How many parallels of latitude have a radius of 2500km?
Assuming the earth is a perfect sphere, and with a radius of 6400 km, there are exactly two parallels with a radius of 2500 km, located at approximately
cos-1(2500/6400)
=67° N or S.
Show that the points (20°N,20°E) and (60°N,160°W) lie on the same great circle.Find their great circle distance apart.
There is at least one great circle that passes through any two non-coincident points lying on a sphere.
If the two points are diametrically located, there is an infinite number of great circles that pass through the two points.
The proof is relatively simple. The two points and the centre of the sphere form a plane that bisects the sphere into two hemispheres.
As for the distance, we are into spherical trigonometry.
There are various formulas to calculate the central angle subtended at the centre. The great circle distance is therefore the radius multiplied by this angle (in radians).
There are three formulas
1. spherical cosine formula.
This formula does not apply to small distances between the two points (say less than 10 km) because numerical round-off errors are magnified by the difference of cosine of small angles.
2. Haversine formula works well for almost all cases except for points which are, or almost, diametrically placed (which is our case in point).
3. A more complicated formula applies to all cases, and that's the Vincenty formula.
Here are the results, using R=6371 km.
(1) 11119.49 km
(2) 9623.09 km (rejected)
(3) 11119.49 km
The respective formulas are, using
R=radius of the earth (assumed a sphere)
p1=latitude of point 1 (radians)
p2=latitude of point 2 (radians)
L =difference in longitude of the two points (in radians)
D=great circle distance
(1)
R*acos(sin(p1)*sin(p2)+cos(p1)*cos(p2)*cos(L))
(2)
2*R*asin(sqrt(sin((p1-p1)/2)^2+cos(p1)*cos(p2)*sin(L/2)^2))
R*atan2(sqrt((cos(p2)*sin(L))^2+(cos(p1)*sin(p2)-sin(p1)*cos(p2)*cos(L))^2),
sin(p1)*sin(p2)+cos(p1)*cos(p2)*cos(L))
Reference:
https://en.wikipedia.org/wiki/Great-circle_distance
Well, I guess there's no latitude-ing this one! If we take the circumference of the Earth to be approximately 40,075 km, we can divide it by 2500 km to find out how many parallels of latitude have that radius. So, 40,075 km divided by 2500 km gives us about 16.03 parallels of latitude. But hey, let's not put too much pressure on those parallels, they're just trying to go with the flow!
To find how many parallels of latitude have a radius of 2500 km, we need to understand that the number of parallels of latitude is determined by how many complete circles can be drawn around the Earth at a given radius.
Here's how to calculate it step by step:
1. Recall that Earth's radius is approximately 6,371 km.
2. Divide the Earth's radius by the given radius of 2500 km: 6371 km / 2500 km = 2.5484.
Therefore, approximately 2.5484 complete circles can be drawn around the Earth at a radius of 2500 km, which means there are about 2 parallels of latitude that have a radius of 2500 km.