A drawing has three horizons

Every perspective drawing is built on a horizon. It is the first line on the sheet, it is where the vanishing points go, and it is where the construction starts.

That is a habit rather than a fact about geometry, and the habit hides a structure that is worth having.

Three sides, three planes #

A box has three edge directions. Each has a vanishing point, and the three points form a triangle. That triangle is the standard route to recovering a camera, and what its sides are is rarely asked.

They are vanishing lines. Specifically: the side joining the vanishing points of axes $i$ and $j$ is the vanishing line of the plane those two axes span.

So the side joining the two horizontal vanishing points is the ground plane’s horizon — the ordinary one. And the other two sides are the vanishing lines of the two vertical coordinate planes, which are perfectly good horizons for anything lying in those planes and which nobody draws.

That the join of two vanishing points is a vanishing line is not a definition — it is checked. A plane’s vanishing line can be computed straight from the camera, as the image of the set of directions lying in the plane, and that computation never mentions a vanishing point. Compared against the join of the two vertices, the two agree to two parts in ten million million.

The orthocentre, which is the fact the site already had #

The principal point is the orthocentre of the triangle, and recovering the camera turns on it: three mutually orthogonal directions give three relations, the three altitudes meet, and the classical fact that they do is what makes the three relations consistent.

Read alongside the sides, the orthocentre says something extra. The altitude from a vertex is perpendicular to the opposite side — which is to say, the perpendicular from the principal point to a plane’s vanishing line passes through the vanishing point of that plane’s normal. That is a sentence about one plane, it is true of every plane, and it is the reason a foot-of-perpendicular keeps appearing in the constructions below.

Each side has its own folded eye #

Every vanishing line carries a rabatted eye: the station point folded flat into the picture about that line.

The rule is the same for all three. Drop a perpendicular from the principal point onto the line, note its length $d$, and the folded eye lands at $\sqrt{f^2 + d^2}$ from the foot, on the far side. For the ground plane’s horizon in a typical layout that comes to 786 pixels against a focal length of 707 and a perpendicular of 343.

Once the folded eye is on a line, every measuring point for a direction in that plane is an arc swung from that direction’s vanishing point. So the three horizons are not decoration: they are the three lines the three measuring points live on, and three-point, laid out with a straightedge is exactly the construction that uses all three.

What the triangle is, as a triangle #

The three vanishing points are treated here as three points that happen to have an orthocentre. They are a triangle, and asking what kind gives two results a draughtsman can use without any camera at all.

It is always acute. The relation $f^{2} = -(\mathbf v_i - \mathbf p)\cdot(\mathbf v_j - \mathbf p)$ must give a positive $f^{2}$ for all three pairs, which requires the orthocentre to see every pair of vertices at an obtuse angle — and that happens exactly when the orthocentre lies inside the triangle, which is exactly when the triangle is acute. So:

If the three vanishing points of a drawn box do not form an acute triangle, no camera made it.

That is a refusal a reader can apply with a straightedge and no arithmetic, and unlike the rectangle’s same-side test it bites on the case that actually occurs: a third vanishing point placed by judgement lands wherever it looks right, and “looks right” has no reason to keep the triangle acute. It is the strongest cheap check this field has on a three-point drawing.

And the focal length is two numbers of the triangle. Writing $R$ for the circumradius and $O$ for the circumcentre, the common value of the three dot products works out to

$$f^{2} \;=\; \tfrac{1}{2}\left(R^{2} - OH^{2}\right) \;=\; 4R^{2}\cos A\cos B\cos C,$$

using Euler’s $OH^{2} = R^{2}(1 - 8\cos A\cos B\cos C)$. So

$$f \;=\; 2R\sqrt{\cos A\cos B\cos C},$$

and the square root is real precisely when all three angles are acute — the reality condition and the acuteness test are the same statement, which is the tidiest way this collection has found to say why a badly placed third point is not merely inaccurate but impossible.

