Light and mirrors
A shadow is a second projection
Shadows as projection from the light, why the shadow of a straight line is straight, and how a point lamp and the sun are the two cases of one construction.
Where shadows vanish
The vanishing point of shadow directions, why it sits on the horizon for sunlight, and how it is used to check a picture's lighting.
A mirror is a second camera
Reflections computed two ways, the handedness reversal between them, and why a mirror swaps neither left-right nor up-down.
The penumbra is the lamp's image
An area light as a family of centres of projection, the penumbra as the source's own image cast through the occluding edge, and the distance at which a hole stops projecting itself and starts projecting the sun.
The shadow of a ball is a conic
A sphere's cast shadow as the section of its tangent cone, with the ellipse-to-hyperbola crossing at the height of the ball's own top.
The lamp, out of the picture
Recovering a lamp's three-dimensional position from a single picture's shadow lines, and the under-determination one post leaves.
A shadow can be un-cast
The shadow map as a homography, inverted to recover an occluder's outline, with the cost of a misplaced lamp and of a curved receiver measured.
A wall does not get darker as it goes away
The cancellation that makes image brightness independent of distance, and the floor illuminance where the inverse square does bite.
A lamp lights less than half a ball
The terminator as a small circle rather than a great one, the lit fraction (1 − R/D)/2, and the bulge that reads the source's distance out of a picture.
The floor that is not a plane
What the shape of the receiving surface costs a shadow's recovery, measured across four floors with the same lamp and the same occluder.
A shadow across an edge
A shadow falling across two planes as two homologies glued along their common line, the kink that measures the crease, and what a single fitted map does to the half it does not describe.
The lamp is the second eye
Depth from a single photograph of a lit scene, treating the lamp as the second camera of a stereo pair, with the conditioning that follows from the angle between the two rays.
A light far enough away
What happens to the recovery of a light's position as the light recedes, why the sun is a limit rather than a special case, and where the picture stops carrying evidence that the light is finite.
Two lamps and one map
The map between the two shadows one object casts from two lamps, why it is a homothety rather than a general projectivity, and what its ratio says about the lights.
A wire with a corner in its shadow
The shadow of a smooth space curve has a cusp wherever the lamp lies on one of the curve's tangent lines, which is a surface rather than an accident.
The edge of a shadow is drawn on the object
The contour generator — where a surface turns away from the lamp — is what casts a shadow's edge, and it is a different curve from the object's drawn silhouette.
A hole is not preserved
What a projection keeps of a caster's shape, in the only terms a projection respects — connectedness survives, holes do not, and the ring's transition has a closed form.
A shadow across a second object
The shadow of a straight edge is a plane, so every point of the curve it draws on a second object is recoverable from its image — and the conditioning is a separate number from the exactness.
The ball stands at a focus
The contact point of a ball and its floor is a focus of the ball's own shadow, and the lamps consistent with that shadow are a hyperbola whose vertices are its two foci.
The residual has a shape
Reading the mispredict of a four-point shadow map as a field rather than a scalar, three shape statistics that separate three receivers at matched worst case, and the obvious statistic that does not.
The floor is a choice of coordinates
Why every measured failure of a curved receiver is a failure of the coordinates rather than of the shadow, and what a crease costs that a curve does not.
The lamp comes out in rays and not in plan
Why one half of the classic lamp recovery is exactly insensitive to the floor's shape and the other half is not, and what a construction that assumes a ground plane costs when there is not one.
The corners a floor cannot add
Why the number of corners in a shadow is a property of the lamp and the caster rather than of the receiver, the crease that is the exception, and the measurement that tells the two apart.
A floor is read along curves
How much of a floor a set of shadows actually touches, why the coverage grows more slowly than the curve length, and what a reading along curves can and cannot say about a surface.
How many lamps a drawing has
Cutting a bundle of drawn shadow lines into pencils, the leverage that makes a pixel of clicking five pixels at the answer, and why a model-selection penalty over-counts where a stated noise level does not.
The distance at which two lamps part
The resolution limit of a shadow-line lamp count, measured as a success rate over repeated drawings rather than as a threshold, and fitted against the noise to a straight proportionality.
A floor cannot fake a second lamp
Why the ray family of a lamp recovery is exactly insensitive to the receiving surface, demonstrated by a search over floors that returns one number, and what can imitate a second light when a floor cannot.
The drawing does not run out of lines
How many lamps a single drawing can support, counted as unknowns against constraints, and why that count is the wrong one — the failure is in the discrete assignment, which no counting of degrees of freedom can see.
A lamp behind the camera
The lamp recovery run on a light behind the camera, why the drawn lines meet at a point the projection would not have produced, and the third case a five-century-old rule about the horizon has no entry for.
Counting shadows is not counting lamps
Why the number of connected shadow regions on a floor is not the number of sources in the room, in both directions, with the arrangement where the two counts agree as the control and the umbra as a third answer.
The arrangement the count cannot see
Whether the count of lamps in a drawing depends on how the posts are arranged, the null result it returns, and the control that makes the null result a finding.
A soft shadow on a curved floor is not the lamp's image
What a dished or stepped floor does to the soft-edged band a lamp with a size casts, measured against the plane floor it recovers exactly.
The lamp's size over its distance, and nothing else
What one straight-edge shadow can and cannot say about a lamp's size, and what a second card at a different height recovers that it could not.
How many lamps make one lamp
How many point lamps it takes to counterfeit an area lamp's soft shadow, and the measured rate at which the imitation converges.
What a point in shadow can see of the sky
How much of an overhead source a shadowed place still receives, computed as a solid angle by the same construction that measures a lamp's penumbra.
The lamp a low shadow cannot locate
Why locating a lamp from posts and their shadows gets rapidly worse as the lamp gets low, measured from the pencil of lines itself.
Where a shadow splits in two
Why a non-convex object's shadow splits into pieces at a specific lamp height, found by a tangency along a curve rather than counted by hand.
How many shadows determine the object
How many outlines it takes to pin down a convex shape, the fitted rate it happens at, and the exact concavity that no number of them ever reaches.
A dent breaks the terminator
Why a dimple in an otherwise convex ball splits its lit boundary into two differently-caused curves, and the elevation at which the second one closes.
A shadow edge read as a profile
Sweeping a straight edge's shadow across an object turns a lamp, a stick and a camera into a depth instrument, and its conditioning is set by one angle.
The lamp and the floor cannot both be recovered
A continuous family of lamp-and-floor pairs that all cast the identical shadow, and the three kinds of outside fact that collapse it to the truth.