Every perspective picture is correct from exactly one point in the room.
It is not a figure of speech. A picture is a projection through a centre, and scaling that centre's distance to the width the picture is actually shown at gives a distance in centimetres. Shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm — which is the whole of why wide photographs are said to distort faces. They do not. They are being looked at from four times too far away. Every figure on this site is projected from a camera with a focal length and states the distance it is correct from, and the camera is recovered from the picture afterwards to prove the picture is a projection of anything at all.
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19 essays
What a projection destroys
A projection loses length, angle, area and the ratio in which a point divides a segment. Exactly one quantity comes through untouched, and almost everything that can be checked about a picture is checked with it — including, as it turns out, some things it cannot check at all.
Constructing a viewOne, two and three point are one construction
The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity.
Where to standThe point you have to stand at
A perspective picture is a projection through a centre, and scaling that centre's distance to the width the picture is actually shown at gives a distance in centimetres. Shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm. Nobody stands there, and that single fact explains most of what gets called distortion.
Light and mirrorsA shadow is a second projection
The construction that puts a shadow on the ground is the construction that puts the scene on the picture plane, with the lamp where the eye was. Shadow drawing is taught as a separate set of recipes and it is one operation with the centre moved, which is why the same code draws both.
Drawn confidentlyThe cube that is a box
The two-point cube every book teaches has a step it supplies no construction for. Place the two far edges symmetrically and the drawing depicts a cube for free; place them eight points apart — a difference invisible on the page — and it depicts a box 1.4 times shallower than it is wide.
The other systemsParallel projection is not primitive perspective
Isometric and oblique drawing are not what people used before they worked perspective out. They are a different answer to a different question, and the difference is one measurable quantity — a parallel projection preserves the ratio in which a point divides a segment, and a perspective projection destroys it by up to a quarter of the segment's length.
What survivesWhere parallel lines meet
They meet at a point that is not in the plane, and the horizon is the line made of all such points. Treating that as bookkeeping misses what it buys — a vanishing point becomes an ordinary point of the picture, findable from the drawn lines, checkable, and enterable into any calculation the others are.
Constructing a viewThe measuring point, and the step the method leaves out
Laying out equal depths correctly needs a second vanishing point that most treatments never introduce — the one belonging to the diagonals. With it the construction lands on the projected divisions to eighty femtopixels. Without it, depth is placed by judgement and the drawing depicts something nobody chose.
Where to standWide angle is not distortion
A wide lens stretches shapes at the edge of the frame by exactly 1/cos θ — 3% at 28° across, 41% at 90°. Every bit of that is what a correct rectilinear projection must do, and every bit of it disappears if the picture is viewed from the point it was made for. Nobody views it from there.
The other systemsWhat isometric actually means
The three axis scales are equal to each other. They are not equal to one. Every unit along every axis is drawn at 0.8165 of its true length, which is √(2/3), and a great deal of confusion about isometric drawing comes from the word promising something it does not deliver.
Light and mirrorsWhere shadows vanish
The shadows of parallel posts under the sun meet at a point, and that point must lie on the horizon. Under a lamp they meet at the lamp's foot instead. Both are checkable in any photograph, and a picture whose shadows fail the check was not lit by anything.
Drawn confidentlyDividing depth by eye
Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score.
What survivesRecovering the camera from the picture it drew
Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments.
Constructing a viewThe horizon is at eye level — if the picture plane is vertical
The horizon cuts every standing figure at the same fraction of its height however far away it is, which is the most immediately usable fact in the subject. It holds when the camera is level, and a twelve-degree tilt is enough to spread the fractions by more than a percentage point.
Where to standAnamorphosis is only a viewpoint
A smear on a wall that resolves into a skull when the viewer stands in one particular place off to the side. It is not a trick added to perspective — it is perspective, with the centre of projection put somewhere nobody would naturally stand, and it is the clearest demonstration there is that every picture has such a point.
The other systemsThe eye taken to infinity
A parallel projection is a photograph from infinitely far away with the lens lengthened to match. That is not an analogy — it is the limit, it can be watched happening, and it explains why a long lens flattens a scene and why an isometric drawing has no viewing distance to state.
Light and mirrorsA mirror is a second camera
Reflect the scene and photograph it, or reflect the camera and photograph the scene. The two routes disagree by 906 px and agree to 6 × 10⁻¹⁴ once one axis of the image is reversed — which is the whole of why a mirror is said to swap left and right, written down.
What survivesThe circle whose centre moves
The image of a circle is an ellipse, and the image of the circle's centre is not the centre of that ellipse. The gap is 3 or 4% of the ellipse's width in an ordinary view — large enough to see, small enough to be dismissed as a slip by anyone not expecting it, and the reason drawn wheels look wrong.
Where to standWhen the picture surface is not flat
A flat picture plane keeps straight lines straight and stretches the edges without bound. A cylindrical one spreads the stretch evenly and bends every straight line that is not through the axis. Neither is the distorted one — they are answers to different questions, and the choice decides what a wide view can be.
Threads running through
themes, not chapters
Projected, not constructed
Every point in every figure here comes out of a camera with a focal length, not out of a line run to a vanishing point that was placed where it looked right. A picture no camera could produce fails to build rather than appearing with nothing to say so.
The viewer is in the geometry
A perspective picture is a projection from a point, and that point is a fact about the picture — computable from its focal length and the width it is shown at. Every figure states the distance it is correct from, because leaving it out is what makes the subject feel like a matter of taste.
The round trip
Draw the picture from a known camera, forget the camera, recover it from the drawn edges alone, and compare. Agreement to fifteen digits is a statement about the geometry; anything less is a statement about the drawing.
What a projection destroys
Length goes, angle goes, area goes, and the ratio in which a point divides a segment goes. Knowing precisely what is lost is what makes the one surviving quantity worth anything, so the losses are measured alongside it rather than mentioned.
One machine used twice
A shadow is a projection from the lamp. A reflection is the view from a camera on the far side of the mirror. A parallel drawing is a photograph from infinitely far away. Three subjects usually taught as three recipes are one operation with the centre moved.
Taught, and never measured
The standard constructions are drawn here exactly as they are taught, and then asked what solid or what spacing they depict. Some answers are fine. The point is that the method itself supplies no way to find out, so nobody drawing knows which case they are in.