Patent ID: 11879985
Assignee: nan
Field: Measurement (Instruments)
Classification: CPC G  A | IPC A  G

Claim 14:
15. A method for determining center coordinates of a circular arc on a spherical surface, comprising:
receiving position coordinates for a plurality of points along the circular arc; and
processing said position coordinates for said plurality of points to determine the center coordinates of the circular arc in a three-dimensional coordinate system without converting the position coordinates into two-dimensional coordinates;
wherein said position coordinates are processed by solving a geometric relationship for a cone inscribed in a sphere, with the sphere representing the Earth, the position coordinates defining points along a base perimeter of the cone, and an apex of the cone being located at the center of the Earth, and wherein said position coordinates are processed using a regression model that relates an angle Ω between a slant height and center height of the cone to the center coordinates and the position coordinates for points along the base perimeter of the cone; and
wherein said position coordinates comprise GNSS latitude and longitude coordinates, the latitude of said GNSS coordinates are converted to spherical coordinates by adding 90° and sign inversion, and said geometric relationship is represented by equation:

cos Ω=sin θb cos φb sin θc cos φc+sin θb sin φb sin θc sin φc+cos θb cos θc 

where Ω is the angle between the slant height and center height of the cone, θb is the updated latitude of a point along the base perimeter of the cone, φb is the longitude of the point along the base perimeter of the cone, θc is the updated latitude of the point located in the center of the base of the cone, and φc is the longitude of the point located in the center of the base of the cone, and
wherein said equation is algebraically manipulated into linearized equation:, cos
   ⁢
   
    θ
    b
   
  
  =
  
   
    
     -
     tan
    
    ⁢
    
     θ
     c
    
    ⁢
    cos
    ⁢
    
     φ
     c
    
    ⁢
    sin
    ⁢
    
     θ
     b
    
    ⁢
    cos
    ⁢
    
     φ
     b
    
   
   -
   
    tan
    ⁢
    
     θ
     c
    
    ⁢
    sin
    ⁢
    
     φ
     c
    
    ⁢
    sin
    ⁢
    
     θ
     b
    
    ⁢
    sin
    ⁢
    
     φ
     b
    
   
   +
   
    
     cos
     ⁢
     Ω
    
    
     cos
     ⁢
     
      θ
      c
     
    
   
  
 

to be used with a plurality of θb, φb pairs in linear regression to produce the center coordinates θc, φc.