Physics Engines

Requires Position, Velocity and Acceleration

The location of an object in space, represented by a vector (2D or 3D)

Amount of change in a position, represented in direction and magnitude (speed)

The amount of change in velocity
Not typically stored, calculated using other factors

Law of Inertia

An object keeps its velocity unless acted upon by some other force

Resistance to change in motion

Force = mass x acceleration

For every action there is an equal and opposite reaction

SUVAT equations
s = Displacement
u = Initial Velocity
v = Final Velocity
a = Acceleration
t = Time

v = u + at
s = ut + 1/2at^2
s = 1/2(u+v)t
v^2 = u^2 + 2as

Quantities that have both a magnitude and a direction.

E.g. [1, 0, 2]

Easily addable, easily multiplicative with a scalar

Normalised direction can be calculated by:
-vector / vector length

Calculation of change in something, makes for easy translation between velocity, acceleration, etc.

The reversing of derivatives, used for figuring out the states of objects in the future.

vn+1 = vn + andt
pn+1 = pn + vndt

Uses the velocity and acceleration of the current frame to get the velocity of the next

Not perfect, we ignore acceleration throughout the timestep, causing inaccuracies over time

vn+1 = vn + an+1dt
pn+1 = pn + vn+1dt

Uses the projected velocity and acceleration of the next frame to get the velocity of the next

More accurate if you have the data, but since we don't due to real time simulation it is wildly inaccurate (prediction)

vn+1 = vn + andt
pn+1 = pn + vn+1dt

Uses the acceleration of the current frame and the projected velocity of the next frame.

Most accurate we can do in real time

Extremely important, as objects need to be consistent with each other

Stands for "vertical synchronization," is used to synchronize the graphics card/framerate with the output on the monitor.

A customizable framerate-independent interval that dictates when physics calculations and FixedUpdate() events are performed, regardless of how fast the machine is running.
The smaller this is the better, as it gives more accurate results.

Objects recognised by a physics engine that it will update.

Easiest representation of a physics body

Has mass and position

Doesn't have orientation

Most common representation of a physics body

Has mass, position, and volume the mess is distributed around, as well as orientation

Doesn't change shape over time

Elastic connectors that bring objects together and push them apart.

Uses Hooke's Law for the calculation (F = -kx) (K = elasticity of spring, X = distance from resting length)

Takes into account spring damping to get the effect of friction (F = -kx - c.v) (c = damping factor, v = velocity along spring axis)

Rigid bodies that can deform, used for cloth-type material.

Typically modelled with a grid of rigid bodies connected via springs with resting length equal to grid spacing.

If a force is applied to one of the masses, the springs attached to it stretch and eventually apply force to all the masses.

Does require extra structural support to maintain their 3D shapes

Integrate Acceleration
Collision Detection
Collision Resolution
Integrate Velocity

Forces that have acted on an object in a timestep are stored for calculation.

a = F/m

The amount of matter in an object

Stored in physics engines as 1/m, in order for a quicker calculation. Allows for infinitely heavy objects (m = 0)

Objects in the world are examined for collisions, and if so they react accordingly.
Done before velocity is calculated so the reactions can be taken into account.
Done after acceleration in order to determine momentum, so it can be conserved.

Uses all previously calculated information to get the velocity of the object (using Semi-Implicit Euler or similar).

Typically game engines use a damping factor to reduce velocity over time (simulation of air resistance)