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Turing Award | A Closer Look Into Edwin Catmull’s PhD Thesis


Figure 1. Relationships between the various terms defined.

A General Algorithm for Displaying Curved Patches

The subdivision algorithm

Figure 2 (top). The grid-like dots are the sample points, while the lines represent the curved edges of the projected object. Recall that each sample point corresponds to a raster element, which is associated with an intensity value; Figure 2 (middle). The algorithm works by dividing the patch of sample points which the object covers into smaller sub-patches that cover a smaller set of sample points; Figure 2 (bottom). This process is continued until each sub-patch covers no more than a single sample point. Compute the intensity of each sub patch and write its value into the corresponding element of the frame buffer.
Figure 3. A rectangular polygon approximation of a patch.

Sub-dividing a Cubic Curve

Extending Cubic Sub-division to Surfaces

The Hidden Surface Problem

Figure 4. The hidden surface problem. Any object being looked at by some observer should only display the parts in which the observer can see. The remaining parts of the object become useless in terms of what is required to be displayed.

A high-level description of the Z-buffer algorithm

Figure 5. The observer is at the negative z-axis looking towards the positive z-axis. Objects with smaller depth are closer and are stored into the final Z-buffer, which is then displayed on the screen.

A high-level description of the modified Newell algorithm

Intensity, Sampling, Rastering, and Aliasing


Sampling, Rastering, and Aliasing

Figure 6 (top). The sample is fast enough that the reconstructed signals will have the same frequency as the original signal; Figure 6 (bottom). The sample is too slow such that the reconstructed wave will appear to have a much lower frequency than the original. This is called an aliased signal.
Figure 7. The rendered image exhibits aliasing.

Concluding Remarks



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