Lost? Have you tried Spherical Navigation?

This video describes how to use spherical trig to solve problems in terrestrial navigation including dead reckoning, distance between geographical points (long/lat) and using the intercept method. You can see more of my math videos at www.math4every1.info

[@ivolol]

Very interesting bit of trig. I appreciate you trying to go through one example of varying types of question that could be posed using these. Would GPS use any form of this math? If you get an AT2020USB+ or Blue Yeti Nano, your audio clarity would go way up.

[@math4every1]

GPS typically uses signals from 3 satellites for positioning and a 4rth when available for confirmation but no, not the same type math.

[@lyingcat9022]

Typically when choosing an Assumed Position a Navigator will NOT use the Dead Reckoning position. They will choose an AP on a whole degree of Latitude closest to the DR and a Longitude that will result in a whole degree of LHA(Local Hour Angle) from the GP(Geographical Position). This allows the Navigator to enter the Sight Reduction Tables with whole degrees, eliminating the need for interpolation… at least for that part of the calculation of the Navigational Triangle :)

[@math4every1]

I agree. Another "in practice" note. However you are assuming they are using tables to do all the calculation whereas I was pushing the solution via spherical trig, so moving to the nearest degree was not really needed, just an accurate calculator.

[@JeremyWilshere]

12:27 distance then bearing # Remain in degrees throughout # Prerequisites: GNU units, linux bash shell # copy pasta: # SNIP------------------------------------------ phi_A=47.6 lambda_A=122.33 phi_B=21.3 lambda_B=157.86 # PART ONE: distance (A,B) = c or d in Gary's diagram a=`units -t "(90 - $phi_B)"` b=`units -t "(90 - $phi_A)"` gamma=`units -t "$lambda_B - $lambda_A"` cos_c=`units -t "cos ($a deg) cos ($b deg) + sin ($a deg) sin ($b deg) cos ($gamma deg)"` c=`units -t "acos($cos_c)" deg` C=`units -t "$c 60 nmile" nmile` C_statute_mile=`units -t "$c 60 nmile" mile` printf "# %5.2f %5.2f %5.2f %5.2f " $phi_A $lambda_A $phi_B $lambda_B printf "# %5.2f %5.2f %5.2f %5.7f %5.4f %5.2f %5.2f " $a $b $gamma $cos_c $c $C $C_statute_mile # 47.60 122.33 21.30 157.86 # 68.70 42.40 35.53 0.7795150 38.7838 2327.03 2677.90 # # Glen Gary: # 12:37 # PART TWO # Bearing (A,B) # Bearing for same trip, theta (A,B) wrt Grid North (North Pole = NP) # instead of theta, I use alpha to preserve notation symmetry # value d (aka c) is substituted in from above # cos_alpha=`units -t "(cos ($a deg) - cos($c deg) cos ($b deg)) / ( sin($c deg) sin ($b deg) )"` alpha=`units -t "acos( $cos_alpha)" deg` bearing=`units -t "360 -$alpha"` printf "# %5.2f %5.2f %5.7f %5.2f %5.2f " $a $b $cos_alpha $alpha $bearing # 68.70 42.40 -0.5028406 120.19 239.81 # SNIP------------------------------------------

[@JeremyWilshere]

12:27 Same as last comment but remaining in radians until the last line # Exactly the same problem. Remain in radians throughout until we output the result in degrees # Prerequisites: GNU units, linux bash shell # For 1 degree on the GCD through Points (A,B) use arithmetic mean of the polar and equatorial radii # copy pasta: # SNIP------------------------------------------ phi_A_r=0.83077672 lambda_A_r=2.13506127 phi_B_r=0.37175513 lambda_B_r=2.75517676 # PART ONE: distance (A,B) = c or d in Gary's diagram a=`units -t "(pi/2 - $phi_B_r)"` b=`units -t "(pi/2 - $phi_A_r)"` gamma_r=`units -t "$lambda_B_r - $lambda_A_r"` cos_c=`units -t "cos ($a) cos ($b) + sin ($a) sin ($b) cos ($gamma_r)"` c=`units -t "acos($cos_c)" radian` C=`units -t "$c ((earthradius_equatorial + earthradius_polar) / 2)" nmile` C_statute_mile=`units -t "$C nmile" mile` printf "# %5.7f %5.7f %5.7f %5.7f " $phi_A_r $lambda_A_r $phi_B_r $lambda_B_r printf "# %-5.2f %-5.2f %-5.2f %5.7f %5.7f %5.2f %5.2f " $a $b $c_deg $gamma_r $cos_c $C $C_statute_mile # 0.8307767 2.1350613 0.3717551 2.7551768 # 1.20 0.74 38.78 0.6201155 0.7795150 2327.30 2678.21 # PART TWO: bearing (A,B) cos_alpha=`units -t "(cos ($a) - cos($c) cos ($b)) / (sin($c) sin ($b))"` alpha=`units -t "acos( $cos_alpha)" deg` bearing=`units -t "360 -$alpha"` printf "# %-5.2f %-5.2f %5.7f %5.2f %5.2f " $a $b $cos_alpha $alpha $bearing # SNIP------------------------------------------ # 1.20 0.74 38.78 0.6201155 0.7795150 2327.30 2678.21