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Question-44497




Question Number 44497 by MrW3 last updated on 30/Sep/18
Commented by ajfour last updated on 30/Sep/18
the concept of greater circle  route has use in this, i am not  able to apply it yet..
$${the}\:{concept}\:{of}\:{greater}\:{circle} \\ $$$${route}\:{has}\:{use}\:{in}\:{this},\:{i}\:{am}\:{not} \\ $$$${able}\:{to}\:{apply}\:{it}\:{yet}.. \\ $$
Commented by MrW3 last updated on 30/Sep/18
A mountain has the shape of semi  sphere. From point A at the foot of  the mountain to point B over point A  a sightseeing path around the mountain  is to be built. Find the shortes length  of the path in terms of R and θ.
$${A}\:{mountain}\:{has}\:{the}\:{shape}\:{of}\:{semi} \\ $$$${sphere}.\:{From}\:{point}\:{A}\:{at}\:{the}\:{foot}\:{of} \\ $$$${the}\:{mountain}\:{to}\:{point}\:{B}\:{over}\:{point}\:{A} \\ $$$${a}\:{sightseeing}\:{path}\:{around}\:{the}\:{mountain} \\ $$$${is}\:{to}\:{be}\:{built}.\:{Find}\:{the}\:{shortes}\:{length} \\ $$$${of}\:{the}\:{path}\:{in}\:{terms}\:{of}\:{R}\:{and}\:\theta. \\ $$
Commented by MrW3 last updated on 30/Sep/18
Commented by MrW3 last updated on 30/Sep/18
I created this question by myself and  have no solution in mind. It is true  that the shortest distance between two  points on the sphere surface is the  path through the great circle. As the  following picture shows all blue pathes  fulfill the requirement. so the green  one is the shortes one. then we have  L_(min) =(π−θ)R
$${I}\:{created}\:{this}\:{question}\:{by}\:{myself}\:{and} \\ $$$${have}\:{no}\:{solution}\:{in}\:{mind}.\:{It}\:{is}\:{true} \\ $$$${that}\:{the}\:{shortest}\:{distance}\:{between}\:{two} \\ $$$${points}\:{on}\:{the}\:{sphere}\:{surface}\:{is}\:{the} \\ $$$${path}\:{through}\:{the}\:{great}\:{circle}.\:{As}\:{the} \\ $$$${following}\:{picture}\:{shows}\:{all}\:{blue}\:{pathes} \\ $$$${fulfill}\:{the}\:{requirement}.\:{so}\:{the}\:{green} \\ $$$${one}\:{is}\:{the}\:{shortes}\:{one}.\:{then}\:{we}\:{have} \\ $$$${L}_{{min}} =\left(\pi−\theta\right){R} \\ $$
Commented by MrW3 last updated on 30/Sep/18
Commented by MJS last updated on 01/Oct/18
the shortest distance is along the greatest  circle through A and B, all routes around the  sphere are longer, so we need another criteria    btw. a sumilar issue occurs at the former  problem with the cone−mountain with  opening angle of the cone ≥ 60°  the train cannot follow a straight line along  the lateral surface of the cone
$$\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{is}\:\mathrm{along}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{circle}\:\mathrm{through}\:{A}\:\mathrm{and}\:{B},\:\mathrm{all}\:\mathrm{routes}\:\mathrm{around}\:\mathrm{the} \\ $$$$\mathrm{sphere}\:\mathrm{are}\:\mathrm{longer},\:\mathrm{so}\:\mathrm{we}\:\mathrm{need}\:\mathrm{another}\:\mathrm{criteria} \\ $$$$ \\ $$$$\mathrm{btw}.\:\mathrm{a}\:\mathrm{sumilar}\:\mathrm{issue}\:\mathrm{occurs}\:\mathrm{at}\:\mathrm{the}\:\mathrm{former} \\ $$$$\mathrm{problem}\:\mathrm{with}\:\mathrm{the}\:\mathrm{cone}−\mathrm{mountain}\:\mathrm{with} \\ $$$$\mathrm{opening}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\geqslant\:\mathrm{60}° \\ $$$$\mathrm{the}\:\mathrm{train}\:\mathrm{cannot}\:\mathrm{follow}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{along} \\ $$$$\mathrm{the}\:\mathrm{lateral}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone} \\ $$
Commented by MrW3 last updated on 01/Oct/18
thanks for this new aspect!
$${thanks}\:{for}\:{this}\:{new}\:{aspect}! \\ $$
Commented by MJS last updated on 01/Oct/18
I tried to approximate the half sphere with  cones, and found the mentioned problem.
$$\mathrm{I}\:\mathrm{tried}\:\mathrm{to}\:\mathrm{approximate}\:\mathrm{the}\:\mathrm{half}\:\mathrm{sphere}\:\mathrm{with} \\ $$$$\mathrm{cones},\:\mathrm{and}\:\mathrm{found}\:\mathrm{the}\:\mathrm{mentioned}\:\mathrm{problem}. \\ $$

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