The near plane can be any plane

What happens behind the eye is about why a renderer clips before it divides: a point behind the camera has a perfectly plausible image, landing through the principal point on the far side of the frame, and a segment crossing the eye plane is drawn straight, inside the frame, and running in exactly the opposite direction.

The clip is done by the third row of the projection matrix, which maps a point’s depth into a code and lets the clipper compare it against a bound. The row is written to make that bound the near plane, perpendicular to the optical axis at a chosen distance.

It does not have to be perpendicular, and it does not have to be the near plane.

One row, rewritten #

The clip test is that a point survives when its clip-space z is at least −w. So a third row of the form

$$\text{row} = k\,\mathbf{plane} - \mathbf{w}$$

clips exactly on the plane, for any positive k: the test becomes k·(plane · P) ≥ 0, which is the plane’s own inequality.

The w row of this collection’s matrix is (0, 0, 1, 0), so the subtraction lands on the third entry. Writing it on the fourth instead gives a matrix that clips on a plane parallel to the one asked for — and passes every check about where marks are, since the first two rows are untouched either way. That is how it was first written.

The scale k is not free. Fixing it by requiring the far plane to stay where it was, read on the optical axis, is one equation with one root.

The picture does not move at all #

Not by a small amount: by zero. Every point’s pixel is identical, bit for bit, because the first two rows of the matrix were not touched and nothing else in the pipeline reads the third.

Half the scene is being discarded and no ruler laid on the picture can say which half. The marks that survive and the marks that do not are interleaved across the frame, because the plane cuts the scene in depth and the frame is a picture of a three-dimensional arrangement.

That is the separation this essay is for. A projection matrix does two things — it says where a point goes, and it says whether and in what order the point is kept — and the two live in different rows. Everything the foundations field is about is the first. Everything the near and far planes are about is the second. Rewriting the third row exercises the second without touching the first, which is as clean a demonstration of the split as the machinery admits.

Which half survives, and how it is stated #

The plane is given as four numbers, (a, b, c, d), in the camera’s own coordinates, with the surviving half-space the one where a·x + b·y + c·z + d is positive. That sign convention is arbitrary and negating all four numbers keeps the same plane and swaps the halves, which is a thing worth checking rather than assuming: a construction that quietly kept the wrong side would look exactly like one that kept the right side of a plane facing the other way.

Two further conditions are needed and both are checkable. The plane has to have a direction — the first three numbers cannot all be zero — and it has to cross the axis in front of the far plane, or the scale that keeps the far clip in place has no positive root. The second is the interesting one: it is the arrangement in which the plane misses the frustum entirely, and the honest answer is that there is no oblique row to write, not that there is one that does nothing.

Where the plane is perpendicular to the axis, the rewritten row reduces to the ordinary one with the near plane moved, which is the sanity check that costs nothing and is worth having. The construction contains the thing it generalises.

What it costs #

The third row was doing something before it was rewritten, and it was doing it well. Mapping depth to a code that is monotone in distance is what lets a buffer decide which of two surfaces is in front.

The oblique row is not a function of the distance along the axis. It is a function of the distance from the plane, and two points at the same distance from the eye at different heights are at different distances from a tilted plane. So the stored depth no longer orders the scene by depth.

Measured over a scene of scattered points, about a tenth of all pairs come out in the wrong order. Not a rounding: a tenth of every comparison the depth buffer would make, reversed.

For the ordinary matrix the figure is zero by construction, because the row is an affine function of one over the distance and an affine function of a monotone function is monotone. That control is what makes the tenth a measurement rather than an artefact of the sampling.

The plane a reader can move #

Moving the plane along its own normal changes only which points survive, and watching that happen is the clearest statement of the split.

Slide it toward the eye and more of the scene is kept; slide it away and less. The marks that remain are in precisely the positions they were in at every step — there is no shifting, no rescaling, no change of framing — and the count changes. A reader watching the figure move is watching a picture in which nothing moves and the contents change, which is not an experience the rest of this collection offers.

Tilting it does the same thing with a different cut. A steeply tilted plane divides the scene by something close to height; a shallow one by something close to depth. In every case the surviving marks are unmoved and the discarded ones simply are not drawn.

That is worth doing once because the natural expectation is the opposite. A near plane is usually met as the thing that makes objects disappear when a camera gets close to them, and the disappearance feels like an event in the picture. It is not; it is an event in the set of things being drawn, and the picture is a bystander.

What it is for #

The construction is not a curiosity, and the case it exists for is one this site already has in a different field.

A mirror in a scene is rendered by reflecting the camera through the mirror’s plane and drawing the scene again — which is exactly a mirror is a second camera, where a reflection is computed by reflecting the scene and by reflecting the camera, and the two agree once one image axis is reversed. The reflected camera sees the whole room, including everything behind the mirror, which must not appear in the reflection.

The thing that must not appear is bounded by the mirror’s own plane, which is at an arbitrary angle to the reflected camera’s axis. So the clip needed is exactly an oblique near plane, and the alternative — clipping the geometry against the plane before drawing it — is more work and has to be redone whenever anything moves.

The same shape of problem arrives with a portal, a doorway, a window cut in a wall, or any arrangement in which one part of a scene is visible only through an opening in another. In each case the boundary is a plane in the world and not a plane the camera chose.

And the reflection has one more twist in it #

A mirror rendered this way carries a second consequence the collection has already measured, and it is worth joining up.

