Time Units Symbols and S Values
S = Seconds per Time Unit
SymbolTime UnitsSDistance Units
SSeconds1Light Seconds
MMinutes60Light Minutes
HHours3600Light Hours
DDays86400Light Days
WWeeks604800Light Weeks
YYears31557600Light Years
When performing the relativistic computations, 1 standard year = 365.25 days.

If the time units selected are years, then distances must be reckoned in Light Years.
If the time units used are days, then distances must be reckoned in Light Days.
etc.




Jay Tanner - https://www.PHPScienceLabs.com

Midpoint
$Length$ = Source Cardboard Length
$Width$ = Source Cardboard Width

$H^1$
$H^2$

$A ≠ 0$

$L$ = Length of Box
$W$ = Width of Box

V = Volume of Box

$x$  =  Depth of Box

$V$ = Volume of Box  =  $(L - 2x)\times(W - 2x) \times x$

$V$ = $L~\times~W~\times~x$

$V$ = $(L - 2x)\times(W - 2x) \times x$

$V$ = $4x^3 - 2(L + W)x^2 ~+~ (L \times W)$



Box floor area = (Box Length)$\times$(Box Width)

Box floor area = $(L - 2x)\times(W - 2x)$

$x$ = Box Depth or Height

Volume = (Box Length)$\times$(Box Width)$\times$(Box Depth or Height)

Solving for the first derivative of (V) with respect to (x) gives:

$dV/dx = 12x^2 ~-~ 4(L ~+~ W)x ~+~ (L ~\times~ W)$

Solve quadratic for ($x$):
$12x^2 ~-~ 4(L ~+~ W)x ~+~ (L ~\times~ W) = 0$

$A$ = $12$

$B$ = $-4(L ~+~ W)$

$C$ = $(L ~\times~ W)$

$x = \left(\Large\frac{-B~±~{\sqrt {B^2 ~-~ 4AC}}}{2A} \right)$

$x = \left(\Large\frac{-B}{2A} \right) ± \left(±\Large\frac{{\sqrt {B^2 ~-~ 4AC}}}{2A} \right)$

$Ax^2 + Bx + C = 0$

$x = \left(\Large\frac{4(L ~+~ W)~±~{\sqrt {-4(L ~+~ W)^2 ~-~ 4~\times~12~\times~(L \times W)}}}{2~\times~12} ~\right)$

$x = \left(\Large\frac{4(L ~+~ W)~±~{\sqrt {-4(L ~+~ W)^2 ~-~ 48~\times~(L ~\times~ W)}}}{24} ~\right)$

This can be simplified by dividing through by 4, to obtain:

$x = \left(\Large\frac{{L ~+~ W ~±~ \sqrt {(L ~~+~~ W)^2 ~-~ 3(L ~\times~ W)}}}{6} ~\right)$


The general solution is to equate the first derivative of
volume (V) with respect to (x) to zero and solve for (x).




