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Danielle Feinberg, Director of Photography, PixarWe use math, science, and code to create these amazing worlds. We use storytelling and art to bring them to life."

While the primary focus is usually on the creativity in animated films, behind the scenes lies the unsung glory of math and how it is significant to an animator.

Math is in everything, and such is the case in animation too. Animation is a visual process that creates the illusion of motion and is all about timing and proportions, which heavily relate to mathematics.

Like our brain, divided into the practical left and creative right sides, math and art equally engineer an animation masterpiece. In animation movies, math is significant in calculating how everything moves and behaves.

This blog does justice by highlighting how math is useful in computer graphics, different math requirements for people in animation, the math animators use to make a complete animation, and answers the question "how do animators use math?"

**How Do Animators Use Math In Their Animation Career**

Animators need to understand math basics to design lifelike or larger-than-life animations. However, more to the question of “how do animators use math” is considering the advanced principles of geometry, trigonometry, linear algebra, and calculus behind the scenes.

**Geometry** addresses the relationships and properties of geometric shapes like lines, surfaces, and solids. Examples of this math in animation are observable in the different shapes and sizes of characters, weapons, and backgrounds.

^{Image by AMSI Calculate}

Using geometry allows animators to make well-balanced and well-defined drawings and representations.

**Trigonometry**, and animators having some knowledge of it, is helpful to make characters move in different ways in an animated film. Understanding trigonometric principles helps animators turn animation characters 180 degrees or 360 degrees, for instance.

Combining geometric and trigonometric calculations is how animators use math to create animations and give them movement.

^{GIF by Wikimedia Commons}

**Linear Algebra** and the knowledge of algebraic equations and solutions is how animators produce exciting and fun to watch animation special effects.

**Calculus** improves animation production, and animators can have a significant advantage by understanding the calculus principles of the software programs animators use to enhance a scene.

**Physics** closely relates to math and is commonly used in computer animation to make tiny details look and feel alive and produce life-like characters that exhibit human-like emotions, movements, and behavior patterns.

^{Image by edX}

An animator can reinforce their natural mathematical abilities by comprehending physics phenomena like Newton's laws, gravity, and energy and motion concepts to make a realistic animation. Thus, proximate to mathematical calculations, physics is fundamental to an animator.

**Anatomy**, like physics, requires a workable grasp of math. Besides obeying the standard laws of nature, animators solve basic math calculations to showcase the movement patterns of their characters.

### Math Tested In 2D And Stop-motion Animation

**2D Animation**

Many traditional animators still use pen and paper and consult geometrical shapes to draw their characters. Unlike 3D animation, 2D animators do not solely use triangles and consult squares and circles to characterize a figure.

^{Image by Darvideo Animation Studio}

Translation, rotation, and enlargement are mathematical skills animators use to keep characters consistent and make them move on the screen.

**Stop-motion**

The animation series Wallace and Gromit is an example of stop-motion using clay models/claymation.

^{Image by Amino App}

Stop-motion animators making these films create every miniature object and character by hand. The animators use math to understand ratios, so they work at the same scale and avoid measurement inaccuracies.

**Math Is Useful To Make Stunning Computer Graphics Simple**

Computer graphics requires algorithms and data structures to simulate physical reality, including light, fluids, and collisions. So, how do animators use math in CGI? Animators use math to understand mass, density, and friction to make objects move believably.

^{Image by The Science Behind Pixar}

Imagine a string puppet with 700 strings; now imagine all those 700 controls working simultaneously to give life to an animation; that's what animators deal with while developing animation features.

Math in animation is necessary to calculate viewpoints, aspect ratios, character movement, artificial lighting, and color balance.

Furthermore, animators use math to calculate movement and frame rates in animation. Frame rates are how many frames per second the animators need to make the character move fluidly with the script.

Looking at a single frame of the animation movie "Inside Out," here's the answer to “how do animators use math in computer graphics?”

**Rendering**

Think of a camera pointing at a 3D frame, and the camera "renders" what it sees into a 2D image. Someone in the animation team needs to compute all the light arriving at the camera lens from the scene, and this is one of the ways lighting artists use math in animation.

The below integral describes the rendering equation, which is a recursive definition of how light moving away in a particular direction from a point is equivalent to how much light it emits in that same direction plus the subtended fraction of how much light it reflects from every other spot.

^{Kajiya Rendering Equation via StackExchange}

Since there are infinite points and infinite recursion in the universe, animators approximate the rendering equation using statistical techniques.

Light has polarization and is not a scalar quantity, so the artist uses math to decide on what basis to represent light. The task also requires computer science and engineering optimizations. All of which reflect how animators use advanced math in CGI.

Although CGI tools do the heavy lifting, it is no cruise through for animators to create seamless transitions, add realistic effects, customize characters with special styling tools, or create advanced engineering projects.

