A Simple Math Problem

Brian Breitsch

May the Fourth Be With You

2018 May 4

Weekly Seminar:

\theta_g
θg\theta_g
\theta_e
θe\theta_e
\theta_g
θg\theta_g
R_E
RER_E
h
hh
R_{RT}
RRTR_{RT}
R_{ET}
RETR_{ET}
\theta_e
θe\theta_e
\theta_g
θg\theta_g
R_E
RER_E
h
hh
R_{RT}
RRTR_{RT}
R_{ET}
RETR_{ET}
\theta_g
θg\theta_g
\left( R_E + h \right)^2 = R_{RS}^2 + R_E^2 - 2R_{RS}R_E\cos\left(\theta_g + {\pi \over 2}\right)
(RE+h)2=RRS2+RE2−2RRSREcos(θg+π2)\left( R_E + h \right)^2 = R_{RS}^2 + R_E^2 - 2R_{RS}R_E\cos\left(\theta_g + {\pi \over 2}\right)
R_{RS}
RRSR_{RS}
R_{ET}^2 = R_{ST}^2 + R_{E}^2 - 2R_{ST}R_E\cos\left(\theta_g + {\pi \over 2}\right)
RET2=RST2+RE2−2RSTREcos(θg+π2)R_{ET}^2 = R_{ST}^2 + R_{E}^2 - 2R_{ST}R_E\cos\left(\theta_g + {\pi \over 2}\right)
R_{ST}
RSTR_{ST}
\theta_s
θs\theta_s
R_{RT}^2 = R_{RS}^2 + R_{ST}^2 - 2R_{RS}R_{ST}\cos\left(\pi - 2\theta_g\right)
RRT2=RRS2+RST2−2RRSRSTcos(π−2θg)R_{RT}^2 = R_{RS}^2 + R_{ST}^2 - 2R_{RS}R_{ST}\cos\left(\pi - 2\theta_g\right)
\cos\left(\theta_g + {\pi \over 2}\right) = -\sin \theta_g = {R_{RS}^2 - 2R_Eh - h^2 \over 2R_{RS}R_E} = {R_{ST}^2 - R_{ET}^2 + R_E^2 \over 2R_{ST} R_E}
cos(θg+π2)=−sinθg=RRS2−2REh−h22RRSRE=RST2−RET2+RE22RSTRE\cos\left(\theta_g + {\pi \over 2}\right) = -\sin \theta_g = {R_{RS}^2 - 2R_Eh - h^2 \over 2R_{RS}R_E} = {R_{ST}^2 - R_{ET}^2 + R_E^2 \over 2R_{ST} R_E}
R_{ST} \left(R_{RS}^2 - 2 R_E h - h^2\right) = R_{RS} \left(R_{ST}^2 - R_{ET}^2 + R_E^2 \right)
RST(RRS2−2REh−h2)=RRS(RST2−RET2+RE2)R_{ST} \left(R_{RS}^2 - 2 R_E h - h^2\right) = R_{RS} \left(R_{ST}^2 - R_{ET}^2 + R_E^2 \right)
0 = R_{RS} R_{ST}^2 - \left(R_{RS}^2 - 2 R_E h - h^2\right) R_{ST} - R_{RS} \left( R_{ET}^2 - R_E^2 \right)
0=RRSRST2−(RRS2−2REh−h2)RST−RRS(RET2−RE2)0 = R_{RS} R_{ST}^2 - \left(R_{RS}^2 - 2 R_E h - h^2\right) R_{ST} - R_{RS} \left( R_{ET}^2 - R_E^2 \right)
R_{ST} = {\left(R_{RS}^2 - 2 R_E h - h^2\right) \pm \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}}
RST=(RRS2−2REh−h2)±(RRS2−2REh−h2)2+4RRS2(RET2−RE2)2RRSR_{ST} = {\left(R_{RS}^2 - 2 R_E h - h^2\right) \pm \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}}
R_{ET}^2 = \left[ {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right]^2 + R_{E}^2 + 2 \left( {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right)R_E\sin\left(\theta_g \right)
RET2=[(RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2)2RRS]2+RE2+2((RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2)2RRS)REsin(θg)R_{ET}^2 = \left[ {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right]^2 + R_{E}^2 + 2 \left( {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right)R_E\sin\left(\theta_g \right)
4 R_{RS}^2 \left(R_{ET}^2 - R_E^2\right) = \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)^2 + 4 R_{RS} \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)R_E\sin\left(\theta_g \right)
