After examining yesterday’s results in SAS, the data outputted by the simulation produced tuples of coordinates that represented a particular velocity, eg [20 30 40]. However, I must admit I was a bit confused what this number represented. As such today I’ve been at the University of Hertfordshire’s library, looking through a number of programming games in 3-D books. As such the notes from which are below:
- From Beginning Math and Physics for Game Programmers, by Wendy Stahler:
- In one dimensions, velocity is a positive or negative number where a sign indicates the direction required.
- In two dimensions: displacement equals the final minus initial positions
- n.b. vectors need to be in component form before they can be subtracted: distance in the x direction and distance in the y direction
- for a diagram of this, see your notes on the 13th of April 2007
- *velocity is displacement divided by time
- n.b. polar coordinates like 10 m per second at 53°, whereas Cartesian coordinates like [ 6 8 ] represent a vector, where 6 and 8 represent the x and y coordinates of the vector respectively
- The x component is equal to the magnitude multiplied by the angle’s cosine
- The y component is equal to the magnitude multiplied by the angle’s sine
- from Beginning 3-D Game Programming by Tom Miller:
- again, see your notes the 13th of April 2007 for a figure indicating the difference between two different vectors
- vectors measure both direction and velocity
- most times a vector is defined as a free vector, which is when the root of the vecto is at the origin
- and therefore the arrowhead in the figure in your notebook indicates the [x,y,z] vector value
- drawing a line between this point in the origin therefore leads to the direction and the velocity of the two points
- nb vectors do not need to be rooted at the origin, but it makes calculations easier
- to calculate the velocity of a vector when you know the arrowhead coordinates, the length of the vector is equal to the square root of (x coordinate squared, + y coordinate squared, + z coordinate squared) — again see your notes for the written equation
- a normalised vector, otherwise known as unit vector, is a vector with a length equal to one: this means you can store direction without regard to the velocity
- to calculate the normalised version of a vector, take each vector component and divide by the vector length
- e.g. with a free vector of (23, 18, 7) the length this vector is approx 30 .0333
- dividing each component by this number leads to a new vector with coordinates: (0.77, 0.60, 0.23): the length of this vector was approximately one, which is expected
- from 3-D Game Programming with C++by John De Goes
- you can describe the position of a three-dimensional object that is moving in a straight line by using its initial velocity, its initial position and its acceleration. Equation:
- P(t) = P(0) – V(0)t + 1/2at^2, where:
- P(t) point of the object
- P(0) initial position
- V(O) initial velocity
- a acceleration
- t time
- this can be expressed equivalently for the x, y, z coordinates:
- Px = P(x0) – V(x0)t + 1/2axt^2
- Py = P(y0) – V(y0)t + 1/2ayt^2
- Pz = P(z0) – V(z0)t + 1/2azt^2
- this equation can also be useful:
- V = V0 + at
- Game Physics by David H Eberly
- looked interesting but probably a little bit too much at this point
- websites of interest:
- gamedev.net
- gamasutra.com
No doubt more will follow, and hopefully tomorrow will have the opportunity to do a bit more investigation in this area.