Both forms are worth having for different reasons. $\tfrac12(R^{2}-OH^{2})$ is a compass construction: swing the circumcircle, mark the orthocentre, and the focal length is a length on the page. The trigonometric form says how the answer behaves — $f$ is largest for a nearly equilateral triangle, where $\cos A\cos B\cos C$ reaches $1/8$ and $f = R/\sqrt2$, and falls to zero as any angle approaches a right angle.

That last remark prices the two-point case as a limit rather than a special case. A two-point drawing has its third vertex at infinity, so the triangle degenerates, $R$ runs to infinity and the product of cosines to zero, and $f$ survives as the finite product of the two — which is why the two-point relation is an equation on one line instead of a triangle. A drawing whose third vanishing point is merely very far away is a triangle with one very acute vertex and two nearly right ones, sitting close to the boundary where the square root fails, and it is therefore the layout in which a small misplacement of the third point flips the drawing from possible to impossible. The nearly-two-point drawing is the fragile one, which is the opposite of how the three cases are ranked for difficulty.

It also says what a draughtsman should do with a third point they are free to place. Put it so the triangle is comfortably acute — as near equilateral as the composition allows — and the drawing has the longest focal length its three directions permit, which by the correct viewing distance is the arrangement a reader can stand furthest back from. A third point placed for looks is a viewing distance chosen by accident; a third point placed to keep the triangle round is the best one available.

What follows about the ordinary horizon #

Three consequences, and each of them is a habit repaired.

The horizon is not privileged. It is the vanishing line of the plane most things stand on. A ramp has its own, a roof has its own, and the site already has the ramp has its own horizon as a special case of a general fact that had not been stated.

“The horizon is at eye level” is a statement about the ground. It says the ground’s vanishing line passes through the principal point, which is true when the picture plane is vertical because the vertical is then parallel to the picture plane and its vanishing point is at infinity — which puts the opposite side through the orthocentre.

Tilt the camera and the vertical vanishing point comes in from infinity, the triangle closes up, and the ground’s horizon stops passing through the principal point. Nothing was lost; the triangle simply became finite on all three vertices.

And a degenerate triangle is a real refusal. Point the camera along one of the axes and that axis’s vanishing point is at the principal point, the other two run to infinity, and the triangle has no interior. The recovery refuses, and the construction refuses too, and both refusals are the same fact: a one-point drawing contains two directions, not three, and no amount of care extracts a third.

The triangle’s shape says what the camera was #

The triangle is not free. Its shape is the camera, and reading it is a one-line exercise once the orthocentre is known.

An acute triangle has its orthocentre inside it, and that is the case in which the three focal lengths come out real. An obtuse one puts the orthocentre outside, and the relation then asks for the square root of a positive number — a refusal, and the honest report is that the three drawn bundles cannot be three mutually perpendicular directions.

So the acuteness of the triangle is a test on a drawing, performable with a straightedge and no arithmetic: if the three altitudes meet outside, the drawing depicts nothing rectangular. That is a strong statement and it is nearly useless in practice, for the reason the quadrilateral no rectangle casts works out in detail — in the layouts anybody draws, the triangle is enormous and violently acute, and it takes a gross distortion to make it otherwise.

A larger triangle means a longer focal length. Push the three vertices out and the orthocentre relation returns a bigger number; bring them in and the camera is wider. That is the same statement both vanishing points on the paper makes about two of them, extended to three, and it is why a compact triangle is a picture with a short correct viewing distance.

The plane at infinity, if it is wanted #

There is a tidier way to say all of this and it is worth one paragraph.

The directions in space form a plane — the plane at infinity — and a camera images it onto the picture, once and for all. A direction’s vanishing point is its image under that map; a plane’s vanishing line is the image of the line where that plane meets infinity; and three mutually perpendicular axes are three points of the plane at infinity forming a self-polar triangle with respect to the absolute conic on it.

Everything above is that one map read in the picture. The triangle is the image of a self-polar triangle, the orthocentre relation is the polarity, and the three horizons are the images of three lines.

That formulation is more powerful and it is not more useful for drawing, which is why the triangle gets an essay and the plane at infinity gets a paragraph. A draughtsman has a straightedge and three vanishing points; the polarity is what explains why what they do works.

