# Help finding coordinate for left of screen in this frustum

I have a frustum with l=-2, r=2, n=0.5, f=10, corresponding to left, right, near, far respectively. I also define top and bottom too.

I've set the camera eye up at (0, 0, -2.5), looking directly at (0, 0, 0) with up (0, 1, 0).

Suppose I want to position an object centred at z=-1. I want to find the x coordinate so that under this frustum it appears exactly centred on the left edge of the screen.

I set up the following diagram to help me:

                       +z
|
\          |          /
-x<_____\_________|_________/______>x
\____x___|-1      /
\       |       /
\      |      /
\_|l|_|     /
\    |    /     |near dist=n=0.5
\   |   /      |
\  |  /       |
\a| /        |
\|/         |
|-2.5
|
-z


So to find x I can use simple trig. Since tan a = |l|/n and tan a=x/1.5 then the formula for x is x=|l|/n*1.5=2/0.5*1.5=6, where |.| is the absolute value.

But when I use this x value to draw an object left of the centre, it does not appear centred at the edge of the screen. I can see the object but it's not on the edge. Moreover, if I increase n then the gap between the object and the left edge increases. What am I doing wrong?

UPDATE

Empirically I have discovered that if I compute x=|l|/n*2.5 then my object is correctly centred on the left edge of the device screen, and works for any n value I choose. Not sure why 2.5 works...

SOLUTION

Not really worthy of an answer, but I found in some legacy code that the renderer was set up to transalte the z forward by 1 unit, so this was undoing my -1 z value, hence why multiplying by 2.5 worked and not 1.5.

Despite this annoying fact, I'm now confident I fully understand how the frustum fits into OpenGL.

• In your text you write "exactly centered on the left edge" and in the drawing you show it centered in the left half of the screen. – Bram Dec 6 '18 at 23:24
• @Bram well, the camera is centred at (0,0,0) so (-x,0,0) should be exactly on the left edge of the screen. As you can see the left side of the frustum skims the left of the near plane so x should correspond to the left edge of the screen for z=-1. – Antinous Dec 7 '18 at 7:52

0.5*left / n = x / 1.5
-1 / 0.5 = x / 1.5
-2 = x / 1.5
-3 = x


Or more elaborately, if you want to go via the angle it self:

tan( a ) = 0.5*left / 0.5
a = atan( -2 )
tan( a ) = x / 1.5
tan( atan( -2 ) ) = x / 1.5
-2 = x / 1.5
-3 = x

• But tan a is not equal to 0.5*left/0.5 ... it is equal to l/n... goimg by tan = opposite / adjacent – Antinous Dec 7 '18 at 7:47
• Maybe the drawing was a little confusing, please see update. Nonetheless the x value does not draw at the exact left edge of the screen. I'm sure my maths is right? So there must be some unkown i dont know about. Will take a look again. – Antinous Dec 7 '18 at 7:59