Affine map — where it appears
Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
The 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.
The second eye is a shear
A picture drawn from two eyes is a picture drawn from one, of a different room. The map that puts the second eye away holds the picture plane still point by point and carries one centre onto the other, and the member of its family that matters turns out to be affine — a shear along the line joining the eyes, growing with depth, which is the same operation an oblique drawing performs.
What one picture of a plane determines
A photograph of a tiled floor fixes that floor's geometry up to a projectivity, and no further. Supplying the horizon buys back the midpoint — exactly, and nothing else. Supplying the image of one circle buys the right angle and the ratio of two lengths at right angles. Three stages, three prices, and a quantity a stage does not determine has no value rather than a wrong one.
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.
A parallel floor under a perspective room
Draw the floor without diminution and the people on it with it, and the picture has a centre at infinity glued to a centre in the room. The same map absorbs it — but there is no shear this time, and there cannot be: bringing a point in from infinity is not something an affine map does, so the room the picture is equally a picture of has its midpoints moved as well as its angles.
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.
What a flat map leaves alone
A projectivity of the plane is eight numbers in a matrix, and reading them tells a reader nothing. What it does is decided by its fixed points, and there are exactly three cases: three isolated fixed points, or a whole line of them with one point off it, or a whole line of them with the point fallen onto it. The middle case has five numbers instead of eight and every point slides along a line, and it is what most of the maps this site builds turn out to be.
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.
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 ball a drawing does not draw round
An orthographic drawing of a sphere is a circle wherever the sphere is, and its centre is the image of the sphere's centre, exactly. A cavalier oblique drawing of the same sphere is an ellipse of aspect exactly √2 — and the drawing office reaches for a circle template. One formula covers both and the camera as well, and only the camera moves the centre.
A picture with no eye
The anamorph field measured which parts of an object one eye can paint, and every finding in it is about a point. A parallel projection has no such point, so the same questions have a different shape — and the two that change are the dimension of the answer and the map onto each face. The one that does not change is occlusion, which is routinely assumed away.
Nothing moves along the direction
A part slid four metres toward the reader along the direction an isometric drawing projects along keeps its drawn place to a ten-thousandth of a pixel, and the same slide seen from a station point moves it forty-four. An exploded drawing is not an approximation that works because the parts do not move far — it is an identity, and it is why cutaways are drawn in parallel systems.
One face, one scale
A design carried from a point onto a flat face varies in scale by nearly six across that one face; the same design carried along a direction varies by 1.000000000. The stretch is reported here as the two singular values of the local map rather than as one directional difference, which is the honest form and which the previous round owed.
Copying square by square
The taught grid workflow sets a pavement's cell corners out exactly and then fills each cell by eye, and the corners are right while the fill is not — 3.30 px on a picture 690 across at eight cells, falling as the square of the cell. On a wall square to the camera the same fill reads 3e-13 px, which is why the method feels reliable.
Named alongside it
The objects these essays reach for when they reach for this one.
Parallel projectioncentre of projectionOblique projectionpoint at infinityDemonstrationDrawing systemHomographyPicture planeFree parameterMidpointOrthographic projectionAnamorphosis