Game Engine Architecture

162 4. 3D Math for Games

of the engines I’ve encountered use either a y-up or a z-up convention. The

y-up convention was probably an extension of the two-dimensional conven-

tion found in most mathematics textbooks, where the y-axis is shown going

up and the x-axis going to the right. The z-up convention is also common, be-

cause it allows a top-down orthographic view of the game world to look like

a traditional two-dimensional xy-plot.

As an example, let’s say that our aircraft ’s left wingtip is at (5, 0, 0) in mod-

el space. (In our game, front vectors correspond to the positive z-axis in model

space with y up, as shown in Figure 4.17.) Now imagine that the jet is facing

down the positive x-axis in world space, with its model-space origin at some

arbitrary location, such as (–25, 50, 8). Because the F vector of the airplane,

which corresponds to +z in model space, is facing down the +x-axis in world

space, we know that the jet has been rotated by 90 degrees about the world

y-axis. So if the aircraft were sitt ing at the world space origin, its left wingtip

would be at (0, 0, –5) in world space. But because the aircraft ’s origin has been

translated to (–25, 50, 8), the fi nal position of the jet’s left wingtip in model

space is (–25, 50, [8 – 5]) = (–25, 50, 3). This is illustrated in Figure 4.18.

We could of course populate our friendly skies with more than one Lear

jet. In that case, all of their left wingtips would have coordinates of (5, 0, 0)

in model space. But in world space, the left wingtips would have all sorts of

interesting coordinates, depending on the orientation and translation of each

aircraft.

4.3.9.3. View Space

View space (also known as camera space) is a coordinate frame fi xed to the cam-

era. The view space origin is placed at the focal point of the camera. Again,

any axis orientation scheme is possible. However, a y-up convention with z

Airport

zW

xW

xM

zM

(5,0,0)M

(–25,50,3)W

(–25,50,8)W

Aircraft:

Left

Wingtip:

Figure 4.18. A lear jet whose left wingtip is at (5, 0, 0) in model space. If the jet is rotated by 90

degrees about the world-space y-axis, and its model-space origin translated to (–25, 50, 8) in

world space, then its left wingtip would end up at (–25, 50, 3) when expressed in world space

coordinates.