function XYproj = ProjectPointsOntoConic(XY,ParA)
%
%   Projecting a given set of points onto a quadratic curve (conic)
%   and computing the distances from the points to the conic

%   The conic is defined by quadratic equation 
%      Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F = 0

%   Input:  ParA = (A,B,C,D,E,F)' is the column vector of the parameters of the conic
%           XY(n,2) is the array of coordinates of n points x(i)=XY(i,1), y(i)=XY(i,2)

%   Output: XYproj is the array of coordinates of projections

%   The algorithm consistes of several steps:
%       1. Determine the type of conic and find its geometric parameters
%       2. Transform the conic to its canonical coordinate system
%       3. Find the projections  in the canonical coordinates
%       4. Transform the projected points back to the original coordinates
%       5. Adjust the coordinates of the projected points

%   Step 1 is done in the function "AtoG"
%   Steps 2,3,4 are done in the conic-specific functions 
%               "ProjectPointsOntoEllipse", "ProjectPointsOntoHyperbola", etc.
%   Step 5 is done here in this function

%        Nikolai Chernov,  February 2012

%  First, determine the type of conic (given by its code)
%  and convert its algebraic parameters ParA to its geometric parameters ParG

[ParG,code] = AtoG(ParA);   %  ParG is vector of geometric parameters

%  if code>2, then it is either imaginary or degenerate conic

if code > 3
    fprintf(1,' Imaginary or degenerate conic (code=%d)\n',code);
    return;
end

if (code==1)     %  conic is an ellipse
    XYproj = ProjectPointsOntoEllipse(XY,ParG);
end

if (code==2)     %  conic is a hyperbola
    XYproj = ProjectPointsOntoHyperbola(XY,ParG);
end

if (code==3)     %  conic is a hyperbola
    XYproj = ProjectPointsOntoParabola(XY,ParG);
end

%   Adjust the projected points on the conic
%     (this step ensures maximal accuracy)

end   %  ProjectPointsOntoConic