$$\left(\frac{t_{2}}{t_{1}}\right)^2 = ~ \left(\frac{d_{2}}{d_{1}}\right)^3$$
$$t_{1} ~ = ~ \large \frac{t_{2}}{ \sqrt{\left(\Large\frac{d_{2}}{d_{1}}\right)^3}}$$
$$t_{2} ~ = ~ {t_{1}\cdot\sqrt{\left(\frac{d_{2}}{d_{1}}\right)^3}}$$
$$d_{1} ~ = ~ \large \frac{d_{2}}{\sqrt[3]{\left(\Large\frac{t_{2}}{t_{1}}\right)^2}}$$
$$d_{2} ~ = ~ \large d_{1} \cdot {\sqrt[3]{\left(\frac{t_{2}}{t_{1}}\right)^2}}$$
$$\sqrt{\left(\frac{d_{2}}{d_{1}}\right)^3}$$
$$T = \frac{JD   -   2451545}{36525}$$
$$t = \frac{JD   -   2451545}{365250} = \frac{T}{10}$$
$$X = \sum_{i=0}^5   x_{i}   =   {x_{0}+x_{1}+x_{2}+x_{3}+x_{4}+x_{5}}$$ $$Y = \sum_{i=0}^5   y_{i}   =   {y_{0}+y_{1}+y_{2}+y_{3}+y_{4}+y_{5}}$$ $$Z = \sum_{i=0}^5   z_{i}   =   {z_{0}+z_{1}+z_{2}+z_{3}+z_{4}+z_{5}}$$
$$x_{n} = \left(\sum_{j=1}^k   A_{X,n,j} \cdot {cos(B_{X,n,j}   +   t \cdot C_{X,n,j})}\right) \cdot t^n$$ $$y_{n} = \left(\sum_{j=1}^k   A_{Y,n,j} \cdot {cos(B_{Y,n,j}   +   t \cdot C_{Y,n,j})}\right) \cdot t^n$$ $$z_{n} = \left(\sum_{j=1}^k   A_{Z,n,j} \cdot {cos(B_{Z,n,j}   +   t \cdot C_{Z,n,j})}\right) \cdot t^n$$
$$X = \sum_{n=0}^5\left(\left(\sum_{j=1}^k   A_{X,n,j} \cdot {cos(B_{X,n,j}   +   t \cdot C_{X,n,j})}\right) \cdot t^n\right)$$ $$Y = \sum_{n=0}^5\left(\left(\sum_{j=1}^k   A_{Y,n,j} \cdot {cos(B_{Y,n,j}   +   t \cdot C_{Y,n,j})}\right) \cdot t^n\right)$$ $$Z = \sum_{n=0}^5\left(\left(\sum_{j=1}^k   A_{Z,n,j} \cdot {cos(B_{Z,n,j}   +   t \cdot C_{Z,n,j})}\right) \cdot t^n\right)$$
$$L = \sum_{i=0}^5   l_{i}   =   {l_{0}+l_{1}+l_{2}+l_{3}+l_{4}+l_{5}}$$ $$B = \sum_{i=0}^5   b_{i}   =   {b_{0}+b_{1}+b_{2}+b_{3}+b_{4}+b_{5}}$$ $$R = \sum_{i=0}^5   r_{i}   =   {r_{0}+r_{1}+r_{2}+r_{3}+r_{4}+r_{5}}$$
$$l_{n} = \left(\sum_{j=1}^k   A_{L,n,j} \cdot {cos(B_{L,n,j}   +   t \cdot C_{L,n,j})}\right) \cdot t^n$$ $$b_{n} = \left(\sum_{j=1}^k   A_{B,n,j} \cdot {cos(B_{B,n,j}   +   t \cdot C_{B,n,j})}\right) \cdot t^n$$ $$r_{n} = \left(\sum_{j=1}^k   A_{R,n,j} \cdot {cos(B_{R,n,j}   +   t \cdot C_{R,n,j})}\right) \cdot t^n$$
