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Computer Graphics

Object space method compares objects and parts of objects to each other in order to decide the visibility.
The different types of object space methods are back face and Painter’s algorithm.
Image space method decides visibility point by point at each pixel position on the view plane. The
different types of image space methods are area subdivision, Z-buffer, and scan line.

Did you know? Image space method requires more work than object space method. Therefore, most
algorithms should theoretically be applied in object space. This is not the case in
general. Image space methods are considered to be more efficient because it is easier to
take advantage of consistency in raster scan implementation of an image space
method.

11.1 Z-Buffer

Z-buffer is one of the simplest and commonly used image space approaches to eliminate hidden
surfaces. This is referred to as the z-buffer, since depth of an object is mainly calculated from the view
plane along the z axis of a coordinate system.
A 3-D shape is a collection of 3-D surfaces or planar surface patches. A planar equation can be used to
estimate each of these surfaces. The planar equation helps to evaluate the Z co-ordinate value of each of
the planar surface points in world co-ordinates.

Once you have defined the Z buffer, Direct3-D will automatically stop rendering
triangles that are hidden by other triangles. The planar equation helps to evaluate the Z
co-ordinate value of each of the planar surface points in world co-ordinates.

When 3-D shapes are projected onto the 2-D computer screen, the position of each pixel on the view
plane is represented by x and y coordinates, whereas the z-value holds the information about depth. For
every (X, Y) screen location, there is one or more 3-D surface points. The different Z co-ordinate values
of these 3-D surface points are defined by the planar equations of the surfaces. These Z values are then
compared to determine the visible surface points. A surface is considered visible if it has the minimum
Z value. It is important to know that in the corresponding (X, Y) location of the screen, the color of the
visible surface point is painted.

Did you know? The Z-buffer algorithm was invented by E. Catmull around the 1970s. It is very simple
and can handle just about any 3-D primitive. Since then, it has become the most
preferred choice for hardware accelerated 3-D boards.

The equation of a planar surface is given by, Ax + By + Cz + D = 0, where vectors (A, B, C) are normal to
the plane and D is constant.
As mentioned earlier, Z-buffer algorithm operates on planar surfaces of a 3-D shape in order to
determine visible surfaces. Therefore, the Z value of the plane at location (X, Y) is given by:
Z = - Ax – By – D/ C …. (Eq.11.1)
Consider a 3-D viewpoint looking at a planar surface, where the Z value of the surface gives the depth
of the surface from the viewpoint. Therefore, this algorithm is also known as depth buffer algorithm.
For planes with known equations, the visibility of planes can be determined by comparing Z values of
these planar points.
Let us assume that Z values are written in a 2-D matrix of the same size as a computer screen and let us
henceforth refer to it as Z-buffer. Initially, the Z-buffer values for all the elements of the entire 2-D
matrix are set to a maximum possible number. Also, initialize a 2-D screen matrix of same size as Z-

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