Winding through the sandy grounds of the Coastal Vivarium and battling Acromantulas in the Forbidden Forest, you’re transported into a magical world in Hogwarts Legacy, where stunning graphics and immersive, free-roam gameplay come to life. But the true enchantment behind the game’s animation lies within the mathematical precision of Bézier curves. Like cheat codes, but for game developers, these curves unlocked the evolution from the 2D characters in the first Legend of Zelda game into the hyperrealistic avatars of today.
In 1959, Paul de Casteljau, a French mathematician working at Citroën, developed a method to model the curved components of their cars numerically. De Casteljau’s algorithm could break down a curve into “anchor points,” the positions of the ends of the curve, and “control points,” areas along the curve that determine its bends. Think of the anchor points as poles that hold up a tent, and control points as the ropes that pull it up into shape. The anchor points provide stability and structure, while the control points determine the overall form of the curve. This method pioneered the advanced curvatures of the body of Citroën’s famous 1966 DS. The industry-leading aerodynamic design contributed significantly to its efficient fuel consumption and optimal high-speed performance.
A few years later, engineer, Pierre Bézier faced a similar problem at Renault. He formalised a function for the smooth curves produced by de Casteljau’s algorithm, eponymously dubbed Bézier curves. The curves come in several types, each with its own position and number of anchor and control points. A linear Bézier curve is a straight line connecting two anchor points, and they’re used for creating the paths of vehicles in games like Fortnite. A quadratic Bézier curve has one control point and creates a parabolic shape, which has been used for the trajectories of weapons in Apex Legends. Finally, a cubic Bézier curve has two control points creating sharp turns or bends, like those of the Neruda Sea shoreline in Breath of the Wild.
Bézier curves enable modeling software to recognise sophisticated shapes beyond circles and lines, making them a highly valued tool in complex rendering and design. As Trevor Savage, a member of the user interface team at WB explains, “Bézier curves have been critical to [Hogwarts Legacy]’s early success,” shaping core animations like spellcasting. More broadly, they played a role in determining in-game lighting, character movement, and graphics quality.
The versatility of Bézier curves has made them popular not only in the Wizarding World, gaming, and the French automobile industry, but in many other fields including string and crochet art, computer-aided design, typesetting, and architecture. The heat shield for NASA’s Mars Reconnaissance Orbiter was designed using their properties. Moreover, researchers at the Newcastle Eye Centre in England published a paper in which they simulated eyelid surgery with Bézier curves!
The ubiquity of bends makes Bézier curves essential to innovation. From building flexible robots to improving people’s vision, the magic of these bends continues to run wild in the digital and physical worlds.
Irregular, curved shapes are ubiquitous in both the online and organic world, giving Bézier curves utility across all spe
Bézier curves are ubiquitous across all spheres of innovation–from lifelike game characters to the organic world. With a method to quantify and design them, their magic continues to run wild, and the possibilities are endless.
About the Author
Okezue is a 17-year-old STEM activist and inventor whose humanitarian efforts through workshops and civic engagement have provided $3.5 million in value to underserved communities and activated over 120,000 individuals. His educational campaigns for quality learning and schooling have reached over 75,000 students across 80 countries. Okezue is the founder of Fidutam, a microlending application for the unbanked, and the inventor of WeArm, a low-cost prosthetic arm for individuals with below-the-elbow amputations. He frequently also consults for large-scale companies on tech, equity, and climate, is a United Nations Association Ambassador, and has conducted intersectional AI and neuroscience research at MIT, Harvard, and more.