$$L = \sum_{n=0}^5\left(\left(\sum_{j=1}^k   A_{L,n,j} \cdot {cos(B_{L,n,j}   +   t \cdot C_{L,n,j})}\right) \cdot t^n\right)$$ $$B = \sum_{n=0}^5\left(\left(\sum_{j=1}^k   A_{B,n,j} \cdot {cos(B_{B,n,j}   +   t \cdot C_{B,n,j})}\right) \cdot t^n\right)$$ $$R = \sum_{n=0}^5\left(\left(\sum_{j=1}^k   A_{R,n,j} \cdot {cos(B_{R,n,j}   +   t \cdot C_{R,n,j})}\right) \cdot t^n\right)$$
$$n! = \prod_{1}^n {x} = {1 \times 2 \times 3 \times \cdot\cdot\cdot \times~n}$$
$$\frac{a}{b} ± \frac{c}{d} = \frac{a \cdot d ~±~ b \cdot c}{b \cdot d}$$ $$\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d ~+~ b \cdot c}{b \cdot d}$$ $$\frac{a}{b} - \frac{c}{d} = \frac{a \cdot d ~-~ b \cdot c}{b \cdot d}$$ $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$ $$\frac{\left(\Large \frac{a}{b}\right)}{\left(\Large \frac{c}{d}\right)} = \frac{a \cdot d}{b \cdot c}$$
$$N_{th} ~ {Root ~of ~ (x) ~ Equation}$$ $$b ~=~ \frac{\frac{\Large x}{\left(\Large a^{\large N-1}\right)} ~+~ a \cdot (N-1)}{N}$$
$$log~{x} = \log_{10}\left({x}\right)$$ $$ln ~ x = \log_{e}\left({x}\right)$$ $$ln ~ x = \log_{e}\left({x}\right)$$
$$The~mean~of~a~population~or~sample~of~(x_{i})~values$$ $$\mu = \frac{1}{N} \cdot \sum_{i=1}^N ~ x_{i} ~=~ \frac{x_{1} ~+~ x_{2} ~+~ x_{3} ~+~ \cdot\cdot\cdot ~+~ x_{N}}{N}$$ $$\bar{x} = \frac{1}{N} \cdot \sum_{i=1}^N ~ x_{i} ~=~ \frac{x_{1} ~+~ x_{2} ~+~ x_{3} ~+~ \cdot\cdot\cdot ~+~ x_{N}}{N}$$
$$Sum~of~squares~of~differences~from~mean$$ $$\sum_{i=1}^N ~ (x_{i} - \mu)^2$$ $$\sum_{i=1}^N ~ (x_{i} - \bar{x})^2$$ $$\sum_{i=1}^N ~ (y_{i} - \mu)^2$$ $$\sum_{i=1}^N ~ (y_{i} - \bar{x})^2$$ $$\sum_{i=1}^N ~ (x_{i} - \mu)^2 ~=~ (x_{1}-\mu)^2 ~+~ (x_{2}-\mu)^2 ~+~ (x_{3}-\mu)^2 ~+~ \cdot\cdot\cdot ~+~ (x_{N}-\mu)^2$$ $$\sum_{i=1}^N ~ (x_{i} - \bar{x})^2 ~=~ (x_{1}-\bar{x})^2 ~+~ (x_{2}-\bar{x})^2 ~+~ (x_{3}-\bar{x})^2 ~+~ \cdot\cdot\cdot ~+~ (x_{N}-\bar{x})^2$$
$$Population~Standard~Deviation$$ $$\sigma = \sqrt{\frac{1}{N} \cdot \sum_{i=1}^N ~ (x_{i} - \mu)^2}$$ $$\sigma = \sqrt{\frac{1}{N} \cdot \sum_{i=1}^N ~ \left(x_{i} - \left({\frac{1}{N} \cdot \sum_{i=1}^N ~ x_{i}}\right)\right)^2}$$ $$\sigma = \sqrt{\frac{1}{N} \cdot \sum_{i=1}^N ~ \left(x_{i} - \left(\frac{x_{1} ~+~ x_{2} ~+~ x_{3} ~+~ \cdot\cdot\cdot ~+~ x_{N}}{N} \right)\right)^2}$$
