r_{sys}^{2}\cdot \left (-1+\dfrac{k_{2}}{2}\right )}}{r_{sys}}\right )$



In [311]:

    
theta_2_min_eqn = atan((-c2/2 + sqrt((c2**2 *(-2+k2))/(4*k2)+r_sys**2 * (-1+k2/2)))/r_sys)
display(trigsimp(theta_2_min_eqn))
display(cancel(theta_2_min_eqn))









    





$$\operatorname{atan}{\left (\frac{1}{r_{{sys}}} \left(- \frac{c_{2}}{2} + \sqrt{\frac{c_{2}^{2} \left(k_{2} - 2\right)}{4 k_{2}} + r_{{sys}}^{2} \left(\frac{k_{2}}{2} - 1\right)}\right) \right )}$$






    





$$- \operatorname{atan}{\left (\frac{c_{2}}{2 r_{{sys}}} - \frac{1}{2 r_{{sys}}} \sqrt{c_{2}^{2} - \frac{2 c_{2}^{2}}{k_{2}} + 2 k_{2} r_{{sys}}^{2} - 4 r_{{sys}}^{2}} \right )}$$

Back Projections

BP Mirror 1:



In [314]:

    
q1, xq1, yq1, zq1, t1, theta1, phi1 = symbols("q_1, x_q1, y_q1, z_q1, t_1, theta_1, phi_1", real=True)
x1_bp_line_eqn = t1*xq1
y1_bp_line_eqn = t1*yq1
z1_bp_line_eqn = t1
mirror1_bp_intersection = mirror_1_eqn_top.subs({x1: x1_bp_line_eqn, y1: y1_bp_line_eqn, z1:z1_bp_line_eqn})
display(mirror1_bp_intersection)
t1_sln = solve([mirror1_bp_intersection], t1)
t1_sln_minus = simplify(t1_sln[0])
display(t1_sln_minus)
t1_sln_plus = together(t1_sln[1])
display(t1_sln_plus)









    





$$- \frac{c_{1}}{2} + t_{1} - \left\lvert{\sqrt{\frac{c_{1}^{2} \left(k_{1} - 2\right)}{4 k_{1}} + \left(\frac{k_{1}}{2} - 1\right) \left(t_{1}^{2} x_{q1}^{2} + t_{1}^{2} y_{q1}^{2}\right)}}\right\rvert$$






    





$$\begin{pmatrix}\frac{c_{1} k_{1} - \sqrt{k_{1} \left(k_{1} - 2\right)} \sqrt{x_{q1}^{2} + y_{q1}^{2} + 1} \left\lvert{c_{1}}\right\rvert}{k_{1} \left(- k_{1} x_{q1}^{2} - k_{1} y_{q1}^{2} + 2 x_{q1}^{2} + 2 y_{q1}^{2} + 2\right)}\end{pmatrix}$$






    





$$\begin{pmatrix}\frac{c_{1} k_{1} + \sqrt{k_{1} \left(k_{1} - 2\right)} \sqrt{x_{q1}^{2} + y_{q1}^{2} + 1} \left\lvert{c_{1}}\right\rvert}{k_{1} \left(- k_{1} x_{q1}^{2} - k_{1} y_{q1}^{2} + 2 x_{q1}^{2} + 2 y_{q1}^{2} + 2\right)}\end{pmatrix}$$

Simplifying the + solution of $t_1$, we get: \begin{align} t_1=\frac{c_1}{k_1-||{q_1}||\sqrt{k_1\cdot(k_1 - 2)}} \end{align}

BP Mirror 2:



In [318]:

    
q2, xq2, yq2, zq2, t2, theta2, phi2 = symbols("q_2, x_q2, y_q2, z_q2, t_2, theta_2, phi_2", real=True)
x2_bp_line_eqn = t2*xq2
y2_bp_line_eqn = t2*yq2
z2_bp_line_eqn = d-t2
mirror2_bp_intersection = mirror_2_eqn_bottom.subs({x2: x2_bp_line_eqn, y2: y2_bp_line_eqn, z2:z2_bp_line_eqn})
display(mirror2_bp_intersection)
t2_sln = solve([mirror2_bp_intersection], t2)
t2_sln_minus = simplify(t2_sln[0])
display(t2_sln_minus)
t2_sln_plus = together(t2_sln[1])
display(t2_sln_plus)









    





$$\frac{c_{2}}{2} - t_{2} + \left\lvert{\sqrt{\frac{c_{2}^{2} \left(k_{2} - 2\right)}{4 k_{2}} + \left(\frac{k_{2}}{2} - 1\right) \left(t_{2}^{2} x_{q2}^{2} + t_{2}^{2} y_{q2}^{2}\right)}}\right\rvert$$






    





$$\begin{pmatrix}\frac{c_{2} k_{2} - \sqrt{k_{2} \left(k_{2} - 2\right)} \sqrt{x_{q2}^{2} + y_{q2}^{2} + 1} \left\lvert{c_{2}}\right\rvert}{k_{2} \left(- k_{2} x_{q2}^{2} - k_{2} y_{q2}^{2} + 2 x_{q2}^{2} + 2 y_{q2}^{2} + 2\right)}\end{pmatrix}$$






    





$$\begin{pmatrix}\frac{c_{2} k_{2} + \sqrt{k_{2} \left(k_{2} - 2\right)} \sqrt{x_{q2}^{2} + y_{q2}^{2} + 1} \left\lvert{c_{2}}\right\rvert}{k_{2} \left(- k_{2} x_{q2}^{2} - k_{2} y_{q2}^{2} + 2 x_{q2}^{2} + 2 y_{q2}^{2} + 2\right)}\end{pmatrix}$$

Test and Plot Forward Projections

Horizontal Depth from Triangulation

From Geometry Expressions:

\begin{align} D=\left |\dfrac{-\left (-d+c_{1}+c_{2}\right )\cdot cos\left (\theta_{1}\right )\cdot cos\left (\theta_{2}\right )}{sin\left (\theta_{2}\right )\cdot cos\left (\theta_{1}\right )+sin\left (\theta_{1}\right )\cdot cos\left (\theta_{2}\right )}\right | \end{align}



In [27]:

    
theta1, theta2 = symbols("theta_1, theta_2", real=True)
D_eqn = simplify(-B*cos(theta1)*cos(theta2)/(sin(theta2)*cos(theta1)+sin(theta1)*cos(theta2)))
display(D_eqn)
D_eqn_2 = B*sin(pi/2-theta1)*sin(pi/2-theta2)/sin(pi-theta1-theta2)
display(D_eqn_2)
D_eqn == D_eqn_2









    





$$- \frac{B \cos{\left (\theta_{1} \right )} \cos{\left (\theta_{2} \right )}}{\sin{\left (\theta_{1} + \theta_{2} \right )}}$$






    





$$\frac{B \cos{\left (\theta_{1} \right )} \cos{\left (\theta_{2} \right )}}{\sin{\left (\theta_{1} + \theta_{2} \right )}}$$






    Out[27]:





False