ax-2-by-2-cz-2-r-2-Point-P-a-b-c-Point-Q-l-m-n-Both-points-lie-on-the-curve-what-is-the-shortest-path-from-point-P-to-Q-along-the-outside-of-the-curve-

Question Number 11309 by FilupS last updated on 20/Mar/17

{a x}^{2} + {b y}^{2} + {c z}^{2} = r^{2}

Point P = (a, b, c)

Point Q = (l, m, n)

Both points lie on the curve

what is the shortest path from point

P to Q, along the outside of the curve ?

Commented by mrW1 last updated on 20/Mar/17

T h i s i s a v e r y d i f f i c u l t q u e s t i o n . I f

a, b, c > 0, {a x}^{2} + {b y}^{2} + {c z}^{2} = r^{2} i s a t r i a x i a l

e l l i p s o i d . Y o u r q u e s t i o n i s t h e n a b o u t

g e o d e s i c s o n a t r i a x i a l e l l i p s o i d .

Commented by FilupS last updated on 21/Mar/17

what if a = b = c = 1 ?

Commented by mrW1 last updated on 21/Mar/17

T h e n i t i s a s p h e r e w i t h r a d i u s r . T h e s h o r t e s t p a t h

o n a s p h e r e i s t h e g r e a t - c i r c l e d i s t a n c e .

P o i n t P (x_{1}, y_{1}, z_{1})

P o i n t Q (x_{2}, y_{2}, z_{2})

V e c t o r p = \vec{O P}, q = \vec{O Q}

A n g l e b e t w e e n p a n d q = Δ α

∣ p ∣= \sqrt{x_{1}^{2} + y_{1}^{2} + z_{1}^{2}} = r

∣ q ∣= \sqrt{x_{2}^{2} + y_{2}^{2} + z_{2}^{2}} = r

\cos Δ α = \frac{p \cdot q}{∣ p ∣ \times ∣ q ∣} = \frac{x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2}}{r^{2}}

Δ α = \cos^{- 1} (\frac{x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2}}{r^{2}})

T h e s h o r t e s t p a t h f r o m P t o Q i s

d = Δ α \times r

Commented by FilupS last updated on 21/Mar/17

fascinating!

fascinating!

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