The circle whose centre moves
Draw a wheel in perspective. The rim is an ellipse — that much every book says, and it is right. Now put the axle in.
Almost everyone puts it at the centre of the ellipse. That is wrong, and the size of the error is the subject of this essay.
Why they are different points
The reason is the same one that runs through the whole of this subject: a projection destroys ratios, and being a centre is a statement about a ratio.
The centre of a circle bisects every diameter. In the picture, the near half of a diameter that runs away from the viewer is magnified relative to the far half — the near end is closer to the eye — so the image of the midpoint is not the midpoint of the image. It sits further from the viewer than halfway, which in a ground-plane circle means nearer the horizon.
That is all there is to it, and it is not a small effect. The offset in the figure above is 3.6% of the ellipse’s width at a comfortable distance, and it grows as the circle comes closer or gets larger relative to its distance.
The one case where they coincide
The offset vanishes exactly when the circle’s plane is parallel to the picture plane. Then the projection restricted to that plane is a pure scaling about a point, ratios along every line in the plane are preserved, and the image of the centre is the centre of the image.
That is why a wheel photographed dead side-on has its axle at the centre of the rim, and why the error is invisible in an elevation drawing. It is also why the error is easy to internalise wrongly: the case where the rule “axle at the centre” works is the case people learn to draw first.
Every other orientation has the offset, and the offset is a function of the tilt. There is no threshold below which it stops mattering; there is a distance beyond which it is small enough not to see, which the curve above locates.
How the measurement is made
Getting a number out of this requires more care than it looks, because both quantities being compared have to come from the drawing rather than from the scene.
The image of the circle’s centre is easy: project the world centre. The centre of the image ellipse is not — it requires the ellipse, and the ellipse has to be recovered from the drawn curve rather than assumed.
So the drawn curve is sampled at 300 points, and a general conic
is fitted to them as the smallest-eigenvector problem of the design matrix, using a Jacobi eigensolver written for the purpose. The conic’s centre then follows from solving a 2×2 system in A, B, C, D, E, and the discriminant 4AC − B² says whether the fit really is an ellipse.
Two details of that turned out to matter.
The points are centred and scaled before fitting. Without it, the design matrix mixes terms of order x² — tens of thousands on a figure-sized canvas — with terms of order 1, its condition number runs into the millions, and the fitted centre wanders by more than the offset being measured. The measurement would then be reporting the conditioning of the fit rather than the geometry.
The fit is checked against its own points. The largest algebraic residual over the sampled curve is required to be below 10⁻⁶, which a genuine conic through genuine conic points achieves easily and a bad fit does not. Without that check a fit that had quietly converged to something else would produce a plausible centre and a plausible number.
Which way it moves, and by how much
The direction is always the same: the image of the centre lies further from the viewer than the ellipse’s centre, which for a circle on the ground means closer to the horizon.
The magnitude depends on two things and not on anything else. It grows with the circle’s angular size — a large circle nearby has more difference between its near and far halves — and it grows as the circle’s plane tilts away from the picture plane.
For the ordinary cases: a wheel of a metre or so, seen from two or three metres, in a 40° view, the offset runs at 3 to 5% of the drawn width. That is several pixels on a screen and a millimetre or two on a page. It is above the threshold of noticing and below the threshold of obviously wrong, which is the worst possible place for an error to sit — visible enough to make a drawing feel subtly off, small enough that the cause is never identified.
What this means for drawing round things
The rule to replace “the axle goes at the centre” is not much harder to apply.
The minor axis of the drawn ellipse points at the vanishing point of the circle’s own axis. For a wheel on the ground, that is the vertical direction, so the minor axis is vertical in a level view. This one is exact and it is the most useful of the three, because it fixes the ellipse’s orientation without any measurement.
The axle sits on the minor axis, offset toward the horizon from the ellipse’s centre. Not at the centre.
The tangent points are not at the ends of the axes. The points where the drawn ellipse is tangent to the lines from the vanishing point are the images of the circle’s extreme points, and they do not coincide with the ellipse’s own axis ends.
Each of those is a consequence of the same fact — that the projection’s effect on the circle is a projective map, not an affine one — and each is routinely got wrong in drawings that are otherwise careful. The third is why a drawn cylinder’s silhouette lines so often fail to meet its end ellipse tangentially.
Pole and polar, which is the proper statement
The projective account of what has happened is cleaner than the arithmetic, and it explains why nothing about the centre survives.
A conic defines a correspondence between points and lines: every point has a polar line, and every line has a pole. The centre of a conic is the pole of its line at infinity. That is a definition entirely in projective terms, and it is preserved by projection — the pole of a line maps to the pole of the mapped line.
So the projection does preserve the pole–polar relation. What it does not preserve is which line is at infinity. In the world, the circle’s centre is the pole of the ground plane’s line at infinity. In the picture, that line has become the horizon, an ordinary line partway up the frame. The image of the circle’s centre is therefore the pole of the horizon with respect to the image ellipse — not the pole of the picture’s own line at infinity, which is what the ellipse’s centre is.
Two different lines, two different poles. The offset between them is the whole phenomenon, and it goes to zero exactly when the horizon of the circle’s plane is itself at infinity — which is the case where the plane is parallel to the picture plane, and is also the case of a parallel projection, where the centre does survive.
That gives a construction as well as an explanation. Given the drawn ellipse and the horizon of the circle’s plane, the image of the centre is the pole of that horizon, which can be constructed with a straightedge from any two chords. No measurement, no arithmetic, and no reliance on the eye.
