The thread: Projected, not constructed — page 2
The penumbra is the lamp's image
The soft edge of a shadow is a picture of the light, projected through the occluder's edge as through a pinhole. That gives its width without any integration — and it is why the dapples under a tree go crescent-shaped during an eclipse.
The other systemsWhich 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.
Drawn confidentlyThe third point put where it looks right
Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.
Where to standAn anamorph at true size, on paper
An anamorph is the shadow of the intended picture, cast from the reader’s own eye. Every claim about a figure on a screen is quoted against an assumed display width, because nobody can know how wide a screen shows it; this one is not, because it ships a sheet in millimetres and states where to put an eye.
What survivesA projection of a projection
Photograph a photograph and the composite map is a homography — four marks determine it and every other lands where they say, to 1e-13 px. Roll the print and the same four mispredict the rest by 30.5 px, which is why straightening a page of a thick book never quite works.
The other systemsAny 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.
Light and mirrorsThe shadow of a ball is a conic
A cone cut by a plane is a conic section — which is the definition of the phrase, not an analogy — and the shadow of a ball is exactly that cut. So the shadow closes or runs to the horizon depending on where the lamp is, and the condition is not what anyone guesses: it is that the lamp is above the top of the ball. The ball's distance appears nowhere in it.
Constructing a viewBrunelleschi drilled a hole in his panel
The first perspective demonstration in the European record came with its viewing point enforced — a hole through the back of the panel, a mirror held out in front, and one place to stand. That distance is computable from the panel's size and the angle the Baptistery subtends, and the answer lands squarely on the arrangement the account describes.
Where to standThe cylindrical mirror unbends it
Put a mirrored cylinder in the middle of the sheet and the light path from the eye to the paper bends once. The map that results is not a homography and not a projection in the plane sense at all — it varies its scale by a factor of seven across the design, which is why the smear is unreadable and why the mirror can put it back.
The other systemsOblique 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.
What survivesThe diagonals find the middle
Three taught methods for spacing a receding row are all wrong, and the best of them misplaces a post by three and a half metres. There is a fourth, it needs no measurement and no vanishing point, and it is exact at every camera and every depth — because 'the diagonals of a rectangle cross at its centre' is a statement about which lines meet where, and that is the one kind of statement a projection cannot damage.
Surfaces that are not flatThe arcs a curvilinear drawing uses
The taught way to draw a very wide view by hand is to run every straight edge of the world as a circular arc. That recipe has been repeated for sixty years without a surface attached to it, and it turns out to name one exactly — fitting a general conic to the image of a straight line returns a circle to nine decimal places under stereographic projection and returns nothing like a circle under any of the other standard picture surfaces.
Drawn confidentlyWhich way the drawn circle leans
Two rules are given for the direction a drawn circle's short axis runs in: along the axis of the cylinder, and pointing at the centre of vision. On the optical axis both are exactly right. At the edge of an ordinary frame the first is out by three and a half degrees and the second by seventy-nine. The statement neither of them is comes out of differencing the projection, and it matches the drawn ellipse to a hundredth of a degree.
What a machine computesFour numbers and a window
A projection matrix is built from six numbers and one of them is not a number at all. Four sides carry the focal length and the principal point; the near and far planes move nothing a reader can see; and the bottom row, (0, 0, 1, 0), is the only place the depth divides — set it to (0, 0, 0, 1) and the same machine draws a parallel projection.
Systems that kept the measureA picture with two eyes in it
Several traditions draw the floor from one place and the people on it from another. No single camera produces both, as an earlier essay showed. What such a picture actually is has a measurement attached: give the rays their world points and ask for the one place they all pass through, and at a stride of separation the best answer misses them by six tenths of a metre.
The rectangle behind the lensThe centre has an area
Every theorem of perspective follows from a projection through a point, and no instrument has one. Give the pupil a radius and each world point images as a disc — whose centre is the pinhole's mark to 5.7 × 10⁻¹⁴ millimetres, at every aperture, every distance and every field angle. The geometry survives exactly; only the sharpness is spent.
What survivesTwo triangles and the line nobody drew
Desargues' theorem is the reason a hand-drawn shadow construction closes. An object and its shadow are two figures in perspective from the lamp; the theorem says their corresponding sides meet, pairwise, on one line — which is the ground line. So the closure a draughtsman treats as confirmation that the work is accurate is a theorem they cannot violate.
Constructing a viewThe plane is a choice
A projection has a centre and a surface, and they move independently. Keep the eye and turn the picture plane and every point of any scene lands where one 3×3 matrix says, to 2.5e-13 px. Move the eye instead and the matrix fitted to four points is exact at those four and out by 32.0 px everywhere else. The first is a homography of the picture; the second is parallax, and nothing about the picture can undo it.
Drawn confidentlyThe forty-five degree shadow
Draw the shadow at forty-five degrees and make it as long as the object is tall. In the plan that is exactly a sun halfway up the sky. Applied on the paper it puts four posts under four different suns — altitudes from sixteen to twenty-nine degrees, azimuths thirty-two degrees apart, and shadows between one and three-quarters and three and a half times the height. No drawing angle brings them together.
What a machine computesA tile is an off-centre frustum
Rendering a picture in tiles is exact, and the way to do it is one line of arithmetic: a tile's sides are the whole frustum's sides read at the tile's own pixel bounds. Aiming the camera at each tile instead is defensible at every step and is a different picture, out by about a tenth of a tile whatever the tile size.
The real instrumentThe hole a scene actually sees
The stop is not the centre of projection. Model a 50 mm lens with its stop 18 mm behind the glass and the chief rays from every object distance cross the axis at one point 28.1 mm on the other side of the lens — 10.1 mm from the stop and 1.56 times its size — to 3.6 × 10⁻¹⁵ mm. That point is the entrance pupil, and it is where a picture is a projection from.
The other systemsThe 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.
Where to standA floor anamorph is three numbers
An anamorph is described everywhere as a picture stretched until only one viewpoint can read it, which says what it looks like and nothing about what it is. Cast one onto a floor and fit a map to the marks, and the map turns out to be a planar homology — a line of fixed points, one point off it, and a single ratio. Those three numbers are not a description of the eye. They are the eye.
Constructing a viewStraightening does not move the eye
Correct a photograph's converging verticals and what comes out agrees with a level camera at the same point — one the correction was never shown — to 3e-13 px, with the verticals parallel to 0e+0°. The cross-ratio of four points along a ground line reads 1.3333 before and after, so the corrected picture measures exactly what the original measured, from exactly where the original was taken and nowhere else.