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A short discussion on mapping 3D objects to a 2D display

, 21 Apr 2002 119.1K 3.7K 87
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This article describes, in simple terms, how to translate three dimensional co-ordinates into the two dimensional co-ordinates to be displayed on a screen
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A short discussion on mapping 3D objects onto a 2D display.

This article describes, in simple terms, how to translate three dimensional co-ordinates into the two dimensional co-ordinates to be displayed on a screen.

It is intended to be an introduction to the basics of rendering 3D images and does not cover topics like rotation or movement.

A few basic definitions and assumptions

The co-ordinate system that we will be working in is the standard Cartesian co-ordinate system as depicted in figure 1.

fig 1: The co-ordinate system

where:    o is the origin
             +x is to the right (-x is to the left)
             +y is up (-y is down)
             +z is forwards (-z is the opposite direction)

A point within this 3D space can have it's location represented as P(x, y, z) representing it's position relative to our origin defined above.  A polygon will simply be a collection of such points joined by lines.

Next, we also need to define where we are looking from (the viewpoint). This will be the position of the camera E.  In order to simplify the explanation a little, the camera will be placed at the origin (0, 0, 0). Later in this article, we will move the camera away from the origin.  Another simplification can be made by assuming that the camera is always looking along the Z axis towards +Z.  The maths is more complicated if this is not so, and I won't discuss that here.

Finally, we need somewhere to project the image onto, screen S. This will always be in a plane that is at right angles to the direction we are viewing and at a distance of Sz from the origin.  This screen can be assumed to extend as far as required. Figure 2 show the current scenario:

fig 2: The setup

Seeing the point

If we view the scene from along the X axis, we get a 2D cross section of the YZ plane (looking towards the origin from the positive side).  Now we draw an imaginary line between the camera and the point we want to show on the screen, we can see that line will intersect the screen at a point S. Figure 3 illustrates this.

fig 3: Intersection with screen

We already know the z co-ordinate for our screen point (Sz, same as the screen) we need to work out it's y value.

Basic Trigonometry can give us the answer here. Using the rules for similar triangles, we can see that the ratio of Sz over Pz is the same as the ratio of Sy over Py; Written as an equation, we have:-

    Sz = Sy

    Pz   Py

and re-arranging the formula gives the following equation for Sy;

    Sy = Py Sz

          Pz

A similar process will result in a similar equation for the X co-ordinate.

    Sx = Px Sz

          Pz

You can see that if we have another point P2, as shown in figure 4, which has the same X and Y values but a different Z value, you can see that it intersects the screen at a different point. This mimics the perspective effect we notice in real life. Objects further away appear to be smaller than those that are nearer.

fig 4: Adding another point with a new Z value

Moving the camera is away from the origin?

The formula given above are valid only when the co-ordinates are presented relative to the the origin (ie (0, 0, 0). If the camera is not at the origin, we need to adjust the formulae to account for this. Figure 5 shows the camera moved along the Y axis and the Z axis.

fig 5: The viewpoint is offset from the origin

If we go back to the first equations and adjust them for the offset, we have:

    (Sz-Ez) = (Sy-Ey)

    (Pz-Ez)   (Py-Ey)

and re-arranging the formula gives us

    (Sy-Ey) = (Py-Ey)*(Sz-Ez)

                 
(Pz-Ez)

so, we end up with

    Sy = (Py-Ey)*( Sz-Ez) + Ey

             ( Pz-Ez)

Similarly doing a similar transformation for Sx,

    Sx = (Px-Ex)*( Sz-Ez) + Ex

             ( Pz-Ez)

About the program

With this being Code Project, I suppose it's useful to supply some code. The demonstration program is something I wrote to show the effect of changing some of the parameters that affect the rendering of the image. I have used Chris Maunder's 3D classes, but most of the functionality in those classes is not used.

Using the program

When you run the program, you are presented with three 'wire frame' houses which are in fact all the same size, but placed at different Z co-ordinates. You can use the toolbar buttons or the keyboard to move the camera around.

The following keys do things in the program:

K/L: Move the position of the screen along the 'Z' axis

A/D: Move the position of the camera along the 'X' axis

W/X: Move the position of the camera along the 'Y' axis

R/T: Move the position of the camera along the 'Z' axis

Pressing the toolbar buttons will move the camera in the direction indicated by the button icon (if you can decode my artistic efforts)

License

This article has no explicit license attached to it but may contain usage terms in the article text or the download files themselves. If in doubt please contact the author via the discussion board below.

A list of licenses authors might use can be found here

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About the Author

Paresh Solanki
Software Developer (Senior)
United Kingdom United Kingdom
No Biography provided

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Comments and Discussions

 
GeneralI don't understand the screen's &quot;z&quot; Pin
Michael Chourdakis6-Aug-07 2:34
memberMichael Chourdakis6-Aug-07 2:34 
GeneralRe: I don't understand the screen's &quot;z&quot; Pin
Paresh Solanki6-Aug-07 3:34
memberParesh Solanki6-Aug-07 3:34 
GeneralRe: I don't understand the screen's &quot;z&quot; Pin
Krindaaru7-Sep-07 20:43
memberKrindaaru7-Sep-07 20:43 
GeneralRe: I don't understand the screen's &quot;z&quot; Pin
Paresh Solanki14-Sep-07 7:28
memberParesh Solanki14-Sep-07 7:28 
GeneralRotation Pin
Georgi Petrov3-Jan-05 1:47
memberGeorgi Petrov3-Jan-05 1:47 
Questionwhat about points... Pin
Roman Nurik29-Jul-02 20:44
memberRoman Nurik29-Jul-02 20:44 
AnswerRe: what about points... Pin
Paresh Solanki29-Jul-02 23:21
memberParesh Solanki29-Jul-02 23:21 
GeneralRe: what about points... Pin
Roman Nurik30-Jul-02 7:00
memberRoman Nurik30-Jul-02 7:00 
GeneralRe: what about points... Pin
Paresh Solanki30-Jul-02 23:27
memberParesh Solanki30-Jul-02 23:27 
GeneralRe: what about points... Pin
Roman Nurik31-Jul-02 7:38
memberRoman Nurik31-Jul-02 7:38 
GeneralRe: what about points... Pin
Roman Nurik31-Jul-02 7:38
memberRoman Nurik31-Jul-02 7:38 
GeneralRe: what about points... Pin
Paresh Solanki6-Aug-02 23:05
memberParesh Solanki6-Aug-02 23:05 
GeneralRe: what about points... Pin
Roman Nurik7-Aug-02 7:10
memberRoman Nurik7-Aug-02 7:10 
GeneralRe: what about points... Pin
Paresh Solanki8-Aug-02 0:16
memberParesh Solanki8-Aug-02 0:16 
GeneralGood start Pin
Igor29-Apr-02 20:54
memberIgor29-Apr-02 20:54 
GeneralRe: Good start Pin
WREY23-Sep-02 0:05
memberWREY23-Sep-02 0:05 
GeneralBook Recomendation Pin
Michael A. Barnhart22-Apr-02 15:22
memberMichael A. Barnhart22-Apr-02 15:22 

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