Animation and CGI

SUMMARY: Animators have become adept at creating realistic products with the help of mathematics.

Animation is the process of creating the illusion of fluid movement from a series of static images. When these images are viewed in sufficiently fast succession, the human eye sees them as continuous motion rather than a sequence of discontinuous still images. From the earliest mechanical devices, through hand-drawn, stop-motion, and computer-assisted film techniques of the twentieth century, up to the latest computer-generated imagery (CGI), the quest has been to create interesting representations of movement and action.

Early Animation Devices

Historically, there have been several mechanical devices that were developed to simulate movement using still pictures. The Phenakistoscope, invented in 1832 by the Belgian physicist Joseph Plateau, consisted of a spindle with two mounted discs, one with slots around its edge and the other with pictures of successive action. With the discs spinning in unison and the picture side facing a mirror, the view through the slots appeared to show a moving drawing. In 1834, a British mathematician named William Horner produced the Zoetrope, a cylinder cut with vertical slits. Pictures of successive action were positioned on the inside opposite the slits. With the cylinder rotating, the image seen through the slits appeared to be in motion. As the Zoetrope used more pictures and could be rotated more quickly, this gave a better illusion of movement. Even in the early twenty-first century, the Zoetrope is used to illustrate the basic idea of animation.

Animation Principles

By the start of the twentieth century, these mechanical devices were superseded by animated films. The principal technique was to hand-draw each frame. In the 1930s, animators at Walt Disney Studios developed what became known as the “12 principles of animation,” many of which remain pertinent in an era of CGI. To illustrate, consider someone throwing a ball so that it bounces along the ground. A thrown ball is known to follow a parabolic path, a form of arc. The “arc” principle of animation is that almost all actions follow some form of arc. Arcs, as the Disney animators were well aware, give animation a more natural appearance. Another principle, “slow in and slow out,” relates to the ball taking time to accelerate and decelerate. The animation looks most realistic if there are more frames near the beginning and end of a movement and fewer in the middle. The flight of the ball and its bounce involve the principle of “squash and stretch.” As the ball falls, animating a slight stretch gives the impression of the ball having speed. Dilation is the mathematical transformation for stretching and shrinking. Animating a squash to the ball as it bounces gives the impression of weight. For the ball to seem real, the animator uses the principle of “solid drawing” by taking into account the form of the ball in three-dimensional space as well as using the geometry of light and shadow.

CGI and Mathematics

CGI is even more mathematically based than hand animation because the images must be mathematically represented in order to be manipulated in the computer environment. Oscar-winning computer scientist Tony DeRose, who has worked for Pixar Animation Studios, said, “. . . different kinds of mathematics are used for different aspects of a film, from the simulation of how light bounces around in an environment (integral calculus) to obtaining smooth surfaces efficiently (subdivision surfaces) and making characters move in a realistic fashion (harmonic coordinates).” Trigonometry and vector algebra are widely used in CGI algorithms for creating and manipulating images. Matrices are a standard algebraic way of representing various transformations. Dilation makes objects larger or smaller in addition to stretching; translations move objects; and rotations turn objects.

One classic CGI method for creating three-dimensional animated objects involves using polygonal meshes, which are collections, or grids, of polygons. This method makes use of the geometry of smooth surfaces. Like animated motion, this method relies on the human eye’s tendency to smooth discontinuous regions. Locally, smooth surfaces look flat, so they can be approximated with small, flat polygons such as triangles or quadrilaterals. Basic three-dimensional shapes such as cubes, cylinders, spheres, and cones may be joined to form composite three-dimensional objects. Interpolation is also used. More complex and smoother-looking three-dimensional objects can be modeled using sophisticated mathematics like spline patches and nonuniform rational basis splines, where a spline is a mathematical function defined piecewise by polynomials. Such techniques have become standard practice in CGI. The mathematical representation of three-dimensional shapes, including layout and materials, is used to compute a two-dimensional image from a given viewpoint, a process called “rendering.” This process entails addressing issues such as visibility from selected viewer angles (including which parts of objects in the scene are hidden) and appearances and how objects look different as the lighting varies. Finally, the motion of each object in the scene has to be specified.

Lucasfilm LTD and animator Kecskemeti B. Zoltan of Ste-One provided mathematician Timothy Chartier with digital models of Yoda from the Star Wars movies to explore in linear algebra classrooms. One of the models had 53,756 vertices, 4040 triangles, and 49,730 quadrilaterals, illustrating that realistic images and their transformations have many more data points and matrix multiplications than is typical as classroom examples. Chartier noted, “More recently, computer animation produced the character’s movement, which required mathematical concepts from such areas as linear algebra, calculus, differential equations, and numerical analysis. Drawing on these popular culture ties in appropriate coursework can pique students’ curiosity and compel further learning.”

Despite the many available mathematical techniques and advances in the computational and visualization power of computer systems, convincing simulation of some physical features, like hair, continues to be challenging. Pixar noted that it took up to 12 hours to render a single frame of the character Sulley in the 2001 movie Monsters, Inc. because of his nearly 3 million individually animated hair strands. Each hair was mathematically modeled as a series of springs connected via hinges. Techniques continued improving into the 2020s. In 2024, a new mathematical algorithm was created to improve the accuracy and visual, as well as the animating speed, of tightly coiled hair for Black characters. It utilizes three-dimensional spirals and allows animators to create more variety in curly hair.

CGI has come a long way since the 1976 movie Futureworld, which many acknowledge as the first use of three-dimensional computer imagery. Even though the first CGI film to win an Oscar was Pixar’s short movie Tin Toy in 1988, the 1995 movie Toy Story was the first full-length, fully CGI feature film. In the late 2010s, animators achieved photorealism using CGI. For example, the 2019 Disney remake of The Lion King was advertised as a live-action remake due to its photorealistic animation. To accomplish this, the movie was filmed using virtual reality.

Bibliography

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Barnhart, Benjamin. "The History of Computer Animation." Linearity, 18 Feb. 2024, www.linearity.io/blog/computer-animation/. Accessed 4 Sept. 2025.

Chartier, Timothy. “Using the Force: Star Wars in the Classroom.” PRIMUS, vol. 17, no. 1, 2007, pp. 8–23, doi:10.1080/10511970601126860. Accessed 4 Sept. 2025.

Desta, Yohana. "The Lion King, Hand-Drawn Animation, and the Problem With Photo-Realism." Vanity Fair, 18 July 2019, www.vanityfair.com/hollywood/2019/07/lion-king-remake-disney-photorealism-animation. Accessed 4 Sept. 2025.

McAdams, A., S. Osher, and J. Teran. “Crashing Waves, Awesome Explosions, Turbulent Smoke, and Beyond: Applied Mathematics and Scientific Computing in the Visual Effects Industry.” Notices of the American Mathematical Society, vol. 57, no. 5, 2010, www.ams.org/jackson/fea-mcadams.pdf. Accessed 4 Sept. 2025.

Ortega, Lizette. "Animations of Coiled Hair for Black Film Characters Improve with New Algorithms." The Washington Post, 30 Nov. 2024, www.washingtonpost.com/science/2024/11/30/coily-hair-animation-black-film-characters/. Accessed 4 Sept. 2025.