4RRS2(RET2−RE2)=((RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2))2+4RRS((RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2))REsin(θg)4 R_{RS}^2 \left(R_{ET}^2 - R_E^2\right) = \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)^2 + 4 R_{RS} \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)R_E\sin\left(\theta_g \right)
= R_{RS}^4 - (2h^2 + 4 R_E^2) R_{RS}^2
=RRS4−(2h2+4RE2)RRS2= R_{RS}^4 - (2h^2 + 4 R_E^2) R_{RS}^2
\left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)^2
((RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2))2\left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)^2
\alpha = \left(R_{RS}^2 - 2 R_E h - h^2\right)
α=(RRS2−2REh−h2)\alpha = \left(R_{RS}^2 - 2 R_E h - h^2\right)
C =
C=C =
\beta = 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)
β=4RRS2(RET2−RE2)\beta = 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)
R_{ST} = {\alpha + \sqrt{\alpha^2 + \beta} \over 2R_{RS}}
RST=α+α2+β2RRSR_{ST} = {\alpha + \sqrt{\alpha^2 + \beta} \over 2R_{RS}}
R_{ET}^2 = \left( {\alpha + \sqrt{\alpha + \beta} \over 2R_{RS}}\right)^2 + R_E^2 + 2 {\alpha + \sqrt{\alpha + \beta} \over 2R_{RS}} R_E \sin \theta_g
RET2=(α+α+β2RRS)2+RE2+2α+α+β2RRSREsinθgR_{ET}^2 = \left( {\alpha + \sqrt{\alpha + \beta} \over 2R_{RS}}\right)^2 + R_E^2 + 2 {\alpha + \sqrt{\alpha + \beta} \over 2R_{RS}} R_E \sin \theta_g
4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right) = \left(\alpha + \sqrt{\alpha + \beta} \right)^2 + 2 R_{RS} R_E \left(\alpha + \sqrt{\alpha + \beta} \right) \sin \theta_g
4RRS2(RET2−RE2)=(α+α+β)2+2RRSRE(α+α+β)sinθg4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right) = \left(\alpha + \sqrt{\alpha + \beta} \right)^2 + 2 R_{RS} R_E \left(\alpha + \sqrt{\alpha + \beta} \right) \sin \theta_g
4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right) = \alpha^2 + 2 \alpha \sqrt{\alpha + \beta} + \alpha + \beta + 2 R_{RS} R_E \left(\alpha + \sqrt{\alpha + \beta} \right) \sin \theta_g = \alpha^2 + \alpha + \beta + 2 R_{RS} R_E \alpha \sin \theta_g + (2\alpha + 2 R_{RS} R_E \sin\theta_g) \sqrt{\alpha + \beta}
4RRS2(RET2−RE2)=α2+2αα+β+α+β+2RRSRE(α+α+β)sinθg=α2+α+β+2RRSREαsinθg+(2α+2RRSREsinθg)α+β4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right) = \alpha^2 + 2 \alpha \sqrt{\alpha + \beta} + \alpha + \beta + 2 R_{RS} R_E \left(\alpha + \sqrt{\alpha + \beta} \right) \sin \theta_g = \alpha^2 + \alpha + \beta + 2 R_{RS} R_E \alpha \sin \theta_g + (2\alpha + 2 R_{RS} R_E \sin\theta_g) \sqrt{\alpha + \beta}
{4 R_{RS}^2 ( R_{ET}^2 - R_E^2) - \alpha^2 - \alpha - \beta - 2 R_{RS} R_E \alpha \sin \theta_g \over 2\alpha + 2 R_{RS} R_E \sin \theta_g} = \sqrt{\alpha + \beta}
