A Basic Guide to How 3D Graphics are Rendered

One of the greatest developments of computer generated imagery came via the discovery that 3D environments could be computationally rendered unto a 2D screen. This concept has revolutionized the entertainment industry leading to the creation of classic animated films such as Pixar’s Toy Story to block buster videogames. The first videogame that I ever played was The Legend of Zelda: Ocarina of Time and it is probably the single largest motivation for me deciding to major in computer science. I was initially amazed at how the game could create such a complex and beautiful 3D world. As I got older, I became more curious about how computers which process 1s and 0s could manipulate information to create this 3D world. Below is a very basic overview of how this occurs.

3D Cartesian Graphs

All 3D models are represented through a series of points represented in a 3D Cartesian space. These points are then linked together to form triangles (or polygons) which make up the faces of the 3D model. Polygons can be added together to mathematically represent almost any object in 3D space and the more polygons a model has, the more detailed it will appear when it is finally rendered (at the cost of a longer render time). Most AAA games process thousands of polygons all at once; the creators of Final Fantasy XV estimate that each character will have about 100,000 polygons each.

3D points are rendered with respect to the origin and this presents a problem with mapping models in 3D space. When multiple models need to be rendered, each one’s points will be rendered with respect to the origin and they will often overlap. To manipulate the location and orientation of where an object is rendered, the points that make up the model need to undergo transformations, specifically translations and rotations.

Translations involve shifting a model’s points from one location to another while still retaining their relative distance and orientation. This can be done simply taking the x, y, and z values of each point needing to be translated and adding (or subtracting) the distance the point will travel to its new location. For example, if a point A at (1, 2, 3) needs to translate to point B at (4, 5, 6), then a value of 3 will need to be added to all three directions. Another way to translate points is to use the translation matrix operation. The MATLAB code to translate a 3D point using the first method would be…

							% Translation can occur using simple addition...
							A = [1 2 3];
							DP = [3 3 3];
							B = A + DP;

							% or using a Translation Matrix Option
							A = [1; 2; 3; 1];
							translation_matrix = [
								1 0 0 3;
								0 1 0 3;
								0 0 1 3;
								0 0 0 1;
								];
							B = translation_matrix * A;
							

Without going into too much detail, rotations involve shifting a model’s orientation around a certain axis. This can be done by multiplying the points of the model by a rotation matrix specific to the axis that is needs to be rotated around. For instance, if I wanted to rotate a 3D model around the z axis, then I would use the following MATLAB code…

					% Rotation occurs by performing a rotation 
					% matrix operation.  should be 
					% a matrix of a all the points in a model but 
					% is just a bunch of random points in this 
					% case for demonstration purposes.
					model_points = rand(3, 100);
					angle = pi/2;
					Rz = [
						cos(angle) -sin(angle) 0; 
						sin(angle) cos(angle) 0;
						0 0 1;
						];
					rotated_model = Rz * model_points;
				

Rendering the Data

Even with all the polygons in place, the computer still needs a way to turn all the abstract data into a 2D image. If we think of the computer screen as a camera peering into the world that the computer is rendering, then the first step is to determine where the camera is placed. In most 3D modeling software, the camera is permanently placed at the origin. When the camera wants to “move” it actually stays in place while the rest of the world is translated in order to give the camera a new view. This is done because it is actually mathematically more efficient to translate the entire world than to translate the camera.

Once the appropriate camera location is determined, then 3D modeling programs will initialize a process known as rasterization where it converts the 3D world into a 2D image. When rasterization occurs, a camera computes which polygons and points it is supposed to see out of the camera and then essentially flattens them into a 2D image. The program will then apply the appropriate textures and voila the abstract 3D world has been converted into a 2D image. This is the essential process by which 3D graphics are rendered.

Conclusion

This article was meant to be an extremely basic overview of how 3D graphics are represented internally by computers and how they are rendered into a 2D image. It would be impossible to discuss every detail of computer graphics in this article. Topics missed include lighting and shading, reflections, animation ect. Still, this extremely simplistic overview still shows how developers have been able to use math ranging from geometry to linear algebra to simulate 3D worlds on a 2D screen.

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