OpenGL : equation of the line going through a point defined by a 4x4 matrix ? (camera for example)

Question

I would like to know what is the set of 3 equations (in the world coordinates) of the line going through my camera (perpendicular to the camera screen). The position and rotation of my camera in the world coordinates being defined by a 4x4 matrix.

Any idea?

[Something like this](https://stackoverflow.com/questions/15697273/how-can-i-get-view-direction-from-the-opengl-modelview-matrix)? — genpfault, Mar 14 '18 at 16:13
Not the exact answer I need :) What does it give under the form ax + by + cz + d = 0 ? — toto_tata, Mar 14 '18 at 16:24
That's a plane equation. Two of that kind make a 3D line as the intersection of two planes. — Ripi2, Mar 14 '18 at 16:48
Not sure how this is directly related to OpenGL. Perhaps the tag `3d` would have been more useful. — meowgoesthedog, Mar 14 '18 at 18:11
I corrected my question. I removed the 'ax + by + cz + d = 0' mention. Indeed, the equation of a line in 3D is a set of 3 equations. — toto_tata, Mar 14 '18 at 18:32
No. A line in 3D is defined by two independant equations. Three equations, if independant, give you a point. — Ripi2, Mar 14 '18 at 20:55

score 1 · Answer 1 · answered Mar 15 '18 at 08:30

parametric line is simple just extract the Z axis direction vector and origin point O from the direct camera matrix (see the link below on how to do it). Then any point P on your line is defined as:

P(t) = O + t*Z

where t is your parameter. The camera view direction is usually -Z for OpenGL perspective in such case:

t = (-inf,0>

Depending on your projection you might want to use:

t = <-z_far,-z_near>

The problem is there are many combinations of conventions. So you need to know if you have row major or column major order of your matrix (so you know if the direction vectors and origins are in rows or columns). Also camera matrix in gfx is usually inverse one so you need to invert it first. For more info about this see:

Understanding 4x4 homogenous transform matrices

OpenGL : equation of the line going through a point defined by a 4x4 matrix ? (camera for example)

1 Answers1