4RRS2(RET2−RE2)−α2−α−β−2RRSREαsinθg2α+2RRSREsinθg=α+β{4 R_{RS}^2 ( R_{ET}^2 - R_E^2) - \alpha^2 - \alpha - \beta - 2 R_{RS} R_E \alpha \sin \theta_g \over 2\alpha + 2 R_{RS} R_E \sin \theta_g} = \sqrt{\alpha + \beta}
\theta_e
θe\theta_e
\theta_g
θg\theta_g
R_E
RER_E
h
hh
R_{RT}
RRTR_{RT}
R_{ET}
RETR_{ET}
\theta_g
θg\theta_g
\left( R_E + h \right)^2 = R_{RS}^2 + R_E^2 - 2R_{RS}R_E\cos\left(\theta_g + {\pi \over 2}\right)
(RE+h)2=RRS2+RE2−2RRSREcos(θg+π2)\left( R_E + h \right)^2 = R_{RS}^2 + R_E^2 - 2R_{RS}R_E\cos\left(\theta_g + {\pi \over 2}\right)
R_{RS}
RRSR_{RS}
R_{ET}^2 = R_{ST}^2 + R_{E}^2 - 2R_{ST}R_E\cos\left(\theta_g + {\pi \over 2}\right)
RET2=RST2+RE2−2RSTREcos(θg+π2)R_{ET}^2 = R_{ST}^2 + R_{E}^2 - 2R_{ST}R_E\cos\left(\theta_g + {\pi \over 2}\right)
R_{ST}
RSTR_{ST}
\theta_s
θs\theta_s
R_{RT}^2 = R_{RS}^2 + R_{ST}^2 - 2R_{RS}R_{ST}\cos\left(\pi - 2\theta_g\right)
RRT2=RRS2+RST2−2RRSRSTcos(π−2θg)R_{RT}^2 = R_{RS}^2 + R_{ST}^2 - 2R_{RS}R_{ST}\cos\left(\pi - 2\theta_g\right)
\cos\left(\theta_g + {\pi \over 2}\right) = -\sin \theta_g = {R_{RS}^2 - 2R_Eh - h^2 \over 2R_{RS}R_E} = {R_{ST}^2 - R_{ET}^2 + R_E^2 \over 2R_{ST} R_E}
cos(θg+π2)=−sinθg=RRS2−2REh−h22RRSRE=RST2−RET2+RE22RSTRE\cos\left(\theta_g + {\pi \over 2}\right) = -\sin \theta_g = {R_{RS}^2 - 2R_Eh - h^2 \over 2R_{RS}R_E} = {R_{ST}^2 - R_{ET}^2 + R_E^2 \over 2R_{ST} R_E}
R_{ST} \left(R_{RS}^2 - 2 R_E h - h^2\right) = R_{RS} \left(R_{ST}^2 - R_{ET}^2 + R_E^2 \right)
RST(RRS2−2REh−h2)=RRS(RST2−RET2+RE2)R_{ST} \left(R_{RS}^2 - 2 R_E h - h^2\right) = R_{RS} \left(R_{ST}^2 - R_{ET}^2 + R_E^2 \right)
0 = R_{RS} R_{ST}^2 - \left(R_{RS}^2 - 2 R_E h - h^2\right) R_{ST} - R_{RS} \left( R_{ET}^2 - R_E^2 \right)
0=RRSRST2−(RRS2−2REh−h2)RST−RRS(RET2−RE2)0 = R_{RS} R_{ST}^2 - \left(R_{RS}^2 - 2 R_E h - h^2\right) R_{ST} - R_{RS} \left( R_{ET}^2 - R_E^2 \right)
R_{ST} = {\left(R_{RS}^2 - 2 R_E h - h^2\right) \pm \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}}
RST=(RRS2−2REh−h2)±(RRS2−2REh−h2)2+4RRS2(RET2−RE2)2RRSR_{ST} = {\left(R_{RS}^2 - 2 R_E h - h^2\right) \pm \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}}
R_{ET}^2 = \left[ {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right]^2 + R_{E}^2 + 2 \left( {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right)R_E\sin\left(\theta_g \right)
RET2=[(RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2)2RRS]2+RE2+2((RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2)2RRS)REsin(θg)R_{ET}^2 = \left[ {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right]^2 + R_{E}^2 + 2 \left( {\left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \over 2R_{RS}} \right)R_E\sin\left(\theta_g \right)
