The Application of Calculus To Video Games Physics: An Integral Part of Game Design How is it that in video games we can move around in realistic virtual environments? What is it that allows a simulation to so closely mimic reality? Why does a non-real entity that we control react to the world much like a real life person would? The answer to all of these questions lies in something known as a physics engine. The History of Physics Engines First developed by military

Used for basic models, such as gravity

Later expanded to weather forecasting

Fluid mechanics for designing air/watercraft The Application of Physics Engines To Video Games First utilized in simple video games like Pong

The physics engines consisted almost entirely of collision detection and response The Expansion of Physics in Video Games http://books.google.com/books?id=xdkjjlNdzBsC&pg=PA43&lpg=PA43&dq=physics+engines+calculus&source=bl&ots=Mraz436NSY&sig=pnW3zalOQZeP09T2qpaWo8sQ3-I&hl=en&sa=X&ei=zLKkUfL0DYXgqgHkpoCABQ&ved=0CDkQ6AEwAw#v=onepage&q=physics%20engines%20calculus&f=false Pong As the video game industry began to expand, so did its use of physics engines These new engines utilized a collision detection system with a physics engine updating at real time to determine the properties of particles This Is Where Calculus Applies The Integrator Focuses on the Calculus of Change Over Time Physics engines focus primarily rates of change

Differential calculus

Integral calculus The Integrator! Physics engines do their complex mathematical computation in something known as an Integrator

This is a chunk of code that is responsible for actively updating the position of objects These rates determine the velocity and acceleration vectors of a particular particle The basic formulas utilized by the integrator to calculate the vector and position of a particle at a certain time are listed below Physics engines multiply these vectors by using vector notation to more quickly compute the end product But What About Physical Laws? In addition to the vectors of particles, each one is assigned a set properties that influences its vectors

These properties include the Newton's basic laws of motion as well as his law of universal gravitation Each particle has a full set of these properties as well as predetermined constants for all other equations, such as f=ma(where the particle is assigned a mass) While a video game is running, each particle on screen including your character is being actively run through the physics engine's integrator which can process anywhere up to one billion points per second

This speed is measured in FLOPS(floating point operations per second) The End Result The end result of the active updating of these force vectors is a realistic looking and behaving simulation of real life

Without calculus this whole process would be impossible and modern video games could not exist This video demonstrates the use of the physics engine to realistically calculate what would happen if one were to drop 20k watermelons on a hill (Skip about halfway into the video) This video illustrates a physics engine's role in video games The End Calculus is not the only math done in the integrator

Trigonometry is also a key aspect of the calculations of the collision detection and response system Valve's simulation of particles falling on a sphere An excerpt from a set of code in a physics engine that simulates the flight path of a projectile fired from a cannon D'Alembert's principle applies here in that all of the acting forces on a particle can be summed into one acting force

When all of the force vectors acting on the particle are assigned and determined, they are summed up into a single acting force vector D'Alembert's Principle

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# The Calculus of Video Games

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