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the horizon correct from 16 cm, at 160 mm wide cross-ratio 1.366025
Helix
1
The stair that turns has a vanishing point that moves
1 essay · construction
plane — line, 5e-18 cylinder — sinusoid, 6e-17 stereographic — circle, 6e-16 equidistant — circle, 9e-4 equal-area — circle, 2e-3 equirect. — sinusoid, 1e-4 tilted 10°, sampled over 140° of azimuth the horizon swings ±10°
Horizonshape
1
The horizon's shape belongs to the surface
1 essay · curved
a plan, looking down: the object on the left, the frame, the hook on the right the hook the frame, edge-on 160 cm plan at 0.25 px per mm · a hook 160 cm behind a 56 cm frame correct from 46 cm at 160 mm wide
Instrumentdrawing
1
The hook is the centre, and the eye is not
1 essay · construction
the horizon the centre, 318 focal 622.396 px true 622.396 px
Involution
4
Perpendicular is a pairing
1 essay · foundations
elevation 0.0000 two equal, orthographic cabinet 0.5000 two equal, oblique cavalier 1.0000 all three equal, oblique dimetric 0.4714 all three different, orthographic isometric 0.8165 all three equal, orthographic ← trimetric 0.5479 all three different, orthographic smallest of the three axis scales measured from each projection
Isometric
3
What isometric actually means
1 essay · parallel
horizon v_z the horizon misses the imaged circle, so the two points are a conjugate pair protractor on the paper: 80.43° · cross-ratio: 90.000000° correct from 21 cm, at 160 mm wide 42° across
Laguerre
4
An angle is a cross-ratio
1 essay · foundations
angle, worst over the sphere 4.4e-8° anisotropy, worst 1.000000023 area scale, largest over smallest ×255 what a reader calls distorted the third row, not the first the disc is 160° of the sphere drawn to 160° off axis the first two rows are conformality
Littleplanet
5
Conformal is not undistorted
1 essay · curved
horizon equal steps halve the taper by worst error, in metres of depth equal steps to the horizon 14.72 m halve the remaining gap 228.13 m taper by eye 350.80 m correct from 26 cm, at 160 mm wide 34° across
Measuring
3
The measuring point, and the step the method leaves out
1 essay · construction
front view · 32 filled top view · 36 filled side view · 36 filled the largest solid with these views — 192 cells and a solid with the same views — 64 cells 6 × 6 × 6 cells the three views are identical; the solids differ by 128 cells
Multiview
3
Three views do not fix the solid
1 essay · parallel
column 62 mm correct from 19 cm, at 160 mm wide 0.966% hidden, to 15.3° from the nadir
Nadir
1
The hole a rig cannot fill
1 essay · curved
to the vertex, 450 px further correct from 22 cm, at 160 mm wide 5 courses · 1e-13 px
Offboard
1
Dividing to a point off the board
1 essay · construction
d = 0 d = 0.5 d = 1 d = 2 d = 4 d = 0 straight 48.06° angle ×17.80 area d = 0.5 4.81% bend 23.08° angle ×4.66 area d = 1 7.21% bend 16.84° angle ×2.65 area d = 2 9.61% bend 23.13° angle ×1.57 area d = 4 11.54% bend 32.71° angle ×1.73 area a general straight line · worst angle · area range, over the field 130° across angular minimum at d = 1.00
Panini
3
One parameter between two surfaces
1 essay · curved
elevation x 1.000 y 1.000 z 0.000 cavalier x 1.000 y 1.000 z 1.000 cabinet x 1.000 y 1.000 z 0.500 isometric x 0.816 y 0.816 z 0.816 dimetric x 0.943 y 0.943 z 0.471 axis scales measured from the drawing all 5 preserve midpoints
Parallel
2
Parallel projection is not primitive perspective
1 essay · parallel
2 4 6 8 -1 -0.500 0 0.500 1 position along the family the two pictures leave free how far the solid is from the scene's own shape (stretch ratio) the scene every member redraws both pictures to 7e-16 m the ambiguity is a family of solids, not a tolerance
Parallelrecovery
5
What two parallel views leave free
1 essay · parallel
paper construction and projected shadow agree to 1e-16 isometric one angle: 41.76°
Parallelshadow
3
The shadow rules that hold here
1 essay · parallel
S₁ S₂ a b c lean 24° · centre at 0.45 along the join probe closes to 1.2e-13 px
Perspectivity
1
Every projectivity is two perspectivities
1 essay · foundations
correct from 20 cm, at 160 mm wide verticals converge 3.59° · horizon 213 px off centre
Planechoice
3
The plane is a choice
1 essay · construction
what was drawn the solid it depicts — reading 1 of 2 x y z looked at 45.00° off the normal cube edge 1.0000 of the drawn unit residual 5e-16
Pohlke
5
Any three lines you draw are a cube
1 essay · parallel
0 0.500 1 10 20 30 40 50 distance from the camera to the wall (m) relative to the value at 1 m the patch, growing as d² the light per unit area, falling as 1/d² their product — what the picture records 2500× the footprint at the far end product flat to 2e-16
Radiance
2
A wall does not get darker as it goes away
1 essay · light
plane — ×24.99 edge to centre cylinder — ×2.40 edge to centre stereographic — ×2.22 edge to centre equidistant — ×1.30 edge to centre equal-area — ×1.00 edge to centre equirect. — ×1.34 edge to centre 36 cells of 377 square degrees, out to 70° equal-area ×1.00 · plane ×25.0
Resolution
1
Where a surface spends its pixels
1 essay · curved
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