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Arc length is the distance between two ... Since it is straightforward to calculate the length of each ... The last equality is proved by the following steps:
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown.. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them.
2 Arc length and curvature. 3 Characteristics. 4 General Archimedean spiral. 5 Applications. 6 Construction methods. ... results in the Cartesian equation + = ...
It approximates the arc length, , to the tunnel distance, , or omits the conversion between arc and chord lengths shown below. The shortest distance between two points in plane is a Cartesian straight line. The Pythagorean theorem is used to calculate the distance between points in a plane.
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
In this equation, the origin is the midpoint of the horizontal range of the projectile, and if the ground is flat, the parabolic arc is plotted in the range . This expression can be obtained by transforming the Cartesian equation as stated above by y = r sin ϕ {\displaystyle y=r\sin \phi } and x = r cos ϕ {\displaystyle x=r\cos \phi } .
This is a 2-d problem in span{^, ^}, which will be solved with the help of the arc length formula above. If the arc length, s 12 {\displaystyle s_{12}} is given then the problem is to find the corresponding change in the central angle θ 12 {\displaystyle \theta _{12}} , so that θ 2 = θ 1 + θ 12 {\displaystyle \theta _{2}=\theta _{1}+\theta ...
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation , it is a special case of a more general formula in spherical trigonometry , the law of haversines , that relates the sides and angles of spherical triangles.