Visual Plane
every picture states the point it is correct from
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The figure library — page 2

Page 2 of 4, continuing through the generators in the same alphabetical order.

fixed-structure

centrefaint dots: before · solid: afterhomologyratio 2.4000

flat-against-curved

flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one

flat-port-narrows-the-view

05010015050100150field of view in air (degrees)field of view behind a flat port, into water (degrees)no port at allthe ceiling: 97.2°114° → 78°90° → 64°63° → 46°46° → 34°28° → 21°n = 1.333a 90° lens becomes a 64° one

floor-anamorph

floorpicture planeeye level — no mark above thiseye · 1.62 m up, 2.40 m backthe ground line, seen from abovethe mark runs to 3.00 ma point 1.62 m up casts no mark at all

forced-perspective

eyefaint: the row it imitates · solid: the row that is builtbuilt row21.0 pxthe deep row28.4 pxdisparity across the row, two eyes 63 mm apartfar column cut to 0.300the eyes read 74.1%, not 22.2%

format-and-angle

full frame · 39.6°APS-C · 26.6°Micro Four Thirds · 19.6°1 inch · 15.0°phone (1/1.7″) · 8.7°one 50 mm lens · the angle is a property of the rectangle behind itMicro Four Thirds: 2.00× diagonal, 2.08× wide, 1.85× tall50 mm across five formats39.6° down to 8.7°

four-centre-vs-true

true ellipse — axis ratio 0.5774four arcs — 5.72% short along the major axisthe four tangent points are exactworst departure 5.72% of the semi-major axis

four-poses-one-of-them-visible

points in front of both camerasR₁, t0 of 44180° from the truthR₁, t reversed0 of 44180° from the truthR₂, t44 of 44chosenR₂, t reversed0 of 44180° from the truththe chosen pose is the true one to 0.0e+0°cheirality, not algebra, picks it

great-circle-image

fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples

half-pixel

one pixel is an areacentrescornersprincipal point moves0.707 pxfocal length changes by4.5e-13 pxan edge-versus-centre viewport1.303 pxa half-pixel convention is a principal-point error; an off-by-one viewport is a focal-length error8 vertices, all shifted by the same 0.7071 pxspread across marks 0.0e+0 px

harmonic-construction

the diagonals against a ruler, at 3.2 mthe diagonals — exactthe ruler — 22.7 px outcorrect from 23 cm, at 160 mm wideharmonic set -1.000000 · 8e-14 px

height-becomes-position

0204060800123how far the figure stands from the point under the eye, metresdrawn length of a 1.7 m figure, pxa figure under the eye is drawn at zeroidentical figures at increasing radiusthe fit is exactly linear

height-error-uncorrected

5.405.505.60-0.400-0.2000k₁ of the lens the photograph was taken withheight recovered from the photograph (m)the true 5.6 m5.48 m5.6 m tall, 11 m away, on a level camera2.15% out — against 0.14% for a 2.2 m object

height-from-one-picture

horizon — the eye's own heightbase — 0 mhorizon crossing — 1.62 mtop — 3.40 m recoveredthe vertical vanishing point is 8586 px above this framerecovered 3.400 m · true 3.400 m1.8 cm per pixel of click error

horizon-is-eye-level

horizon = eye level, 1.62 m91.01%correct from 26 cm, at 160 mm widespread 0

how-wrong-is-the-cube

0.50011.502-10010difference between the two by-eye placements (points)side ratio of the box the drawing depicts (1 = a cube)a cubeeven-handedthe one free choice in the taught methodand it decides the whole solid

interpolation-error

-0.400-0.200000.2000.4000.6000.8001position across the drawn surfacehow far along the real surface, minus how far along the drawn one(√k−1)/(√k+1) = 0.5195at the page's midpoint, 40.9%depth ratio 10 : 1peak 0.5195 at s = 0.760

k1-k2-valley

01020-0.200-0.10000.1000.200distance from the fitted coefficients, along each directionstraightness residual (px)stiff directionsoft directioncondition number 813k₁ and k₂ correlate at -0.9969

keystone-quad

15° of yaw, 6° of pitch, 1.50 throw ratio81.3% of the panel reaches the corrected rectangleouter: the thrown quadrilateral · inner: what correction can keep18.7% of the panel discarded

lamp-from-the-shadows

the shadow lines meet below the horizon — a lamp in the roomhorizoncorrect from 25 cm, at 160 mm wide3 posts · foot 351 px below the horizon

lamp-is-a-second-eye

correct from 26 cm, at 160 mm wide34° across

lit-fraction-curve

0204051015distance from the source to the ball, in radiifraction of the ball's surface that is lit (%)one half — the source at infinity25.0% at 2 radiicurve: the closed form · dots: quadratureagreeing to 5e-4

