The Slant Range from Subtended Anglecalculator computes the slant range (R) from a ground station to an object (satellite or aircraft) based on the station and object altitudes and the subtended angle (β). 
INSTRUCTIONS: Choose units and enter the following:
- (sA) Station Altitude above Mean Earth Radius
- (β) Subtended Angle (aka Look Angle)
- (oA) Object Altitude above Mean Earth Radius
Slant Range (R): The range is returned in kilometers. However this can be automatically converted to compatible units via the pull-down menu.
The Math / Science
The formula for the slant range uses both the law of sines and cosines. In the obtuse triangle below:
- b = Re + sA
- a = Re + oA
- c is the slant range
- CA is the Earth Central Angle
- BA is the subtended angle (β)
- AA is elevation angle (α) + 90°
where:
- asin(AA)=bsin(BA)=csin(CA)
- CA = π - BA - AA
- c2=a2+b2-2a⋅b⋅cos(CA)
- Therefore: c=√a2+b2-2a⋅b⋅cos(CA)
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- XYZ to Latitude, Longitude, Altitude
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- Earth WGS-84 Polar radius in meters: 6356752.31424518 m
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k⋅V - scalar multiplication- V/k - scalar division
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- |V| - Computes the magnitude of a vector
- U + V - Vector addition
- U - V - Vector subtraction
- |U - V| - Distance between vector endpoints.
- |U + V| - Magnitude of vector sum.
- V • U - Computes the dot product of two vectors
- V x U - Computes the cross product of two vectors
- V x U • W - Computes the mixed product of three vectors
- Vector Angle - Computes the angle between two vectors
- Vector Area - Computes the area between two vectors
- Vector Projection - Compute the vector projection of V onto U.
- Vector Rotation - Compute the result vector after rotating around an axis.
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- (ρ, θ, φ) to (x,y,z) - Spherical to Cartesian coordinates
- (x,y,z) to (ρ, θ, φ) - Cartesian to Spherical coordinates
- (r, θ, z) to (x,y,z) - Cylindrical to Cartesian coordinates
- (x,y,z) to (r, θ, z) - Cartesian to Cylindrical coordinates
- (x,y) to (r, θ) - Cartesian to Polar
- (r, θ) to (x,y) - Polar to Cartesian
- Vector Normal to a Plane Defined by Three Points
