NOTE :
vectors are represented by capitals, scalars are represented by lower case.
The x, y, and z components of a vector are denoted by '.x', '.y', and '.z'
respectively. For example, the x component of the vector V would be represented
by 'V.x'.
The
Coordinate System and the Camera
Below is a diagram of the coordinate system employed.
The camera is always :
-located at the origin
-pointing in a positive z-direction
Projecting
a 3D Point Onto the Screen
Below is the BASIC formula for projecting the 3D point represented by the
vector P, onto the screen, into the 2D vector S :
There are a couple of problems with the above formula. Firstly, any point
lying on the z-axis (x=0, y=0, z=?) will be projected onto (0,0) on the
screen, - the top left corner. A point on the z-axis should be projected
onto the centre of the screen, so if w and h are the screen's width and
height respectively, so the coordinated should be (w/2,h/2), so this makes
the formula :
Secondly, the field of view is too large. In the above formula, the field
of view will also depend on the screen resolution used, - the large greater
the resolution, the larger the field of view. There is no way of controlling
the field of view. Below is a formula for implementing variable horizontal
field of view :
'f' is a scalar, which will alter the horizontal field of view. To calculate 'f', given the horizontal field of view 'a' degrees, use the formula
Below are the formulae required to calculate 'f' and 'g' using 'a' and 'b' as the horizontal and vertical fields of view respectively, and then project point P onto the screen, with the result being the vector S.
Reverse Perspective Projection
In applications such as raytracing, it may be necessary to produce a vector of a ray for a pixel. To do this, we simple reverse the formula. Firstly, the formule for the x-coordinate will be reversed. Note that the 'f' and 'g' from normal prespective projection described above can be used here. We know S, (the screen coordinates) and we want to find P, however, we must set a value for P.z. We will set it to 1, to keep things simple. Here is the process by which the perspective projection formula can be rearranged :
This means that (since the same rules apply to x and y) :
It is likely
that the vector P would be most useful if it were normalized. To do this
divide each of the x, y, and z components of the vector by the vector's
length.
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