Computer graphics is a powerful way to learn statistics concepts like MCMC methods, importance sampling, statistical mechanics, and the bias-variance tradeoff.

**Simulation**

Big animation studios like Pixar invest significantly in math because creating animations that reflect real life requires significant accuracy in depicting an animation frame.

^{Image by Juliano via Pinterest}

In the top picture, the characters are playing atop molten lava. They are wearing clothing of different "fabrics," and their hair deforms in motion as they move.

The cloth fabrics behave like dampened spring systems where each fabric thread is a spring, simulated to behave as they would in real life when the characters move in the animation. Their hair is also a system of springs that absorb kinetic energy from their head movements and the influence of the wind.

Lava is complex to animate because of its fluid advection behavior and texture, so it is simpler to simulate it rather than have an artist animate it. Reflecting on animation design according to physical reality and the simulation of physical reality to animation shows how animators use math in their careers.

**Scene Representation**

The use of linear algebra is ubiquitous in scene geometry. Transformations are constant in animation, and linear algebra is fundamental for the purpose, whether by modelers adjusting vertices on a character's face mesh or a set dresser rearranging items.

Rotations, transformations, and scalings are simple, but linear algebra involves inverse kinematics problems. e.g., animation, and numerical solvers for simulation, i.e., finding numerical solutions to large equation systems, spring mechanics, or preserving volumes when characters move.

**Shading**

In simulating reality, shading portrays how each patch of geometry in the scene reacts to incoming light and goes hand-in-hand with rendering.

Some animators use math modules including machine learning, linear regression, and other parametric models to approximate diverse surfaces. A flat surface is noticeably distinct from a bumpy surface depending on how it reflects light, and shading compensates for this heterogenous geometry.

^{Image by Thomas J. Fellers via Olympus}

Computer graphics facilitate many facets of maths not seen in other places. For instance, with enough data, machine learning methods can do regression on the light field equation, i.e., "guess" what the light integral turns out to be without path tracing.

**Signal Processing**

Spatial frequency is crucial from an artistic standpoint since compositions with a scale of spatial frequencies are usually more visually compelling. Set dressers intersperse a scene with large blocks, followed by progressive detail refinement at higher frequencies. Animators use math procedurally to compose multiple spatial frequency octaves to produce impressions like hair and shrubs.

**Everything You Need To Grasp The Basic Math Quickly**

^{GIF by Giflytics via Giphy}

##### Imagine Art, Then Make It Practical With Skill

Mathematics and art are two seemingly disparate fields, but many visual artists make mathematics a focus of their work. How do animators use math in an artistic expression?

Animators who are primarily artistic but are curious to see how artists use math can find inspiration in some common themes in Visual Mathematical Art.

Mathematical artists use several mathematical themes in art. These include polyhedra, tessellations, impossible figures, Mobius bands, distorted or unusual perspective systems, and fractals.

##### Think Like A Creative Director, Stick To The Fundamentals

Creative talent and technical proficiency make a successful animator. Animation studios and freelancers need creative people with business skills. Here, animators use math in management and administrative roles like finance, data analysis, and record-keeping.

As David, the Creative Director at Radix Communications, highlights, a professional understanding of division, multiplication, addition, and subtraction is sufficient for a career in the animation industry.

Creatives looking at “how do animators use math” to contribute to their business skills can relate to David's answer to the question, what is it about your work that is mathematical?

##### Know It All; Math Is As Simple As ABC

Put the lessons of this post to practice with an introduction to an animation tool to make simple animations and graphs to explore simple math expressions or make short presentation-style animations.

To further understand the mathematics of animation, Varun Vachhar covers topics like trigonometry, coordinate systems, and curves and vectors while exploring some basic mathematical concepts often encountered when creating animations.

Animators can improve and expand their skills by enrolling in online animation courses. For example, animators can learn about fundamental math concepts, vector space in 2D and 3D, and projectile motion mathematics from the "Math For Video Games: The Fastest Way To Get Smarter At Math" course.

**How Do Animators Use Math To Unlock New Animation Skills**

How do animators use math, you ask? An animator is responsible for implementing math in animation, and this blog encourages artists to consider learning mathematical formulas to reinforce their creative competencies and natural genius in math.

To be an animation specialist, animators should be passionate about math, equal to their creative knack. Mathematical formulas work at the back end to process and complete the visuals audiences see in the front end.

Math in animation is a critical behind-the-scenes role, without which shapes wouldn't appear smooth or act according to the artist's command.

Animators can formulate unique math to realize new concepts and breakthroughs, especially for animators that dislike authority. In that case, an animator can choose to be independent and have all the freedom necessary.

Visit our blog on How to Start an Animation Studio, check out the Business of Animation free masterclass, and download a copy of our free marketing handbook to grow your animation skills.