Physical Bodies

A state of a rigid body at any moment is specified with its position, orientation in space (with respect to some reference point — center of mass) and velocity. There are two types of body motion and therefore two velocity components:

• Linear motion. If we imagine that the orientation of the body is fixed, then the only movement the body can undergo is translation motion — change in linear position. This change is performed with a linear velocity.
• Angular motion. On the other hand, if we freeze the center of mass of our body in space, the only movement the body will be able to perform is rotation, which is described by an angular velocity

As the body moves, its linear and angular momenta change. The linear momentum can be thought of as an extent, to which the body will continue to move along a straight path. It is the product of the mass and linear velocity of the body:

p = m * v.

The body will keep on moving forever, unless affected by an external force or an impulse. Force equals mass of the body multiplied by acceleration:

F = m * a.

By causing the body to undergo an acceleration (i.e. to change its velocity over time), force controls its velocity and position indirectly.

Impulse is the integral of force over time. It may be be regarded as the change in momentum of an object to which a resultant force is applied. For example, when two bodies collide, they exchange impulses that are equal and opposite, as Newton's third law applies, and in the result objects move apart.

Similarly to the linear momentum, the angular momentum is the measure of the "quantity of rotation motion" and can be thought of as an extent, to which the body will continue to rotate around an axis of symmetry. It is expressed as the product of the inertia tensor of the body and its angular velocity.

Rotation continues until torque, a rotational force, is exerted on this body. Torque is the cross product of the radius vector (a vector from the center of mass to the point where torque is applied) and the force vector (the magnitude of the force). Loosely speaking, it acts like a lever affecting rotational speed.

Forces and impulses can also be applied to an arbitrary point of the body and may cause body rotation, when this point is not the center of mass. In this case, force is computed as cross product of force vector and radius vector (from a center of mass to the necessary point) and is added to the torque. And the other way around, torque applied not to the center of mass increases force.

To sum it up, the movement is characterized by the following basic parameters:

Linear Motion	Angular Motion
Mass (scalar)	Inertia tensor (mat3)
Linear velocity (vec3)	Angular velocity (vec3)
Force(vec3) Impulse ( vec3)	Torque (vec3)

Adjustable body parameters, that determine its behavior in the framework of rigid body dynamics, are as follows:

Mass of the object multiplied by gravity specified for the world defines its weight:

The center of mass is automatically calculated as the mean location of mass of all the shapes that approximate the object. It serves as a reference point for linear motion and rotation, as well as application of external force and torque.

Mass parameters of the body can be set up manually or determined automatically using the shape-based parameters. It is convenient when a body has several shapes.

Density of the objects is defined as its mass per unit volume:

Density value evidently depends on the mass value and vice versa: the higher the values are, the heavier and more dense the object is.

Density determines buoyancy of the body in accordance with Archimedes' principle. The higher the density, the less tendency a body has to float.

When the object begins moving in a definite direction, the linear damping force slows it down up to a complete stop. Similar to linear damping, angular damping reduces angular velocity of objects over time, so that their rotation ceases. To the linear damping of the body the global Linear damp is added, and the exponential function is calculated. In exactly the same way, to angular damping of the body the global Angular damp is added.

These two parameters ensure that objects smoothly come to a stop and no calculations are done for unnecessary motion.

Maximum linear and angular velocity define the maximum possible velocities of the body. Velocities that exceed this limit will be clipped. For example, maximum linear velocity parameter can help to avoid tunneling (penetration) effect.

Coefficient of friction allows to model more rough rubbing of surfaces and is opposite to the body's movement direction. Friction parameter of both surfaces being in contact are considered:

• The higher the value, the less tendency the body has to slide.

The resulting calculated friction depends on the objects' masses and gravity, and the angle between contacting surfaces. E.g. if a body slides down an inclined plane, the friction is lower, because less is the force of gravity that is perpendicular to the face of the surface.

Friction is calculated on contact between physical bodies.

Coefficient of restitution determines the degree of relative kinetic energy retained after a collision. It defines how bouncy the object is by contacting with another object. It depends on the elasticity of the materials of colliding bodies. The simulated restitution, like friction, considers the total value for both objects being in contact.

• The maximum value of 1 models elastic collision. Objects bounce off according to the impulse they get by contact.
• The minimum value of 0 models inelastic collision. Objects do not bounce at all.

Again, just like friction, restitution is calculated by the contact between physical bodies.