Projective invariance — where it appears
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
Carrying a height across the room
A known height at one place on the floor, and the same height wanted at another: two lines settle it, and they settle it exactly, at every camera and every pair of positions. What the recipes never mention is that one of those two lines has to be drawn to a point that is usually not on the paper — 3,300 canvas widths away in the case drawn here — and that the repair is not to extend it further.
The circle in the square wants a number
Every manual draws a circle in perspective by inscribing it in a square, crossing the diagonals, and marking four more points about seven tenths of the way out. Seven tenths is right — along the diagonal of the real square. Along the drawn one it is 54 mm off a circle two and a half metres across, and the drawn diagonal is the only diagonal on the paper.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Cross-ratioPicture planeStraightedge constructionAffine mapConditioningConicdegrees of freedomDepth cueDrawing systemDyadic rationalForeshorteningFree parameter