Games Demystified: Super Mario Galaxy

Welcome
to the first article in a new Gamasutra series called *Games
Demystified*. In this series I'll be deconstructing games with unique or distinguishing mechanics
that may have left some folks scratching their heads, including reproducing effects with code samples, examining -- just how did
they accomplish that? The first game we'll be covering is *Super
Mario Galaxy* for the
Nintendo Wii.

Upon
release, *Super Mario Galaxy*
enchanted the gaming community because it allowed
gamers to walk upside down and inside out -- all while feeling
completely intuitive. Especially geeky message boards and blogs have lit up with
people attempting to apply real-world physics equations for gravity
to each of the game's tiny planetoids.

Obviously, real world physics have a place in today's games. However, they take a backseat to psychology when it comes to making real world gameplay. In reality, mass distorts the fabric of time and space causing gravity, the force that keeps us stuck to Earth.

People such as Isaac Newton have studied and derived a great deal about gravity, but no one actually knows the true mechanics behind the force. The important thing for the game is that we expect to get pulled toward planets -- and in fact, Mario does get pulled to each and every strangely shaped planetoid in the game.

Normally, gravity is calculated as a force between two objects that is directly proportional to the product of their respective masses and inversely proportional to the square of the distance from the objects' centers.

Force of
Gravity = Gravitational Constant * ( ( mass1 * mass2 ) / distance^{2})

To simplify, gravity follows the inverse square law.

Intensity
of Gravity = 1 / Distance^{2}

This means that as an object moves closer to a planet, the gravity between them

increases dramatically due to the exponential effect applied by distance. Conversely, as an object moves away
from a planet the gravity between them weakens very quickly.

Traditionally,
the force of gravity is applied using the radius of the more massive
body. For instance, if we were to calculate the gravity holding a
person to Earth we'd plug the Earth's radius into the above equation.
Additionally, if the mass is great enough to hold a person down, the
celestial body will be round. This is where reality and *Super
Mario Galaxy* differ.

In
*Galaxy,*
players are pulled toward each and every arbitrarily shaped planetoid
they encounter. At this point all real world physics equations break
down and tell us nothing about the gravity in the game, which as in
most cases is a trick. Much like the magicians in *The
Prestige,* game developers
are always attempting to one up each other with new tricks. As we all
know, Nintendo stole the show with its latest.

To accomplish this "trick", the developers use the surface normals of whatever planetoid Mario inhabits to keep him running and jumping happily along. This is how gravity is distributed through completely irregular bodies. A surface normal is a ray or unit vector that runs perpendicular to a plane.

In the illustration above we can see an arrow pointing up from a plane, the arrow represents the surface normal of a plane that points straight up. If we're using a Y-up right handed coordinate system, that normal would be a (0, 1, 0) which means it's not pointing along the x or z axes at all and that the full length of the vector is all in the up direction.

A unit vector always has a length of 1, which is then distributed amongst the x, y, and z directions. The longer one direction is the shorter the others must become.