A subsidiary of Walt Disney Studios, Pixar is regarded as a leading pioneer in computer animation famous for award-winning feature films such as Finding Nemo (2003), The Incredibles (2004), WALL-E (2008), Inside Out (2015) and Toy Story 4 (2019). What may be lesser known is some of the mathematics underlying their revolutionary animations.
In the 2014 TED-Ed presentation Pixar: the math behind the movies, Tony DeRose, a computer scientist at Pixar Animation Studios, stated that “artists and designers think in terms of shapes and images whereas computers think in terms of numbers and equations”. Thus, computers use mathematics to make sense of human ideas and animate them.
Creating animated films generally starts with artists sketching thousands of hand-drawn characters and scenes either using pen and paper or with the help of computer software like Photoshop. These blueprints transfer into animating programs where they are brought to life by underlying maths, including geometric coordinates (to map a character’s place in space using an X and Y axis graph), trigonometry (to move characters), linear algebra (to show the way that an object is rotated and shifted and made larger and smaller), and even integral calculus (to accurately stimulate the movement of lighting).
Typically, millions of mathematical equations are solved for every frame of animated film!
Commonly utilised by Pixar studios are harmonic coordinates and subdivision surfaces. Harmonic coordinates efficiently map movement and create and control volume deformations used to articulate characters in a simplified fashion, helping to make character movement appear more realistic. Subdivision surfaces allow the efficient smoothing of complex geometric surfaces and shapes, dividing polygons or other two-dimensional shapes into smaller pieces to make a geometric model appear far more realistic and detailed.
Subdivision surfaces were first successfully used by Pixar for their animated short film Geri’s Game (1997), winning an Academy Award in 1998. The plot follows elderly man Geri playing chess against himself in a park with Autumn foliage fluttering in the background. Acting as both competitors, Geri distinctively switches persona as he repeatedly runs to either side of the table, displaying an impressive range of facial expressions throughout the story. Eighteen animators, including Tony DeRose, created the film. They used Tony’s curve-generating algorithm, which helped to perfect face shapes and clothing movements.
ORIGO Education worked with Cetati Studios, whose veteran artists hail from beloved creativity centres like Pixar, to create the fun and interactive Animated Big Books, an award-winning series of engaging large-format mathematics storybooks for young learners. The animations in ORIGO Animated Big Books exist because of the underlying maths mentioned!