Line with variable width and Cdata coloring

I am able to plot a line with a separate data vector specifying the CData by plotting a surface and viewing it from the side:

h = surface('XData',[x(:) x(:)], 'YData',[y(:) y(:)], 'ZData',zeros(length(x(:)),2), 'CData',[c(:) c(:)], 'FaceColor','none', 'EdgeColor','flat', 'Marker','none');

I would now like to vary the width of the line according to the CData as well. Is there any function or suggested way to do this?

[Robin]

I am able to change the vertical width using surface as below, but this is not quite right - because vertical width is not the same as line width when the line is not horizontal.

Finding the direction of the line and expanding the surface segment perpendicular to the line seems like it will be quite difficult.

widthrange = maxwidth-minwidth;
width = ((c ./ max(c))*widthrange) + minwidth;

h = surface(...
  'XData',[x(:) x(:)],...
  'YData',[y(:)-(width/2) y(:)+(width/2)],...
  'ZData',zeros(length(x(:)),2),...
  'CData',[c(:) c(:)],...
  'FaceColor','interp',...
  'EdgeColor','interp',...
  'Marker','none');