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Question Number 44497 by MrW3 last updated on 30/Sep/18

Commented by ajfour last updated on 30/Sep/18

$t h e c o n c e p t o f g r e a t e r c i r c l e$ $r o u t e h a s u s e i n t h i s, i a m n o t$ $a b l e t o a p p l y i t y e t . .$
Commented by MrW3 last updated on 30/Sep/18

$A m o u n t a i n h a s t h e s h a p e o f s e m i$ $s p h e r e . F r o m p o i n t A a t t h e f o o t o f$ $t h e m o u n t a i n t o p o i n t B o v e r p o i n t A$ $a s i g h t s e e i n g p a t h a r o u n d t h e m o u n t a i n$ $i s t o b e b u i l t . F i n d t h e s h o r t e s l e n g t h$ $o f t h e p a t h i n t e r m s o f R a n d θ .$
Commented by MrW3 last updated on 30/Sep/18

Commented by MrW3 last updated on 30/Sep/18

$I c r e a t e d t h i s q u e s t i o n b y m y s e l f a n d$ $h a v e n o s o l u t i o n i n m i n d . I t i s t r u e$ $t h a t t h e s h o r t e s t d i s t a n c e b e t w e e n t w o$ $p o i n t s o n t h e s p h e r e s u r f a c e i s t h e$ $p a t h t h r o u g h t h e g r e a t c i r c l e . A s t h e$ $f o l l o w i n g p i c t u r e s h o w s a l l b l u e p a t h e s$ $f u l f i l l t h e r e q u i r e m e n t . s o t h e g r e e n$ $o n e i s t h e s h o r t e s o n e . t h e n w e h a v e$ $L_{m i n} = (π - θ) R$
Commented by MrW3 last updated on 30/Sep/18

Commented by MJS last updated on 01/Oct/18

$the shortest distance is along the greatest$ $circle through A and B, all routes around the$ $sphere are longer, so we need another criteria$ $btw . a sumilar issue occurs at the former$ $problem with the cone - mountain with$ $opening angle of the cone ⩾ 60 °$ $the train cannot follow a straight line along$ $the lateral surface of the cone$
Commented by MrW3 last updated on 01/Oct/18

$t h a n k s f o r t h i s n e w a s p e c t!$
Commented by MJS last updated on 01/Oct/18

$I tried to approximate the half sphere with$ $cones, and found the mentioned problem .$
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