Two of the three horizons are usually off the paper #

Worth stating plainly, because it explains why the structure is invisible in practice.

In an ordinary tilted view — a camera looking up at a building, say — the two horizontal vanishing points are far out to the sides and the vertical one is far above or below. The ground’s horizon runs across the picture and is the only one of the three sides that crosses the frame at all. The other two run from a point far to the left up to a point far above, passing nowhere near the sheet.

So a draughtsman never sees two of the three. What they see is one horizon and a third vanishing point, and the natural reading is that the third point is a separate object placed by judgement — which is exactly how it is taught and exactly the mistake the third point put where it looks right measures.

What this does not say #

It says nothing about finding the triangle on a real photograph. Every vanishing point here is exact; a fitted one has a residual, and a triangle fitted from three bundles of drawn edges has three, whose spread recovering the camera reports rather than averaging away.

It says nothing about which side to build on. All three are equally valid vanishing lines and a construction has to choose one per run; the choice is decided by which plane the run lies in, and where a run lies in two planes at once — every axis does — either of the two sides through its vertex will serve and they give the same answer.

It says nothing about scenes without boxes. A picture of a hillside has no three orthogonal bundles and therefore no triangle, and a picture with nothing straight in it is what to do instead.

And it does not claim the two vertical horizons are useful to draw. They are useful to have, because the measuring points sit on them; drawing them puts two more lines across a sheet that already has a great many, and a draughtsman constructs the measuring point and leaves the line implicit.

Where the three horizons are already being used without a name #

Three constructions in the drawing office turn out to be this structure, and each is taught as its own recipe.

Shadow vanishing points. The vanishing point of a shadow direction and the vanishing point of the light direction lie on a line through the vanishing point of the plane the shadow falls on — which is to say, on the receiver’s own vanishing line. Where shadows vanish computes the arrangement; read here, it is one more plane with one more horizon.

Reflections in a floor or a wall. A reflection is the view from a camera on the far side of the mirror, and the mirror plane’s vanishing line is where the two views’ shared directions land. It is a horizon that belongs to the mirror rather than to the ground.

And the true shape of a cut. An auxiliary view is a picture on a plane chosen to be parallel to something; choosing it is choosing which vanishing line is to run along the sheet, which is the same decision as choosing which of a triangle’s sides to fold the eye about.

None of those three needed the triangle to be drawn, and none of them is usually presented as being about a vanishing line at all. What they share is that a construction which looks like a rule about points turns out to be a rule about a line, and the line belongs to a plane in the scene rather than to the sheet of paper. Once that is seen, the recipe stops needing to be remembered: the plane is in the picture, its vanishing line is where its directions land, and everything the recipe does follows.

A fourth line, which is not a side #

One more line in the picture deserves naming, because it is the one a construction is set out along and it is not a vanishing line.

The trace of a plane is where that plane meets the picture plane. Its image is itself — every point of it is in the picture plane already — and it is drawn at true scale, which is what makes it the line a ruler is entitled to be used on.

A trace and a vanishing line are parallel in the picture and they are different objects. The vanishing line is where the plane’s directions land; the trace is where the plane itself crosses the glass. A plane has one of each, and a construction uses both: true lengths along the trace, directions to points on the vanishing line.

So a fully described plane in a perspective drawing has three things attached to it: a trace, a vanishing line, and a rabatted eye on the vanishing line. The first says where the plane is, the second says which way it faces, and the third sets the scale on which measurements are carried between them.

The transferable form #

A construction’s scaffolding is usually an object. Ask what the scaffolding is and the special case you were taught turns into one member of a family, with the same rule on every member.

The horizon was scaffolding — the line drawn first, where the vanishing points go. Asking what it is makes it the ground plane’s vanishing line, which makes every plane have one, which makes the rabatted eye and the measuring point rules general rather than ground-specific, which is what let the third axis be constructed at all.

The same move is available on the rest of the drawing office’s furniture. The ground line is the trace of a plane. The distance point is a measuring point for the diagonal. The station point is a rabatted eye. Each of them is a member of a family with a rule, and each of them is normally taught as one line with a name.