“I wasn’t trying to reinvent the wheel. I was just trying to roll it my way.”

So I was sitting with my notebook, trying to understand how the orthographic projection matrix works. I knew that it was just a mapping to NDC

I had seen the matrix in OpenGL docs a multiple dozen times:

| 2/(R-L)      0           0          -(R+L)/(R-L) |
| 0           2/(T-B)      0          -(T+B)/(T-B) |
| 0            0         -2/(F-N)     -(F+N)/(F-N) |
| 0            0           0               1       |

But instead of just copying it blindly, I wanted to feel where it comes from. So I thought:

“Let me just map near and far (N and F) to -1 and 1… that should work.”

So here’s what I tried:

  1. I wanted the near plane N to go to -1, and the far plane F to go to +1.
  2. To do that, I:
    • Added half the range: (F - N)/2
    • Subtracted F
    • Divided by (F - N)/2

Let’s verify with our example values:

  • For z = 2 (near):
    • Add half range: 2 + 2 = 4
    • Subtract F: 4 - 6 = -2
    • Divide: -2 / 2 = -1 ✅
  • For z = 6 (far):
    • Add half range: 6 + 2 = 8
    • Subtract F: 8 - 6 = 2
    • Divide: 2 / 2 = 1 ✅

On paper it looked like this:

For mapping a value z from [N, F] to [-1, 1]:

\[z_{ndc} = \frac{2z - (F + N)}{F - N}\]

if you expand this even further:

\[z_{ndc} = \frac{2}{F - N} * z - \frac{(F + N)}{(F - N)}\]

Then I realized…

Wait — this is exactly what that orthographic matrix is doing, just written in matrix form so the GPU can crunch it in real-time.

Here’s How You Actually Derive the Z-Mapping Row

Let’s say you want f(z) to map:

  • N → -1
  • F → +1

So:

\[f(z) = Az + B\]

Solve two equations:

  • -1 = A*N + B
  • 1 = A*F + B

Subtract to eliminate B:

\[2 = A*(F - N) \Rightarrow A = \frac{2}{F - N}\]

Plug back in to find B:

\[B = -1 - A*N = -\frac{F + N}{F - N}\]

Boom:

\[z_{ndc} = \frac{2}{F - N} * z - \frac{F + N}{F - N}\]

Which matches the orthographic matrix’s Z-row.

Final Thoughts

Sometimes, detours make the best lessons.
I just wanted to understand what the heck it was doing.

Stay curious. Keep breaking things.
Even projection matrices. 😄