The related error in ellipse templates
The commonest way circles get drawn in perspective by hand is with an ellipse template, chosen by “degree” — the angle at which the circle is being viewed — and this carries an error of its own worth naming.
A template ellipse has a fixed axis ratio, which corresponds to a circle seen under parallel projection at a stated tilt. Under perspective the image is still an ellipse, but its axis ratio varies across the picture: the same physical circle at the edge of a wide frame images with a different ratio than at the centre, because the angle between its plane and the line of sight is different there.
So a template gives the right shape only where the circle is near the optical axis, and the discrepancy grows toward the frame edge for exactly the reason a wide frame stretches shapes at its edges. In a narrow view the two agree closely and the template is fine; in a wide one it is not, and the wheels at the sides of the drawing come out too round.
A note on what is not claimed
The offset described here is geometric. It says where the axle must be drawn for the picture to be a correct projection, and nothing about whether a viewer will notice if it is not.
Very often they will not. Pictorial depth perception is tolerant, the viewer is rarely at the correct viewpoint anyway, and a wheel with its axle at the ellipse’s centre reads as a wheel. Draughtsmen have got this wrong for centuries and produced work nobody complains about.
What the geometry says is narrower and still worth having: that the drawing is not a projection of the object it appears to depict, that the departure is computable, and that its size is 3 to 5% in ordinary cases rather than an unknown quantity to be argued about. Whether that matters is a decision, and it is a decision that can only be made after the number exists.
Arches, vaults and the reason this was noticed early
The people who found this out were not drawing wheels. They were drawing buildings, and specifically the round-headed arch, which is the commonest circular object in European architecture and the one that most obviously fails when it is drawn wrong.
An arch in perspective is a semicircle in a vertical plane, and the same offset applies with the roles turned: the image of the arch’s centre is displaced along the minor axis away from the viewer, and the springing points — where the arch meets its piers — are not at the ends of the drawn ellipse’s major axis. A draughtsman who constructs the arch by drawing a plausible half-ellipse between the piers gets the curve’s shape roughly right and its relation to the piers wrong, which shows as an arch that seems to sit unhappily on its supports.
The construction that fixes it is old and is the plane version of what the whole of this site does. Enclose the circle in a square, project the square — its diagonals, its midlines and its centre all being straight-line constructions that a projection preserves — and use the projected square to place the eight points where the circle meets the square’s sides and diagonals. The circle is then drawn through eight correctly placed points rather than fitted between two.
The reason it works is exactly the reason cross-ratio works. A square’s diagonals meet at its centre, its midlines pass through the meeting point, and each of those is an incidence, which is what a projection preserves. Nothing in the construction requires a length or a midpoint to survive, so nothing in it breaks.
Cylinders, and the tangency nobody checks
Combining the above gives the most-failed drawing in technical illustration: a cylinder seen at an angle.
The cylinder has two end circles, each imaging to an ellipse, and two silhouette lines running between them. The silhouette lines are the images of the two lines on the cylinder’s surface that are tangent to the eye — and they must be tangent to both end ellipses, at points that are not the ends of either ellipse’s major axis.
That last clause is where drawings fail. Joining the major-axis ends of the two ellipses gives a shape that reads as a cylinder and whose sides cut across both ends rather than touching them. The error is small in a narrow view and unmistakable in a wide one, and it compounds with the edge stretch of a wide frame so that cylinders at the sides of a wide drawing look distinctly wrong even to people who cannot say why.
The correct tangent points are found the same way as everything else here: they are the images of the tangency points on the world cylinder, which are computed and then projected. Constructed rather than judged, the silhouette meets each ellipse at one point and the drawing closes.
What to take from it
Three things, in decreasing order of how often they are needed.
The minor axis of a drawn circle points at the vanishing point of the circle’s axis, exactly, always. This is the single most useful fact about drawing round things in perspective and it costs nothing to apply.
The centre of the drawn ellipse is not the image of the circle’s centre, and the gap is 3 to 5% of the drawn width in ordinary views. Anything that has to sit at the circle’s centre — an axle, a keystone, a boss — goes at the pole of the horizon, not at the middle of the shape.
And the general form, which is the reason both of the above are true: the projection preserves the pole–polar relation and does not preserve which line is at infinity. Every property of a conic defined without reference to the line at infinity survives the journey; every property defined with reference to it, which includes the centre, the axes and the distinction between ellipse, parabola and hyperbola, does not.
Where this sits among the other losses
The circle’s centre is one item on a list, and it is worth putting it back beside the others because the list has a pattern.
A projection destroys length, angle, area, the ratio in which a point divides a segment, the property of being a midpoint, and the property of being a centre. Every one of those is defined by a ratio, and a ratio is what the perspective divide destroys — which is the whole content of the invariant essay.
What survives is defined by incidence: three points being collinear, three lines being concurrent, a conic passing through five points, and the cross-ratio, which is a ratio of ratios built so the divides cancel.
So the wheel’s axle joins the midpoint of a segment and the fraction of a figure’s height cut by the horizon as things that are only where they look under a parallel projection or a level picture plane. And the repair is the same repair every time: express the thing wanted as an incidence, and construct it. The pole of the horizon is the axle, the diagonals of a square meet at its centre, and neither construction needs a length to survive anything.
That is why this site’s checks are built from four operations and no others. There is nothing else a projection leaves intact to build with, and the ones it does leave are enough — including, as it turns out, enough to recover the camera that made the picture.