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## -*- texinfo -*-
## @deftypefn  {} {} surfnorm (@var{x}, @var{y}, @var{z})
## @deftypefnx {} {} surfnorm (@var{z})
## @deftypefnx {} {} surfnorm (@dots{}, @var{prop}, @var{val}, @dots{})
## @deftypefnx {} {} surfnorm (@var{hax}, @dots{})
## @deftypefnx {} {[@var{Nx}, @var{Ny}, @var{Nz}] =} surfnorm (@dots{})
## Find the vectors normal to a meshgridded surface.
##
## If @var{x} and @var{y} are vectors, then a typical vertex is
## (@var{x}(j), @var{y}(i), @var{z}(i,j)).  Thus, columns of @var{z} correspond
## to different @var{x} values and rows of @var{z} correspond to different
## @var{y} values.  If only a single input @var{z} is given then @var{x} is
## taken to be @code{1:columns (@var{z})} and @var{y} is
## @code{1:rows (@var{z})}.
##
## If no return arguments are requested, a surface plot with the normal
## vectors to the surface is plotted.
##
## Any property/value input pairs are assigned to the surface object.  The full
## list of properties is documented at @ref{Surface Properties}.
##
## If the first argument @var{hax} is an axes handle, then plot into this axes,
## rather than the current axes returned by @code{gca}.
##
## If output arguments are requested then the components of the normal
## vectors are returned in @var{Nx}, @var{Ny}, and @var{Nz} and no plot is
## made.  The normal vectors are unnormalized (magnitude != 1).  To normalize,
## use
##
## @example
## @group
## len = sqrt (nx.^2 + ny.^2 + nz.^2);
## nx ./= len;  ny ./= len;  nz ./= len;
## @end group
## @end example
##
## An example of the use of @code{surfnorm} is
##
## @example
## surfnorm (peaks (25));
## @end example
##
## Algorithm: The normal vectors are calculated by taking the cross product
## of the diagonals of each of the quadrilateral faces in the meshgrid to find
## the normal vectors at the center of each face.  Next, for each meshgrid
## point the four nearest normal vectors are averaged to obtain the final
## normal to the surface at the meshgrid point.
##
## For surface objects, the @qcode{"VertexNormals"} property contains
## equivalent information, except possibly near the boundary of the surface
## where different interpolation schemes may yield slightly different values.
##
## @seealso{isonormals, quiver3, surf, meshgrid}
## @end deftypefn

function [Nx, Ny, Nz] = surfnorm (varargin)

  [hax, varargin, nargin] = __plt_get_axis_arg__ ("surfnorm", varargin{:});

  if (nargin == 0 || nargin == 2)
    print_usage ();
  endif

  if (nargin == 1)
    z = varargin{1};
    [x, y] = meshgrid (1:columns (z), 1:rows (z));
    ioff = 2;
  else
    x = varargin{1};
    y = varargin{2};
    z = varargin{3};
    ioff = 4;
  endif

  if (iscomplex (z) || iscomplex (x) || iscomplex (y))
    error ("surfnorm: X, Y, and Z must be 2-D real matrices");
  endif
  if (! size_equal (x, y, z))
    error ("surfnorm: X, Y, and Z must have the same dimensions");
  endif

  ## FIXME: Matlab uses a bicubic interpolation, not linear, along the boundary.
  ## Do a linear extrapolation for mesh points on the boundary so that the mesh
  ## is increased by 1 on each side.  This allows each original meshgrid point
  ## to be surrounded by four quadrilaterals and the same calculation can be
  ## used for interior and boundary points.  The extrapolation works badly for
  ## closed surfaces like spheres.
  xx = [2 * x(:,1) - x(:,2), x, 2 * x(:,end) - x(:,end-1)];
  xx = [2 * xx(1,:) - xx(2,:); xx; 2 * xx(end,:) - xx(end-1,:)];
  yy = [2 * y(:,1) - y(:,2), y, 2 * y(:,end) - y(:,end-1)];
  yy = [2 * yy(1,:) - yy(2,:); yy; 2 * yy(end,:) - yy(end-1,:)];
  zz = [2 * z(:,1) - z(:,2), z, 2 * z(:,end) - z(:,end-1)];
  zz = [2 * zz(1,:) - zz(2,:); zz; 2 * zz(end,:) - zz(end-1,:)];

