Themes
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.
The surface is a choice
A picture has to be cast onto something, and the something is decided rather than given. Flat keeps straight lines straight and cannot reach 180°; a cylinder reaches all the way round and bows every horizontal; stereographic keeps every angle and no scale. The price of each is a number, and the corner where a surface pays nothing is empty.
One picture, one length
A photograph gives up every ratio in the scene and no size at all: scale the world and the eye together and not a pixel moves. So a measurement from a single view is a ratio until the reader supplies one length, and which length that is decides how good the answer can be.
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.
Not a projection at all
Every theorem quoted on this site is a theorem about a map through a centre. Water is not one; a real lens is not one. Rather than mention that and move on, these essays measure what survives and what does not — a cross-ratio 1.2% out where the pinhole is exact to the last bit, a bundle of rays that misses its own centre by ten millimetres, and the two things that survive anyway.
Fitted, not assumed
A distortion coefficient recovered from bent lines with no calibration target. A principal point computed instead of guessed to be the middle of the frame. A pivot offset read off the parallax it leaves behind. Each is the site's round trip pointed at a quantity that is usually filled in from a manual, and each comes with the conditioning number that says how well the data actually determined it.
What a second picture adds
One view fixes a ray and a set of ratios. A second fixes a point, the whole three-dimensional shape, and where the other eye was — recovered from the two drawings and nothing else. It does not fix a size, and it does not fix itself when the eyes are close together. Each of those is measured here rather than stated.
Determined up to something
Every recovery on this site returns an answer with a qualifier attached. One picture gives ratios and no size. Two give shape and no size. Many give a scene and a camera track up to seven numbers no quantity of pictures supplies. The qualifier is not a shortcoming to apologise for; it is the exact statement of what the pictures contain, and here it is counted, walked along and shown to be free rather than merely small.
Projected twice
A photograph is a projection onto a sensor and then a projection from a screen into a room, and the second one has a correct point exactly as the first does. Every essay before this quoted its viewing distance against an assumed figure width; here the assumption is replaced by a chain — focal length, sensor width, display width — and the two projections are compared. Almost nobody is standing where the picture says to, and what that does is already measured.
Made by something finite
The departures in these fields are not optical. Depth is kept in a fixed number of bits, the picture is a grid of samples that need not be square, and the frame is read over an interval and mostly a row at a time. None of that changes what a projection is; all of it changes what this particular picture is a projection of, and each one has a closed form and a control.