The reflected camera is a camera on the far side of the mirror plane, and a reflection reverses handedness — so the reflected camera’s basis is left-handed where the original’s was right-handed. A mirror is a second camera records what happens when that is missed: the two routes to a reflection, reflecting the scene and reflecting the camera, disagree by hundreds of pixels until one image axis is reversed.

That reversal and the oblique clip are independent and both are needed. The reversal is about which way the picture runs; the clip is about what is in it. A renderer that reflects the camera, gets the handedness right and forgets the clip draws the room behind the mirror as though it were in front of it — the mirror shows the wall it is hanging on. A renderer that clips correctly and forgets the handedness draws the right things, mirrored.

The two failures look nothing alike, which is fortunate, and they arise from the same operation, which is why they are usually met together.

And why the cost is acceptable there #

A tenth of depth comparisons reversed sounds fatal and is not, in the case the construction is for, and the reason is worth having because it is a good example of a cost that is real and irrelevant.

Behind a mirror or through a portal, the visible scene is entirely on one side of the plane, and it is bounded. The depth range that has to be ordered is the range from the plane outward, not from the eye outward — and the reversals counted above are between points on opposite sides of a tilt, most of which are not both in the visible set.

The residual cost is a loss of precision rather than of order: the codes are no longer spread over the range the way the ordinary row spreads them, so nearby surfaces are harder to separate. That is the ordinary depth-buffer complaint, and the precision a depth buffer has left prices it in the ordinary case. Here it is worse by a factor that depends on the tilt, and it is worse in a way that shows up as flickering between two coplanar surfaces rather than as objects appearing through each other.

Read as a change of what the picture is a picture of #

There is a way of describing the oblique clip that makes both its virtue and its cost obvious at once, and it is the description this collection’s premise suggests.

Every picture on this site states the point it is correct from. The oblique clip does not change that point, does not change the focal length, and does not change the window — so the picture is correct from exactly where it was. What it changes is the scene: the arrangement being projected is now the original with a half-space removed.

So the honest reading is that the matrix is projecting a different scene from the same eye, and the depth row has been spent on saying which scene rather than on ordering it. A projection matrix cannot do both jobs with one row, and the choice of which to do is the whole of the construction.

That framing also says what would happen if the clip were done the other way, by cutting the geometry against the plane before drawing it. The picture would be identical, the depth ordering would be intact, and the cost would be moved from the matrix to the scene — which is exactly the trade, and is why the matrix version is preferred wherever the scene is large and the plane moves.

There is a third reading available too, and it is the one that connects this essay to the field’s own premise. A frustum is a window; an oblique near plane is a second window, at an angle, cut into the same view. The picture is what the eye sees through both, and the second window is the mirror’s frame or the portal’s opening. Nothing about that description needs a matrix, and it is the description a person in the room would give.

The check, and the two things it has to say #

Three claims, and each one fails differently if the row is wrong.

No point moves. If a mark shifts by even a fraction of a pixel, something has leaked out of the third row into the first two, which in a hand-written matrix is entirely possible.

Points on the wrong side really are dropped, and points on the right side really are kept. Both halves, because a row that clipped everything or nothing would satisfy the first claim perfectly.

And the order is genuinely disturbed. If the reversal rate came out at zero, the row would be a re-parametrisation of the original rather than the oblique clip it claims to be — and there is a version of the construction, the one that subtracts on the fourth entry, that produces exactly that.

The third claim also needed its own convergence check, and it got one in the collection’s convergence gate. A rate computed over a scattering of points is a statistic, and a statistic taken over forty points and over nine hundred has to settle before it can be quoted. It does, within a few tenths of a per cent, which is what says the tenth is a property of the arrangement rather than of the sample.

The far plane could have been the oblique one instead #

Symmetry is worth checking, and here it half holds.

The same rewriting could be aimed at the far clip rather than the near one — a plane beyond which nothing is drawn, at an arbitrary angle. The arithmetic is the mirror image and the scale would be fixed by holding the near plane instead.

What is not symmetric is the cost. The near plane is where a depth buffer spends most of its codes, so a row rewritten to serve the near clip disturbs the region the precision was concentrated in. A row rewritten for the far clip would disturb the far region, where the codes are already spread thin and the ordering already coarse — so the same trick costs much less there.

It is also wanted much less. The cases the construction exists for are all near boundaries: a mirror’s surface, a portal’s opening, a wall the camera is looking through. Nothing common is bounded by an arbitrary plane at the far end. So the asymmetry in the cost and the asymmetry in the demand point opposite ways, which is a mildly unfortunate arrangement and is the situation.

What the row can and cannot be made to do #

One limit is worth naming, because the construction looks more general than it is.

The row clips on a plane and only on a plane. It cannot clip on a sphere, a cylinder, a box or any other surface, because a row of a matrix applied to a homogeneous point is a linear form and a linear form’s zero set is a plane. Everything else has to be done by cutting the geometry or by discarding points later.

And there is exactly one such row. A matrix has one third row, so one oblique plane is available, and a scene needing two — a portal seen through a mirror — needs the second one handled another way. That is a hard limit rather than an implementation detail: the projection matrix has four rows and three of them are spoken for.

Which is a small illustration of the accounting four numbers and a window sets out. The matrix is a fixed budget of numbers, each with a job; taking one over for something else means the original job is not being done. Here the original job was ordering the scene, and about a tenth of it is gone.