Every essay — page 12
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
Mirrors that are not cameras
A flat mirror is a second camera and a curved one is not a camera at all: continue the lines of sight behind a mirror ball and they pass through no common point, so a photograph of it is a projection of nothing from anywhere. What they have instead is a caustic — and one shape, the paraboloid, that does focus exactly, for one bundle of rays and no other.
The corner that answers every eye
Three mirrors at right angles compose into the point inversion, so a ray entering leaves antiparallel to itself whatever direction it arrived from — a picture surface with no correct viewpoint because every viewpoint is correct. Tilt one face half a degree and the worst returning ray is out by exactly one degree, twice the error; the best is out by a twelfth of it, which is why a tolerance quoted from one measured ray is a statement about that ray.
Two mirrors show fewer images than they make
Two mirrors at 55° generate seventy-one images of a point and an eye between them can reach six. The count the field teaches — three hundred and sixty over the angle, less one — is out by as much as sixty-six against the orbit and never by a whole image against what a viewer standing on the bisector actually sees. It is a correct rule about the eye, quoted as a rule about the mirrors.
Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
A symmetric object is its own stereo pair
A building with a plane of symmetry photographed once gives fourteen correspondences whose joining lines meet at one point to 1.9 × 10⁻¹² pixels, a skew-symmetric matrix, and the object's whole shape to fifteen digits — with no mirror anywhere and no second exposure. What it does not give is the size, and the instrument that decides whether any of it applies is the same meeting point, which opens to 12.7 pixels when the symmetry is half a per cent out.
Closer than they appear, by a factor with a number in it
A wing mirror of one metre radius held eighty centimetres from the eye reports 1.30 metres for an object at sixty-four, because the image of anything distant sits half a radius behind the glass. The size such an object subtends reads as a distance 2.60 times the true one, and the factor is exactly one plus twice the eye's distance over the radius — so the warning is a number, and it is larger for the mirror that is further away.
The other systems
Isometric, oblique, elevation. Not perspective done badly by people who had not worked it out yet, but a different answer to a different question — and the difference is measurable.
Parallel 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 7% of the segment's drawn length at a comfortable depth, rising to 13% over the range the slider covers.
What 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.
The 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.
Which axis scales are possible
An orthographic projection's three foreshortening ratios always satisfy one identity — their squares sum to two. Isometric's famous 0.8165 is forced by it rather than chosen, and cavalier's 1, 1, 1 sums to three, which is the arithmetic saying cavalier is not the ORTHOGRAPHIC projection of anything. A later rung shows what the departure is a measurement of.
Any three lines you draw are a cube
Four earlier essays said that cavalier projection is not the projection of anything — because its axis scales sum to three where every orthographic projection sums to two. That is true of orthographic projection and false of projection. Three lines from a point, drawn by hand, are a picture of an actual cube seen from an actual direction, and the cube and the direction come out of the drawing in closed form.
Oblique is a shear, and the shear is the whole system
Cavalier and cabinet are usually introduced as easy perspective for people with a set square. They are not a simplification of anything — they are the answer to a demand no orthographic projection can meet, which is a front face at true size and a depth axis at full length at the same time. The two are locked on a unit circle, and buying both costs exactly 45° of obliquity.
A ruler on an isometric drawing
Isometric drawing has one scale — 0.8165 — and every account of it stops there. But that number is about three directions and a drawing has infinitely many, so a length measured off the paper and divided by 0.8165 comes back anywhere between √(1/2) and √(3/2) of the truth: 29.3% short to 22.5% long, with nothing in the picture to say which.
The drawing does not say which corner is nearer
The Necker cube is filed under optical illusion, as though the flipping were something the eye does. It is not: a parallel drawing of a cube is a drawing of exactly two cubes, mirror images of each other, and they project to the identical picture to the last bit. Perspective rules the second one out at a rate exactly inverse in the eye's distance, and never entirely.
The ellipse the drawing office draws
Every isometric ellipse template is cut to a construction of four circular arcs, and every account of it calls the result an ellipse. It is not one and cannot be: a circular arc has constant curvature and a conic's varies, so the two can agree at four points and nowhere between. The four-centre curve reaches 2√2/3 of the true semi-major axis — 5.72% short — and its minor axis is 3.53% too long.
Three views do not fix the solid
A stepped block on a six-cell grid draws a front, a top and a side view. So does a solid with a third of its material, and so does one with more than the block has — 192 cells against 64, every filled square in all three views identical. The drawing office's triple bounds a part between two solids and does not determine it, and the gap runs to a factor of n.
The view that makes a line a point
Descriptive geometry's first drill is to look at an edge from a direction perpendicular to it, so it draws at true length, and then along it, so it draws as a point. Both are the same identity — the imaged length is the true length times the sine of the angle to the ray — and the sine holds to 1e-12 across the whole sweep of directions.
Nothing moves when the object does
Slide a box 3.2 m across the world and its parallel drawing is the same drawing translated 131.5 px — every edge the same length to 4e-14 px. The perspective drawing of the same move changes its edge lengths by 87.2%. One family's pictures depend on where a thing is; the other's do not, and almost everything the two families disagree about follows from that.
The true shape of a cut
A plane through a box makes a hexagon of 2.8996 m². The front view draws it at 1.9966 m² — the true area times the cosine, 0.6886 — and gets its corners wrong as well, the worst by 15.60°, because a foreshortening scales one direction and not the other. The area is recoverable with one number and the angles are not, which is why a section gets a view of its own.
What two parallel views leave free
Two perspective pictures give shape and no size. Two parallel pictures do not even give shape: the metric upgrade has six unknowns, two views supply six equations, and the system comes out rank five — a one-parameter family of solids that redraw both pictures to 7e-16 m. A third view closes it, and the mirror image survives every count.
The dimetric the set square draws
An orthographic direction has two parameters and produces three axis scales, so the achievable triples are a surface rather than a list. The drawing office's dimetric — one axis at 1 in 8, the other at 7 in 8 — has the right three scales exactly and the wrong two angles, and the picture it makes is an oblique projection of a cube rather than an orthographic one.
What one oblique drawing shows
A parallel projection is a linear map from three dimensions to two, so it has a direction it throws away. Slide any point along that direction and its mark does not move — which makes every parallel drawing the drawing of a three-parameter family of solids, and the depth scale a convention chooses is one direction through the family rather than a boundary of it.
A circle off the coordinate planes
An ellipse template is cut at one ratio, and the ratio is the cosine of one angle: between a coordinate plane's normal and the direction of projection. A face tilted out of that plane needs a different ratio and — the half that gets drawn wrong even when the ratio is close — a major axis pointing somewhere else, perpendicular to the drawn normal rather than to any edge.
The shadow rules that hold here
Drop the foot, run a line from the top at forty-five degrees, take the intersection: the drawing manual's shadow construction is exact in a parallel drawing, to arithmetic noise, at every point of the picture and with one set square. It is where the rule came from, and carrying it into a perspective picture is what broke it.
The drawing that gives the solid back
One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.