$$The~mean~of~a~population~of~(y_{i})~values$$ $$\mu = \frac{1}{N} \cdot \sum_{i=1}^N ~ y_{i} ~=~ \frac{y_{1} ~+~ y_{2} ~+~ y_{3} ~+~ \cdot\cdot\cdot ~+~ y_{N}}{N}$$
$$Population~Standard~Deviation$$ $$\sigma = \sqrt{\frac{1}{N} \cdot \sum_{i=1}^N ~ (y_{i} - \mu)^2}$$ $$\sigma = \sqrt{\frac{1}{N} \cdot \sum_{i=1}^N ~ \left(y_{i} - \left({\frac{1}{N} \cdot \sum_{i=1}^N ~ y_{i}}\right)\right)^2}$$ $$\sigma = \sqrt{\frac{1}{N} \cdot \sum_{i=1}^N ~ \left(y_{i} - \left(\frac{y_{1} ~+~ y_{2} ~+~ y_{3} ~+~ \cdot\cdot\cdot ~+~ y_{N}}{N} \right)\right)^2}$$
$$P(N,R) ~=~ \frac{N!}{(N-R)!}$$
$$C(N,R) ~=~ \frac{N!}{R!(N-R)!}$$
$$y ~=~ \sum_{i~=~1}^{N}\left(y_{i} \cdot \prod_{j~=~1, ~~~ j~\ne~{i}}^{N} \left(\frac{x - x_{j}}{x_{i} - x_{j}}\right)\right)$$
$$k ~=~ \sqrt[5]{100} ~=~ 2.5118864315095801$$ $$b ~=~ (\sqrt[5]{100}) ^ {~m_{2} ~-~ m_{1}} ~=~ k ~ ^ {~(m_{2} ~-~ m_{1})}$$ $$m_{~2} ~=~ 5\times\log_{10} (\frac{d_{~2}}{d_{~1}}) + m_{~1}$$ $$m_{~1} ~=~ m_{~2} ~-~ 5\times\log_{10} (\frac{d_{~2}}{d_{~1}})$$ $$m_{~2} ~-~ m_{~1} ~=~ 5\times\log_{10} (\frac{d_{~2}}{d_{~1}})$$ $$\frac{1}{5} \times (m_{~2} ~-~ m_{~1}) ~=~ \log_{10} (\frac{d_{~2}}{d_{~1}})$$ $${10 ^ {\frac{1}{5} \times (m_{~2} ~-~ m_{~1})}} ~=~ \frac{d_{~2}}{d_{~1}}$$ $$d_{2} ~=~ d_{1} \times {10^{\frac{1}{5} \times (m_{~2} ~-~ m_{~1})}}$$ $$d_{1} ~=~ \frac{d_{2}}{10^{\frac{1}{5} \times (m_{~2} ~-~ m_{~1})}}$$ $${\large 10 ^ {\frac{1}{5} \cdot (m_{~2} ~-~ m_{~1})}} ~=~ \frac{d_{~2}}{d_{~1}}$$ $$m ~=~ 5\cdot\log_{10} (d_{AU}) ~-~ 26.74$$ $$m ~=~ 5\cdot\log_{10} (d_{LY}) ~-~ 2.735$$ $$m ~=~ 5\cdot\log_{10} (d_{pc}) ~-~ 0.1679$$ $M$
$$r ~=~ \frac{d}{\frac{\Large 1}{\Large sin(\LARGE\frac{a}{2})} ~-~ 1}$$
$$d ~=~ r ~\cdot~ \left(\frac{1}{sin(\Large\frac{a}{2})} ~-~ 1 \right)$$
$$y = \sum_{i=0}^{n-1}\left(y_{i} \cdot \prod_{j=0, ~~ j\ne{i}}^{n-1} \left(\frac{x - x_{j}}{x_{i} - x_{j}}\right)\right)$$
$$\alpha = 2 \times tan{^{-1}}\left( \frac{{R}}{D} \right) $$ $$Where:~~ R > 0 ~~and~~ D > R$$
$\alpha = 2 \times cos{^{-1}}\left(\frac{{\sqrt {D~^2 ~-~ R~^2}}}{D} \right)$

