Tuesday, 31 January 2012

Projecting 3D points to a 2D screen coordinate system

In this article I will put forward a small function to project 3D points to a 2D coordinate system. I have used VLO framework to display all the points after the calculation of different common functions. The most difficult part is to correctly project 3D points into a 2D perspective. Basically the way to display the points is by using a viewport and setting it up correctly. Once all the points are calculated, we need to apply the transformations matrix to position the viewport and then draw the 3D object projected into a 2D canvas from the viewport perspective. The following function will use rotation matrix to correctly position the points. 

Sample code:
procedure T3DMesh.CalcPoints();
function d3Dto2D (viewport: T3DViewPort; point : T3DPoint; centre : T2DPoint; zoom : double) : T2DPoint;
var
  p : T3DPoint;
  t : T3DPoint;
  d2d : T2DPoint;
begin
  p.x := viewport.x + point.x;
  p.y := viewport.y + point.y;
  p.z := viewport.z + point.z;

  t.x := (x * cos(viewport.roll)) - (z * sin(viewport.roll));
  t.z := (x * sin(viewport.roll)) + (z * cos(viewport.roll));
  t.y := (y * cos(viewport.pitch)) - (t.z * sin(viewport.pitch));
  z := (t.y * cos(viewport.pitch)) - (t.z * sin(viewport.pitch));
  x := (t.x * cos(viewport.yaw)) - (t.y * sin(viewport.yaw));
  y := (t.x * sin(viewport.yaw)) + (t.y * cos(viewport.yaw));
  d2d := nil;
  if z > 0 then
  begin
      d2d := T2DPoint.Create(((x / z) * zoom) + centre.x, ((y / z) * zoom) + centre.y);
  end;
  result := d2d;
end ;

var
  listPoint : TObjectList;
  i: Integer;
  j: Integer;
  x, y, z: Double;
  d2dpoint : T2DPoint;
begin
  listPoint := TObjectList.Create;
  listPoint2 := TObjectList.Create;

  x :=-2.0;
  y := -2.0;
  z := 0.0;
  for i := 0 to 40 do
  begin
    y := -2.0;
    for j := 0 to 40 do
    begin
      z := cos((x * x + y * y) * 2) * exp(-1 * ((x * x) + (y * y)));
      listPoint.Add(T3DPoint.Create(x,y,z));
      y := y + 0.1;
    end;
    x := x + 0.1;
  end;

  for i := 0 to listPoint.count - 1 do
  begin
    d2dpoint := d3Dto2D(viewport, T3DPoint(listPoint[i]), centre, zoom);
    if Assigned(d2dpoint) then
      listPoint2.Add(d2dpoint);
  end;
  listPoint.Free;
end;


Examples:
z = sin(x) * cos (y):

z = cos((x^2+y^2)*2) * exp-(((x^2)+(y^2))):

z = x^2*exp(-x*x)*y^2*exp(-y*y):


With parameters:

Related links:

2 comments:

  1. Is there a particular reason why you're not going down the usual approach of preparing a projection and model view matrix and using them for all the points?
    Your d3Dto2D is redoing the computations from scratch for each point, which is neither efficient not flexible.

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    Replies
    1. Hi, I'm sure you are right. Could you please provide an example with source code?. Maybe the post is not clear enough, but I had to recalc point to point as the view model is a simple canvas with (x,y) coordinates and with origin (0,0) on the top-left corner. So every point needs to be displaced and rotated. I know when I work with OpenGL that I can apply my transformatiom matrix to my whole model, but with delphi this is the way I found it works to calc a simple 3d function and then display it into a particular Tcanvas component

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