Simple web graphics program

Topology refers to how points which we generally call vertices are connected to each other to form faces or flat surfaces. These faces are also called polygons. The box would be made of six faces or six polygons and the set of polygons forms what we call a polygonal mesh or simply a mesh. The second thing we are missing, is a system to create an image of that box. This requires to actually project the corners of the box onto an imaginary canvas, a process we call perspective projection.

Figure 4: if you connect the corners of the canvas to the eye which by default is aligned with our Cartesian coordinate system, and extend the lines further away into the scene, you get some sort of pyramid which we call a viewing frustum.

Any object contained within the frustum or overlapping it is visible and will show up on the image. The process of projecting 3D point of the surface of the canvas, actually involves a special matrix called the perspective matrix don't worry if you don't know what a matrix is.

Using this matrix to project point is not absolutely necessary but makes things much easier. However, you don't really need mathematics and matrices to figure out how it works. You can see an image, or a canvas as some sort of flat surface placed some distance away from the eye. Trace four lines all starting from the eye to each one of the four corners of the canvas and extend these lines further away into the world as far as you can see. You get a pyramid which we call a viewing frustum and not frustrum.

The viewing frustum defines some sort of volume in 3D space and the canvas itself it just a plane cutting of this volume perpendicular to the eye line of sight. Place your box in front of the canvas. Next, trace a line from each corner of the box to the eye and mark a dot where the line intersects the canvas. Find out on the canvas, the dots corresponding to each of the twelve edges of the box, and trace a line between these dots.

What do you see? An image of the box. Figure 5: the box is move in front of our camera setup. The coordinates of the box corners are expressed with respect to this Cartesian coordinate system. Figure 7: the intersection points between these lines and the canvas are the projection of the box corners onto the canvas. By connecting these points to each other, an wireframe image of the box is created. The three rulers used to measure the coordinates of the box corner form what we call a coordinate system.

It's a system in which points can be measured to. All points' coordinates relate to this coordinate system. Note that a coordinate can either be positive or negative or zero depending on whether it's located on the right or the left of the ruler's origin the value 0. In CG, this coordinate system is often called the world coordinate system, and the point 0,0,0 , the origin.

Let's move the apex of the viewing frustum at the origin and orient the line of sight the view direction along the negative z-axis figure 3. Many graphics applications use this configuration as their default "viewing system". Keep in mind that the top of the pyramid is actually the point from which we will be looking at the scene. Let's also move the canvas one unit away from the origin. Finally, let's move the box some distance away from the origin, so that it is fully contained within the volume of the frustum.

Because the box is in a new position we moved it , the coordinates of its eight corners changed and we need to measure them again. Note that because the box is on left side of the ruler's origin from which we measure the object's depth, all depth coordinates which also called z-coordinates will be negative. Four of the corners are also below the reference point used to measure the object's height, and will have a negative height or y-coordinate.

Finally, four of the corners will be to the left of the ruler's origin measuring the object's width: their width, or x-coordinates will also be negative. The new coordinates of the box's corners are:. Figure 8: the coordinates of the point P', the projection of P on the canvas can be computed using simple geometry. Let's look at our setup from the side and trace a line from one of the corners to the origin the viewpoint. As you can see these two triangles have the same origin A.

They are also somehow copies of each other, in the sense that the angle defined by the edges AB and AC is the same as the angle defined by the edge AB', AC'. Such triangles are said to be similar. Similar triangles have an interesting property: the ratio between their adjacent and opposite sides is the same. In other words:. Because the canvas is 1 unit away from the origin, we know that AB' equals 1. We also know the position of B and C which are the z depth and y coordinates height respectively of the corner.

If we substitute these numbers in the above equation, we get:. Where y' is actually the y coordinate of the point where the line going from the corner to the viewpoint intersects the canvas, which is, was we said earlier, the dot from which we can draw an image of the box on the canvas. As you can see, the projection of the corner's y-coordinate on the canvas, is nothing more than the corner's y-coordinate divided by its depth the z-coordinate.

This is probably one the simplest and most fundamental relation in computer graphics, known as the z or perspective divide. The exact same principle applies to the x coordinate. The projected point x coordinate x' is the corner's x coordinate divided by its z coordinate. Note though that because the z-coordinate of P is negative in our example we will explain why this is always the case in the lesson from the Foundations of 3D Rendering section dedicated to the perspective projection matrix , when the x-coordinate is positive, the projected point's x-coordinate will become negative similarly, if P.

The same problem happens with the y-coordinate. As a result the image of the 3D object is mirrored both vertically and horizontally, which is not an effect we want. Thus, to avoid this problem, we will divide the P. We finally get:. We now have a method to compute the actual positions of the corners as they appear on the surface of the canvas. These are the two-dimensional coordinates of the points projected on the canvas.

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