$Where:~~ R > 0 ~~and~~ D > R$
$\alpha = 2 \times cos{^{-1}}\left(\Large\frac{{\sqrt {d^2 ~+~ 2Rd}}}{R ~+~ d} \right)$

$Where:~~ R > 0 ~~and~~ d > R$
$f ~=~ \frac{d}{2(R+d)}$

$Where:~~ R > 0 ~~and~~ d > R$

For the moon

$f ~=~ \frac{d ~~ mi}{2158.7 ~+~ 2d}$

$f ~=~ \frac{d ~~ km}{3474.1 ~+~ 2d}$

$f ~=~ \frac{D ~-~ R}{2D}$

$Where:~~ R > 0 ~~and~~ D > R$
$f ~=~ \frac{D ~-~ R}{2D}$

For the moon

$f ~=~ \frac{D ~-~ 1079.35~mi}{2D}$

$f ~=~ \frac{D ~-~ 1737.05~km}{2D}$
$$Spheres$$ $C ~=~ 2\pi R$

$A ~=~ 4\pi R^2$

$V ~=~ \frac{4}{3}\pi R^3$

$R ~=~ \large\frac{C} {2\pi}$

$R ~=~ \sqrt{\large\frac{A}{4\pi}}$

$R ~=~ \sqrt[\Large 3]{\large\frac{3V}{4\pi}}$
$c = 299792458 ~ m/s$

$\beta ~=~ \Large\frac{v}{c}$

$\beta^2 ~=~ \large(\frac{v}{c})^2 ~=~ \Large\frac{v^2}{c^2}$

$1 - \beta^2 ~=~ 1 - \large(\frac{v}{c})^2 ~=~ 1 - \Large\frac{v^2}{c^2}$

$\frac{\Large t}{T} ~=~ \sqrt{1 - \beta^2} ~=~ \sqrt{1 - \large(\frac{v}{c})^2} ~=~ \sqrt{1 - \Large\frac{v^2}{c^2}}$

$R^2 ~=~ x^2 ~+~ y^2$

$\frac{\Large t}{T} ~=~ xxxxxx$

$1 ~=~ (\large\frac{v}{c})^2 + (\frac{t}{T})^2$


$v ~=~ \sqrt[]{\frac{GM}{R}}$

$v^{~2} ~=~ \frac{GM}{R}$

$Rv^{~2} ~=~ GM$

$M ~=~ \frac{\large Rv^{~2}}{G}$


$f = \frac{xxx}{yyy}$


$d = \frac{\Large at^2}{2}$

$t ~=~ \sqrt[]{\large\frac{2d}{a}}$

$a ~=~ \large\frac{2d}{t^2}$

$v ~=~ at$


$F ~=~ \frac{GMm}{R^2 }$

$FR^{~2} ~=~ GMm$

$\large e ~=~ E_{1}\times(\frac{D_{1}}{D})^2$


$B = CamLat \times \Large \frac{\pi}{180}$

$L = (CamLng - 180) \times \Large \frac{\pi}{180}$

$CamX = -CamDist \times sin(L) \times cos(B)$

$CamY = CamDist \times sin(B)$

$CamZ = CamDist \times cos(L) \times cos(B)$
Spherical and corresponding rectangular coordinates of
the virtual sun on the selenocentric phase sphere:

Given PhaseAlt and PhaseAng in degrees, compute
the corresponding working values in radians.

$B = PhaseAlt\Large\times\frac{\pi}{180}$

$L = PhaseAng\Large\times\frac{\pi}{180}$

Then, the POV-Ray rectangular coordinates of the sun are:

$SunX = SunDist \times sin(L) \times cos(B)$

$SunY = SunDist \times sin(B)$

$SunZ = SunDist \times cos(B) \times cos(L)$
Bearing (AKA Forward Azimuth)
Starting at point-1 and moving continuously forward in initial
direction (FwdAzim) along a great circle until reaching point-2.

Negative Longitude = West

$Lat_{1} =$ Starting point latitude in degrees
$Lng_{1} =$ Starting point longitude in degrees
$Lat_{2} =$ Ending point latitude in degrees
$Lng_{2} =$ Ending point longitude in degrees


Convert latitude and longitude working values into radians.

$B_{1} = Lat_{1}\times\Large\frac{\pi}{180}$

$L_{1} = Lng_{1}\times\Large\frac{\pi}{180}$

$B_{2} = Lat_{2}\times\Large\frac{\pi}{180}$

$L_{2} = Lng_{2}\times\Large\frac{\pi}{180}$



$dL = L_{2} - L_{1}$

$y = sin(dL) \times cos(B_{2})$

$x = cos(B_{1}) \times sin(B_{2}) ~~-~~ sin(B_{1}) \times cos(B_{2}) \times cos(dL)$

$FwdAzim = atan\left(\Large\frac{y}{x}\right)$

$FwdAzim = atan\left(\Large\frac{sin(dL) ~\times~ cos(B_{2})}{cos(B_{1}) \times sin(B_{2}) ~~-~~ sin(B_{1}) \times cos(B_{2}) \times cos(dL)}\right)$

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