function [ n2, pp, thetamin, thetamax ] = sphere_imp_line_project_3d ( r, ...
center, n, p, maxpnt2 )
%*****************************************************************************80
%
%% sphere_imp_line_project_3d() projects a line onto an implicit sphere in 3D.
%
% Discussion:
%
% An implicit sphere in 3D satisfies the equation:
%
% sum ( ( P(1:3) - CENTER(1:3) )^2 ) = R^2
%
% The line to be projected is specified as a sequence of points.
% If two successive points subtend a small angle, then the second
% point is essentially dropped. If two successive points subtend
% a large angle, then intermediate points are inserted, so that
% the projected line stays closer to the sphere.
%
% Note that if any P coincides with the center of the sphere, then
% its projection is mathematically undefined. PP will
% be returned as the center.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 12 January 2021
%
% Author:
%
% John Burkardt
%
% Input:
%
% real R, the radius of the sphere. If R is
% zero, PP will be returned as the center, and if R is
% negative, points will end up diametrically opposite from where
% you would expect them for a positive R.
%
% real CENTER(3,1), the center of the sphere.
%
% integer N, the number of points on the line that is
% to be projected.
%
% real P(3,N), the coordinates of
% the points on the line that is to be projected.
%
% integer MAXPNT2, the maximum number of points on the projected
% line. Even if the routine thinks that more points are needed,
% no more than MAXPNT2 will be generated.
%
% Output:
%
% integer N2, the number of points on the projected line.
% N2 can be zero, if the line has an angular projection of less
% than THETAMIN radians.
%
% real PP(3,N2), the coordinates
% of the points representing the projected line. These points lie on the
% sphere. Successive points are separated by at least THETAMIN
% radians, and by no more than THETAMAX radians.
%
% real THETAMIN, THETAMAX, the minimum and maximum
% angular projections allowed between successive projected points.
% If two successive points on the original line have projections
% separated by more than THETAMAX radians, then intermediate points
% will be inserted, in an attempt to keep the line closer to the
% sphere. If two successive points are separated by less than
% THETAMIN radians, then the second point is dropped, and the
% line from the first point to the next point is considered.
%
%
% Check the input.
%
if ( r == 0.0 )
n2 = 0;
return
end
p1(1:3,1) = center(1:3,1);
p2(1:3,1) = center(1:3,1);
n2 = 0;
for i = 1 : n
if ( p(1:3,i) == center(1:3,1) )
else
p1(1:3,1) = p2(1:3,1);
alpha = sqrt ( sum ( ( p(1:3,i) - center(1:3,1) ).^2 ) );
p2(1:3,1) = center(1:3,1) + r * ( p(1:3,i) - center(1:3,1) ) / alpha;
%
% If we haven't gotten any points yet, take this point as our start.
%
if ( n2 == 0 )
n2 = n2 + 1;
pp(1:3,n2) = p2(1:3,1);
%
% Compute the angular projection of P1 to P2.
%
elseif ( 1 <= n2 )
dot = ( p1(1:3,1) - center(1:3,1) )' * ( p2(1:3,1) - center(1:3,1) );
ang3d = acos ( dot / ( r * r ) );
%
% If the angle is at least THETAMIN, (or it's the last point),
% then we will draw a line segment.
%
if ( thetamin < abs ( ang3d ) | i == n )
%
% Now we check to see if the line segment is too long.
%
if ( thetamax < abs ( ang3d ) )
nfill = floor ( abs ( ang3d ) / thetamax );
for j = 1 : nfill-1
pi(1:3,1) = ( ( nfill - j ) * ( p1(1:3,1) - center(1:3,1) ) ...
+ ( j ) * ( p2(1:3,1) - center(1:3,1) ) );
tnorm = sqrt ( sum ( pi(1:3,1).^2 ) );
if ( tnorm ~= 0.0 )
pi(1:3,1) = center(1:3,1) + r * pi(1:3,1) / tnorm;
n2 = n2 + 1;
pp(1:3,n2) = pi(1:3,1);
end
end
end
%
% Now tack on the projection of point 2.
%
n2 = n2 + 1;
pp(1:3,n2) = p2(1:3,1);
end
end
end
end
return
end