little-planet

angle, worst over the sphere4.4e-8°anisotropy, worst1.000000023area scale, largest over smallest×255what a reader calls distortedthe third row, not the firstthe disc is 160° of the spheredrawn to 160° off axisthe first two rows are conformality

loosening-the-measure-test

02.5057.50-8-6-4-20log₁₀ of the tolerance on midpoint driftsystems counted as preserving measureperspective admitted at 15.6%the exclusion, sweptthe boundary is measured, not chosen

matrix-and-pinhole

x/z, y/z — the pinholeM·p, then divide by wworst disagreement 4.0e-14 px over 8 verticescorrect from 21 cm, at 160 mm wide42° across · near 0.1 m, far 1000 m

measuring-point

24VPcorrect from 26 cm, at 160 mm wide34° across

midpoint-is-the-difference

halfway along the drawn linethe actual midpointcorrect from 26 cm, at 160 mm wide22 px apart

mirror-anamorph

the sheet, seen from abovemirrorthe eye, 2.4 radii upwhat stands up in the mirrorthe light path reverses to the eye to 1e-15scale varies 7.2× across the design

miss-against-section

0246801020metres of the eye's track the section coversrms miss of the best centre, mmeasuredthe closed formleast-squares fit against √12 of the track lengththey agree to 1e-9 m

one-bad-match

00.50011.50010203040correspondenceepipolar error at every OTHER point, after one match is movedmedian 0.54 pxthe moved oneevery point wrong: median 0.54 px, worst 1.7 pxclean fit 2.3e-13 px

one-ray-two-points

where the two points landthrough the tank: 102.2 px apartthrough nothing: one marknear point, 0.9 m105.8 pxfar point, 6.0 m208.0 pxthe two, apart102.2 pxat 30° off axis, through 12 mm of glass into waterno single viewpoint — the rays miss by 102.2 px of splitno warp of the image can undo a depth-dependent shift

one-two-three-point

horizon3 vanishing points at a finite distance2564 px · 8129 px · 531 px

panini-family

d = 0d = 0.5d = 1d = 2d = 4d = 0straight48.06° angle×17.80 aread = 0.54.81% bend23.08° angle×4.66 aread = 17.21% bend16.84° angle×2.65 aread = 29.61% bend23.13° angle×1.57 aread = 411.54% bend32.71° angle×1.73 areaa general straight line · worst angle · area range, over the field130° acrossangular minimum at d = 1.00

parallax-from-the-wrong-centre

the far field — where the stitch was fitted2.2 m — 3.2 px out24 m — 0.3 px outthe sky registers to 1e-13 pxthe foreground does not — up to 3.2 px

parallel-does-not-move

horizonisometric — the same drawing, movedperspective — a different drawingcorrect from 23 cm, at 160 mm wideparallel: 4e-14 px · perspective: 87.2%

parallel-has-no-station-point

isometric — the limit2.5 m6 m20 m200 msame box, same drawn sizethe eye recedes

penumbra-from-an-area-light

source, 35 cmthe occluder's edgefraction of the source visiblepenumbra 17.5 cmprojection: 17.50 cmsampled: 17.41 cm

photograph-of-a-photograph

an anchorcorrect from 19 cm, at 160 mm wideflat 1e-13 px · rolled 30.5 px

picture-runs-away

0246850100150field of view across the picture (degrees)half-width of the picture, in focal lengthsplanecylinderstereographicequidistantcut off at eight focal lengthsthe plane crosses it at 166°

pinhole-crossover

00.2500.5000.7500246distance from the hole to the ground (m)how much of the patch of light is an image of the suncrossover at 0.54 mthe sun subtends 9.30 mradpatch 61 mm wide at 6 m

pixel-aspect

pixels 2.00 : 1recovered focal length219.9 pxthe camera's actual one436.4 pxwhat the recovery returnsspread across three estimates 0.0e+0worst bundle residual 9.1e-13 pxboth are what a wrong picture would trippixel aspect 2.00 unmodelledfocal length 49.6% short, every diagnostic clean

plan-and-picture

plan, looking downpicture planeone camera, two views of it34° across

plane-fidelity-table

elevation1.0000a circlecavalier1.0000a circleisometric0.57741 : 1.732military0.57741 : 1.732the xy plane's drawn ellipseratio of the ellipse's axes, sampled

plumbline-fit

fitted k₁ = -0.280000true -0.280000, off by 5e-15

prewarp-pair

k₁ = -0.32 · the round trip closes to 1.2e-12 pxpale: what the eye receives · dark: what the renderer drawsthe inner 86% of a 72° frame, where the inverse is exact

rays-that-do-not-meet

the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m

receiver-shapes

a flat floor3e-13 mma homology — four points determine ita dished floor5.67 mmnot a homologya ridged floor9.07 mmnot a homologya floor with a step74.95 mmnot a homologyworst error of the predicted pointsexact at the four fitted, on every surface

recover-the-camera

recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across