What survives
What a projection destroys
The cross-ratio as the invariant of projection, what is destroyed alongside it, and why a necessary condition is not a test.
Where parallel lines meet
Points at infinity, the horizon as the image of the line at infinity, and how a vanishing point is found from drawn lines rather than assumed.
Recovering the camera from the picture it drew
Recovering focal length and principal point from three vanishing points, and what the round trip proves that a projection cannot.
The circle whose centre moves
Why the image of a circle's centre differs from the centre of the image ellipse, how large the offset is, and what it means for drawing wheels, arches and cylinders.
A projection of a projection
The composition of two projections through a plane is a projective map, the theorem that makes rectification legitimate, and the counter-case that shows what the flatness of the middle surface is doing.
The diagonals find the middle
The complete-quadrangle construction as the one by-eye method that survives projection, with the harmonic set at cross-ratio −1.
Two triangles and the line nobody drew
Desargues' theorem checked over forty configurations, and read as the statement that makes shadow and reflection constructions close.
What one picture of a plane determines
The projective–affine–metric upgrade run as three measurements on one photograph, with the cross-ratio as the column that holds throughout.
Four lines have a cross-ratio
Projective duality made to do work — the cross-ratio of a pencil, measurable when the points themselves have left the picture.
The centre, got back out of the picture
Recovering the image of a circle's centre by the pole–polar construction, exact where the ellipse's own centre is nine pixels wrong.
An angle is a cross-ratio
Laguerre's formula reads a world angle out of a photograph as the cross-ratio of two lines with the two lines to the imaged circular points.
The two points a picture hides
The imaged circular points are the whole metric upgrade, and the rectification built from them alone recovers angles and ratios and prints a dash for length.
One conic calibrates the camera
The image of the absolute conic is the same statement as the site's own focal recovery, and it supplies every plane's circular points at once.
What a flat map leaves alone
The three classes of planar projectivity — general, homology and elation — read out of the eigen-decomposition, with the repeated root that comes back to half precision.
Three constructions, one map
A census of the site's plane maps by fixed structure, finding shadows, anamorphs and mirrors to be homologies and rectification not.
The conic a circle becomes
Which conic a circle images as, decided by one incidence with the plane through the eye — measured against the discriminant of the image over a sweep that crosses it, with the parabola found by bisection.
Two lines at infinity
The horizon and the picture's own line at infinity are different lines. Which questions each answers, why the imaged circular points belong to the plane rather than to the circle, and a claim about circles that had to be withdrawn.
Five marks and the sixth
Why five marks determine a conic with nothing left over, what the withheld sixth mark proves, when the solver refuses, and what changes the moment a sixth is added to the fit.
Two circles, one picture
The two-fold pose ambiguity of a circle seen once, derived from the circular sections of the cone through the eye, exhibited as two circles that draw one picture, and controlled against a plane one degree away.
The polar with a straightedge
The complete-quadrangle construction of a polar, checked against the matrix product, controlled against a moved point, and followed to its three refusals — the centre, the inside, and the secant that misses.
The ball at the edge of the frame
The image of a sphere is a conic, elongated radially off the axis, whose centre is not the image of the sphere's centre — computed from the quadric and checked against the traced outline.
Every quadric has one outline
The outline of a quadric as C = adj(P adj(Q) Pᵀ), checked point by point against the traced contour generator, and the singular quadrics where the dual route loses the answer and the point route keeps it.
A line is a space of its own
The projective line as an object in its own right — three pairs determine a projectivity, the fourth point is predicted, and a fourth pair with a different cross-ratio is refused.
The map a row of posts is
The three kinds of projectivity of a line, each realised by an operation the drawing office performs — a bay advanced, a distance doubled, and perpendicular directions paired.
Perpendicular is a pairing
Perpendicularity on a plane's vanishing line is an involution; two pairs determine it, and its fixed points are the imaged circular points, so two ground rectangles calibrate a level camera.
An error with two terms
An instrument that separates the part of an error that responds to effort from the part that does not, run over seven of this collection's own error laws.
What a null result is worth in decades
Measuring what a fit that found no floor is actually claiming, as the largest floor its own sweep cannot rule out, and how that interval closes with the length of the sweep.
The ladder of assumptions is a ladder of conditioning
Measuring what one pixel of noise costs a projective, an affine and a metric quantity computed through the same recovered plane map, against the obliquity of the view.
The bias out of reach
Measuring the bias in a single-view height recovery, why it exists, why it is negligible, and what it takes to turn that into a number rather than a guess.
A floor with a referent
A camera recovery whose error stops falling at a level proportional to the lens distortion the pinhole model cannot see, and why more measurement makes the answer worse.
A point and a line are one object
Duality as a matrix rather than a remark, with a configuration and its dual drawn side by side and every incidence re-measured after the exchange.
The horizon has a pole
The image of the absolute conic as the one polarity a picture does not choose, checked against the camera that drew the picture and then inverted to recover its focal length.
Five tangents name the same conic
The dual conic fitted from tangent lines rather than points, checked against the point fit, and the conditioning of the two compared at equal marking precision.
Desargues read the other way
Desargues' configuration dualised, with the counts, the incidences and the converse all measured on the dual figure rather than argued.
The quadrilateral that finds the middle
The complete quadrilateral as the dual of the complete quadrangle, run on a range, checked against the cross-ratio it must produce and against the input it must refuse.
Four points on a conic look the same from anywhere on it
Chasles' theorem measured on a photographed circle over twenty-two viewpoints, with a fifth point just off the curve as the control.
Three conics are one conic and a choice of horizon
One circle photographed at five pitches draws all three conic types, and the deciding line is not the horizon but the picture's own.
Pascal's line, and the theorem underneath Pappus
Pascal's theorem measured down a family of conics ending in two lines, with Pappus as its continuous limit and six points off a conic as the control.
The triangle a camera cannot move
The self-polar triangle of three perpendicular vanishing points, measured against its departure from square, and the spread that cannot fail.
A curved surface made of straight lines
The two rulings of a hyperboloid of one sheet, their straightness in the picture, and the one conic both families' vanishing points lie on.
A line is a closed curve
Why a projective line is topologically a circle, and what that costs a reader who wants to say one point lies between two others in a photograph.
Three kinds of map on a row of posts
The classification of one-dimensional projectivities, read off three pictures rather than off a discriminant, and what each kind does to a point it is iterated on.
Every projectivity is two perspectivities
Steiner's decomposition carried out with a straightedge, and why it is the reason a straightedge is enough for the constructions this collection is about.
The picture contains what is behind the camera
Why every point of an image plane is the image of two opposite rays, and what that costs a reconstruction that has to decide which side of the eye a point was on.
The theorem that is obvious one dimension up
Why the plane Desargues configuration is the projection of a spatial one, what the lift does not determine, and why collinearity and the axis are two claims rather than one.