At Pixar, storytelling is at the heart of everything we do. However, a lesser-known story is the significant role mathematics plays in bringing our films to life. The math you learn in middle and high school is integral to our animation processes. Let’s explore how these mathematical concepts are applied in creating the beloved characters and scenes in Pixar films.
Consider a simple example: Woody from Toy Story walking across the stage. This seemingly straightforward action involves a wealth of mathematics. While artists and designers think in terms of shapes and images, computers interpret these through numbers and equations. To bridge this gap, we use coordinate geometry. By establishing a coordinate system, with ‘x’ indicating horizontal movement and ‘y’ indicating vertical movement, we can precisely describe Woody’s position at any moment. For instance, if Woody’s x-coordinate starts at one and ends at five, the mathematics of translation—essentially addition—comes into play.
Scaling, or resizing an object, involves multiplication. To double the size of an object, we multiply its x and y coordinates by two. Similarly, rotation relies on trigonometry. Whenever you see a character spin in our films, trigonometric equations are at work. These concepts, often introduced in middle school, are foundational to our animation techniques.
Beyond basic transformations, Pixar employs advanced mathematics to create complex shapes and movements. Characters like Dory from Finding Nemo and Al McWhiggin from Al’s Toy Barn have intricate designs that require sophisticated mathematical modeling. One technique we use is subdivision, which involves splitting and averaging points to create smooth curves and surfaces.
For example, starting with a simple four-point polygon, we add midpoints to each edge (split) and then move these points to the midpoint of their neighbors (average). Repeating this process, known as subdividing, transforms the polygon into a smooth curve. This method is crucial for animating characters, as it allows us to manipulate the original points to create dynamic movements.
Subdivision surfaces were first utilized in the short film Geri’s Game and have since become a staple in Pixar’s character creation. Geri’s hands, face, and jacket in Toy Story 2 were among the first to use this technique. By smoothing out facets, subdivision creates the seamless surfaces seen on screen. Characters like Merida from Brave are brought to life using these methods, with her dress, hands, and face all modeled as subdivision surfaces.
Throughout Pixar’s films, mathematics such as addition, multiplication, trigonometry, and geometry are ever-present. With more time, we could delve into how linear algebra, differential calculus, and integral calculus also play roles in our animation processes. The key takeaway is that the math you learn in school is not just theoretical; it is actively used every day at Pixar to create the magic you see on screen.
So, next time you watch a Pixar film, remember the mathematical foundations that make these stories possible. Whether it’s Woody’s walk or Merida’s flowing dress, math is the unseen hero behind the scenes.
Using graph paper or a coordinate plane app, plot the movement of a character like Woody from Toy Story. Start by marking his initial position and then plot his path as he moves across the stage. Use the x and y coordinates to describe his movement and create a simple animation by connecting the dots.
Take a simple shape, such as a triangle or square, and practice scaling it by multiplying its coordinates. Then, use trigonometric functions to rotate the shape around a point. You can do this on graph paper or using a geometry software tool. Observe how the shape changes with each transformation.
Create a basic polygon, like a square or triangle, and practice the subdivision technique. Split each edge by adding midpoints and then average these points to create smoother curves. Repeat the process several times to see how the shape evolves. This can be done using drawing software or by hand on paper.
Using modeling clay or a 3D modeling software, create a simple character. Apply the subdivision technique to smooth out the surfaces. Start with a basic shape and gradually refine it by adding and averaging points. Notice how the character becomes more lifelike with each iteration.
Watch a short Pixar film or clip and identify scenes where mathematics is likely used. Discuss with your classmates or write a short essay on how concepts like coordinate geometry, scaling, rotation, and subdivision might have been applied. Reflect on the importance of math in creating these animations.
Mathematics – The study of numbers, quantities, shapes, and patterns and how they relate to each other. – In mathematics class, we learned how to solve equations using algebra.
Geometry – The branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and shapes. – We used geometry to calculate the area of the triangle in our project.
Trigonometry – A branch of mathematics that studies the relationships between the angles and sides of triangles. – Trigonometry helps us understand how to measure the height of a tree using angles.
Scaling – The process of increasing or decreasing the size of an object while maintaining its proportions. – In art class, we learned about scaling by enlarging a small drawing to fit a larger canvas.
Rotation – The action of turning around a center or an axis. – In geometry, we studied how rotation can change the position of a shape on a plane.
Coordinates – A set of values that show an exact position on a grid or in space, usually defined by numbers. – We plotted the coordinates (3, 4) on the graph to find the location of the point.
Addition – The mathematical process of finding the total or sum by combining two or more numbers. – We practiced addition by combining different sets of numbers to find their total.
Multiplication – The mathematical operation of scaling one number by another. – In multiplication, we learned that 4 times 3 equals 12.
Subdivision – The process of dividing something into smaller parts or sections. – In computer graphics, subdivision is used to create more detailed models by dividing surfaces into smaller polygons.
Animation – The process of creating the illusion of motion by displaying a series of images or frames. – We used animation software to make a short film about a bouncing ball.
