Computer generated imagery is a pretty cool thing, its like the virtual world and real world came together.
First off, how does math and science tie into cgi? Well, computers are just really strong calculators, so finding a way to calculate shapes let alone 3d objects with math is extremely difficult. Coding is definitely its own type of math, giving directions to the computer on how to control lighting and color that is accurate to what is seen in life is a challenge.
There are two main ways of rendering 3D, Projection Rasterization and Ray Casting. Projection rasterization works by sending rays from an object's vertices to the camera, then filling in the shape it creates. This is still used in modern gaming as it is very fast. Ray Casting works by sending rays from the camera and detecting when they intersect with an object, then goes towards the light source seeing for another intersection to detect shadows.
What are the issues with Rasterization and Ray Casting? First off, rasterization creates limited results. One of the issues is that whatever is drawn last is always on top, so even though an object may be behind an object. This problem is overcome with depth maps, drawing based upon distance from camera. Ray Casting's issue is that every ray must be checked against every object. This makes renders take an extremely long time, but creates lifelike results.
The difference between Direct and Indirect lighting, is that they are two different sources of light. Direct lighting is directly from a light source for example a light bulb! Indirect lighting is light that is bounced off an object onto another object, so an orange may have a blue tone to its edge when its next to a blue wall.
Indirect Illumination's issue is that it makes even bigger calculations. This is a challenge in Cgi as you need to create even more rays to detect if another object is bouncing light upon it. So every object you add to a scene needs to be checked against every other object in the scene.
How does Moore's Law and Blinn's Law relate? Moore's law is that the number of transistors on integrated circuits doubles every year. This relates saying that computers become more powerful every year, this means that rendering times should become faster, and easier on a computer's hardware. Blinn's law states that as technology improves render times remain constant. This is due to the fact as the stronger a computer becomes the more we throw at it.
Is the art of CGI different from other art forms? I say Yes and No. It is similar to other art forms in the way we try to mimic realism with it. In all art forms we try to create our art in the way we observe our surroundings. It differs from other art forms as we use a computer to do the manual labor, of the recreation. In no way am I discrediting CGI as an art form, and anyone who does is quite honestly ignorant to the thought that goes into its usage. It still is all down to the artist to decide how they are going to use CGI in what they create.
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