4 R_{RS}^2 R_{ET}^2 - R_E^2 = \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)^2 + 4 R_{RS} \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)R_E\sin\left(\theta_g \right)
4RRS2RET2−RE2=((RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2))2+4RRS((RRS2−2REh−h2)+(RRS2−2REh−h2)2+4RRS2(RET2−RE2))REsin(θg)4 R_{RS}^2 R_{ET}^2 - R_E^2 = \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)^2 + 4 R_{RS} \left( \left(R_{RS}^2 - 2 R_E h - h^2\right) + \sqrt{ \left(R_{RS}^2 - 2 R_E h - h^2\right) ^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right)} \right)R_E\sin\left(\theta_g \right)
\left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right) = R_{RS}^4 + 4R_E^2h^2 + h^4 - 4 R_{RS}^2 R_E h - 2R_{RS}^2h^2 + 4 R_E h^3 + 4 R_{RS}^2R_{ET}^2 - 4 R_{RS}^2R_E^2
(RRS2−2REh−h2)2+4RRS2(RET2−RE2)=RRS4+4RE2h2+h4−4RRS2REh−2RRS2h2+4REh3+4RRS2RET2−4RRS2RE2\left(R_{RS}^2 - 2 R_E h - h^2\right)^2 + 4 R_{RS}^2 \left(R_{ET}^2 - R_E^2 \right) = R_{RS}^4 + 4R_E^2h^2 + h^4 - 4 R_{RS}^2 R_E h - 2R_{RS}^2h^2 + 4 R_E h^3 + 4 R_{RS}^2R_{ET}^2 - 4 R_{RS}^2R_E^2
= R_{RS}^4 - (2h^2 + 4 R_E^2) R_{RS}^2
=RRS4−(2h2+4RE2)RRS2= R_{RS}^4 - (2h^2 + 4 R_E^2) R_{RS}^2
= R_{RS}^4 + (-4 R_E h - 2 h^2 + 4 R_{ET}^2 - 4 R_E^2) R_{RS}^2 + 4 R_E^2h^2 + h^4 + 4 R_E h^3
=RRS4+(−4REh−2h2+4RET2−4RE2)RRS2+4RE2h2+h4+4REh3= R_{RS}^4 + (-4 R_E h - 2 h^2 + 4 R_{ET}^2 - 4 R_E^2) R_{RS}^2 + 4 R_E^2h^2 + h^4 + 4 R_E h^3
\sqrt{x^4 + b x^2 + c}
x4+bx2+c\sqrt{x^4 + b x^2 + c}
(x^2 + \alpha)^2 = x^4 + b x^2 + c
(x2+α)2=x4+bx2+c(x^2 + \alpha)^2 = x^4 + b x^2 + c
b = 2 \alpha, \ \ \ \ \ c = \alpha^2
b=2α,     c=α2b = 2 \alpha, \ \ \ \ \ c = \alpha^2
\alpha = 2R_{ET}^2 - 2R_E^2 - 2 R_E h - h^2
α=2RET2−2RE2−2REh−h2\alpha = 2R_{ET}^2 - 2R_E^2 - 2 R_E h - h^2
\alpha^2 - \left(4 R_E^2h^2 + h^4 + 4 R_E h^3\right) = 4R_{ET}^4 + 4 R_E^4 + 4R_E^2h^2 + h^4 - 8 R_{ET}^2 R_E^2 - 8 R_{ET}^2R_E h - 4 R_{ET}^2h^2 + 8 R_E^3h + 4R_E^2h^2 + 4 R_Eh^3 - 4 R_E^2h^2 - h^4 - 4 R_Eh^3
α2−(4RE2h2+h4+4REh3)=4RET4+4RE4+4RE2h2+h4−8RET2RE2−8RET2REh−4RET2h2+8RE3h+4RE2h2+4REh3−4RE2h2−h4−4REh3\alpha^2 - \left(4 R_E^2h^2 + h^4 + 4 R_E h^3\right) = 4R_{ET}^4 + 4 R_E^4 + 4R_E^2h^2 + h^4 - 8 R_{ET}^2 R_E^2 - 8 R_{ET}^2R_E h - 4 R_{ET}^2h^2 + 8 R_E^3h + 4R_E^2h^2 + 4 R_Eh^3 - 4 R_E^2h^2 - h^4 - 4 R_Eh^3
= 4R_{ET}^4 + 4 R_E^4 + 4R_E^2h^2 - 8 R_{ET}^2 R_E^2 - 8 R_{ET}^2R_E h - 4 R_{ET}^2h^2 + 8 R_E^3h
=4RET4+4RE4+4RE2h2−8RET2RE2−8RET2REh−4RET2h2+8RE3h= 4R_{ET}^4 + 4 R_E^4 + 4R_E^2h^2 - 8 R_{ET}^2 R_E^2 - 8 R_{ET}^2R_E h - 4 R_{ET}^2h^2 + 8 R_E^3h
= 4\left(R_{ET}^4 + R_E^4 + R_E^2h^2 - 2 R_{ET}^2 R_E^2 - 2 R_{ET}^2R_E h - R_{ET}^2h^2 + 2 R_E^3h\right)
=4(RET4+RE4+RE2h2−2RET2RE2−2RET2REh−RET2h2+2RE3h)= 4\left(R_{ET}^4 + R_E^4 + R_E^2h^2 - 2 R_{ET}^2 R_E^2 - 2 R_{ET}^2R_E h - R_{ET}^2h^2 + 2 R_E^3h\right)