  u.x = xx(1:end-1,1:end-1) - xx(2:end,2:end);
  u.y = yy(1:end-1,1:end-1) - yy(2:end,2:end);
  u.z = zz(1:end-1,1:end-1) - zz(2:end,2:end);
  v.x = xx(1:end-1,2:end) - xx(2:end,1:end-1);
  v.y = yy(1:end-1,2:end) - yy(2:end,1:end-1);
  v.z = zz(1:end-1,2:end) - zz(2:end,1:end-1);

  c = cross ([u.x(:), u.y(:), u.z(:)], [v.x(:), v.y(:), v.z(:)]);
  w.x = reshape (c(:,1), size (u.x));
  w.y = reshape (c(:,2), size (u.y));
  w.z = reshape (c(:,3), size (u.z));

  ## Create normal vectors as mesh vectices from normals at mesh centers
  nx = (w.x(1:end-1,1:end-1) + w.x(1:end-1,2:end) +
        w.x(2:end,1:end-1) + w.x(2:end,2:end)) / 4;
  ny = (w.y(1:end-1,1:end-1) + w.y(1:end-1,2:end) +
        w.y(2:end,1:end-1) + w.y(2:end,2:end)) / 4;
  nz = (w.z(1:end-1,1:end-1) + w.z(1:end-1,2:end) +
        w.z(2:end,1:end-1) + w.z(2:end,2:end)) / 4;

  if (nargout == 0)
    hax = newplot (hax);

    surf (hax, x, y, z, varargin{ioff:end});
    old_hold_state = get (hax, "nextplot");
    unwind_protect
      set (hax, "nextplot", "add");

      ## Normalize the normal vectors
      nmag = sqrt (nx.^2 + ny.^2 + nz.^2);

      ## And correct for the aspect ratio of the display
      daratio = daspect (hax);
      damag = sqrt (sumsq (daratio));

      ## FIXME: May also want to normalize the vectors relative to the size
      ##        of the diagonal.

      nx ./= nmag / (daratio(1)^2 / damag);
      ny ./= nmag / (daratio(2)^2 / damag);
      nz ./= nmag / (daratio(3)^2 / damag);

      plot3 (hax, [x(:).'; x(:).' + nx(:).' ; NaN(size(x(:).'))](:),
                  [y(:).'; y(:).' + ny(:).' ; NaN(size(y(:).'))](:),
                  [z(:).'; z(:).' + nz(:).' ; NaN(size(z(:).'))](:),
                  "r");

    unwind_protect_cleanup
      set (hax, "nextplot", old_hold_state);
    end_unwind_protect

  else
    Nx = nx;
    Ny = ny;
    Nz = nz;
  endif

endfunction


%!demo
%! clf;
%! colormap ("default");
%! surfnorm (peaks (19));
%! shading faceted;
%! title ({"surfnorm() shows surface and normals at each vertex", ...
%!         "peaks() function with 19 faces"});

%!demo
%! clf;
%! colormap ("default");
%! [x, y, z] = sombrero (10);
%! surfnorm (x, y, z);
%! title ({"surfnorm() shows surface and normals at each vertex", ...
%!         "sombrero() function with 10 faces"});

## Test input validation
%!error <Invalid call> surfnorm ()
%!error <Invalid call> surfnorm (1,2)
%!error <X, Y, and Z must be 2-D real matrices> surfnorm (i)
%!error <X, Y, and Z must be 2-D real matrices> surfnorm (i, 1, 1)
%!error <X, Y, and Z must be 2-D real matrices> surfnorm (1, i, 1)
%!error <X, Y, and Z must be 2-D real matrices> surfnorm (1, 1, i)
%!error <X, Y, and Z must have the same dimensions> surfnorm ([1 2], 1, 1)
%!error <X, Y, and Z must have the same dimensions> surfnorm (1, [1 2], 1)
%!error <X, Y, and Z must have the same dimensions> surfnorm (1, 1, [1 2])
