Roll Any Point on the Sphere to Any Desired Latitude-Longitude Coordinates with One Straight-Line Roll

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Pick a point on a sphere (in green). This point has certain latitude-longitude coordinates. In one straight-line roll, we can move this point to any desired latitude-longitude coordinates (in red). This Demonstration shows the shortest such roll.
Contributed by: Aaron Becker (October 2012)
Open content licensed under CC BY-NC-SA
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Details
A roll in the horizontal plane of length about an axis at angle
with the
axis produces the following change in orientation:
.
We want to choose the pair that moves a point from the starting latitude-longitude pair to the desired latitude-longitude pair.
First, we compute the 3D location of the point from the latitude-longitude pair :
.
Let and
be the start and end coordinates.
The angle to roll along is given by , which is undefined if the
coordinates overlap. If they overlap and
any angle works. Otherwise, set
.
The length of the roll is determined by the angle between the start and ending latitude-longitude pair with respect to the intersection of the line in the
-
plane between the start and end points and a perpendicular line to the origin. If
is the squared
-
distance between the start and end points, this intersection point is given by
.
If , the intersection point is the origin.
If and
are the vectors to the starting and ending north poles from this intersection point, then we can use a property of the dot product to calculate the angle
, which satisfies
.
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