Calculations

Calculations

• Unit changes on reading

  Standard Surface values converted to the app's units.

  thin film thickness ÷ 1000; emission ÷ 0.02; opacity = (r + g + b) ÷ 3

  MaterialX files: numbers written and read shading/mtlx.js

  • BimCraft
  • ModelCraft
  64-bit

• Tile size from uvtiling

  The material's tile size in meters from the uvtiling shared by a file's images.

  width = 1 ÷ tiling_u; height = 1 ÷ tiling_v

  MaterialX files: numbers written and read shading/mtlx.js

  • BimCraft
  • ModelCraft
  64-bit

• Mean of an input that varies over the tile

  The single value given to a shader input the renderers and the file cannot vary across the surface.

  mean = Σ f(uᵢ, vⱼ) ÷ 64², a 64 × 64 grid of texel centers

  MaterialX files: numbers written and read shading/bake.js

  • BimCraft
  • ModelCraft
  64-bit

• Page box and rotation matrix

  The matrix from PDF user space to the page as displayed: origin at the lower-left corner, y up, the page's rotation applied.

  0°: [1 0 0 1 −x₀ −y₀]; 90°: [0 −1 1 0 −y₀ W + x₀]; 180°: [−1 0 0 −1 W + x₀ H + y₀]; 270°: [0 1 −1 0 H + y₀ −x₀]

  PDF page vectors and snapping to them pdf-vector.js

  • CadCraft
  • BimCraft
  64-bit

• Matrix product

  Combines the current matrix with one the page's content applies.

  [a b c d e f] = m · k for 2 × 3 affine matrices

  PDF page vectors and snapping to them pdf-vector.js

  • CadCraft
  • BimCraft
  64-bit

• Path points

  Each move, line and curve point of a path, taken through the current matrix into displayed-page points.

  x′ = a·x + c·y + e; y′ = b·x + d·y + f; quadratic to cubic: c₁ = p + ⅔(q − p), c₂ = x + ⅔(q − x)

  PDF page vectors and snapping to them pdf-vector.js

  • CadCraft
  • BimCraft
  32-bit in part

• Text position, height and angle

  Where each run of text starts, how tall it is, its angle and its width, in displayed-page points.

  height = |M·(t₂, t₃)|; angle = atan2 of M·(t₀, t₁); width = |w| × |(m₀, m₁)|

  PDF page vectors and snapping to them pdf-vector.js

  • CadCraft
  • BimCraft
  64-bit

• Flattening curves to segments

  Turns every line and curve of the page into straight segments for snapping.

  B(t) = u³p₀ + 3u²t·c₁ + 3u·t²·c₂ + t³p₃, u = 1 − t, t = 1⁄8 … 8⁄8

  PDF page vectors and snapping to them pdf-vector.js

  • CadCraft
  • BimCraft
  32-bit in part

• Nearest point on a segment

  The point of a segment closest to the pointer.

  t = ((p − a)·(b − a)) ÷ |b − a|², held to [0, 1]; q = a + t(b − a)

  PDF page vectors and snapping to them pdf-snap.js

  • CadCraft
  • BimCraft
  32-bit in part

• Crossing of two segments

  The point where two segments of the page cross, or nothing when they are parallel, apart, or only meet end to end.

  den = r × s; u = ((c − a) × s) ÷ den; v = ((c − a) × r) ÷ den; point = a + u·r

  PDF page vectors and snapping to them pdf-snap.js

  • CadCraft
  • BimCraft
  32-bit in part

• Choosing the snap point

  Picks the end, crossing, middle or nearest point within the snap distance of the pointer.

  midpoint = (a + b) ÷ 2; candidate kept when |candidate − p| ≤ tol

  PDF page vectors and snapping to them pdf-snap.js

  • CadCraft
  • BimCraft
  32-bit in part

• Drawing units per PDF point

  The scale that makes a PDF page its true size in the drawing's units.

  1 pt = 1⁄72 in; inches: 1 ÷ 72; feet: (1 ÷ 72) ÷ 12; millimeters: 25.4 ÷ 72; meters: 0.0254 ÷ 72

  PDF underlay placement and PDF import pdfmodel.js

  • CadCraft
  64-bit

• Underlay scale and rotation for saving

  The insertion point, x and y scale and rotation of an underlay as a DWG stores them.

  sx = |(m₀, m₁)|; sy = |(m₂, m₃)| × sign(det); rotation = atan2(m₁, m₀)

  PDF underlay placement and PDF import pdfmodel.js

  • CadCraft
  64-bit

• Underlay read from a file

  Builds the underlay block and its placing reference from a file's PDF underlay record.

  PDF underlay placement and PDF import pdfmodel.js

  • CadCraft
  64-bit

• Scale typed when attaching

  The scale an attached page gets: its true size, or the number typed.

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Inverse of a placement

  Takes clip points picked in the drawing back into the page's own points.

  det = a·d − b·c; inverse = [d −b −c a (c·f − d·e) (b·e − a·f)] ÷ det

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Is this curve a circular arc

  Decides whether one cubic curve of the page is an arc of a circle, and if so gives its center, radius and angles.

  circle through B(0), B(½), B(1); arc when | |B(t) − c| − r | ≤ 0.002·r at t = 0.25, 0.75, 0.1, 0.9

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  32-bit in part

• Circle through three points

  Center and radius of the circle through three points.

  d = 2(aₓ(b_y − c_y) + bₓ(c_y − a_y) + cₓ(a_y − b_y)); center from |a|², |b|², |c|² ÷ d

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Outline points of an imported shape

  The vertices of each imported path in drawing coordinates, with curves replaced by straight pieces.

  B(t) at t = 0.1 … 1.0, then (x, y) → placement matrix

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  32-bit in part

• Stroke width kept on imported objects

  The line width stored on each imported line, arc and polyline.

  lw = round(width × scale × 100) ÷ 100

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Arcs and circles from curves

  Turns a path made only of circular curves into a circle or an arc object.

  same circle when |rᵢ − r₀| < 1e-3·r₀ and |cᵢ − c₀| < 1e-3·r₀; circle when Σ sweep = 2π within 1e-3

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  32-bit in part

• Imported text

  Position, height and angle of each imported text.

  h = PDF height × scale ÷ FONT_EM; angle = (PDF angle + placement rotation) mod 2π

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Scale from two points and a real distance

  Rescales an imported page so that two picked points are a stated distance apart.

  f = real ÷ |p₂ − p₁|

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Scaling an object about a point

  Moves and resizes one imported object by the factor f about the insertion point.

  x′ = aₓ + (x − aₓ)·f; r′ = r·f; h′ = h·f

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Rigid fit from point pairs

  The rotation and shift that best take the picked points of one scan onto their targets, with the error left at each pair.

  R from the top eigenvector (a quaternion) of Horn's 4 × 4 matrix; t = c_dst − R·c_src; rms = √(Σ rᵢ² ÷ n)

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Eigenvalues of a symmetric matrix

  Eigenvalues and eigenvectors of the 3 × 3 and 4 × 4 symmetric matrices the fit and the normals use.

  Jacobi rotations: θ = (a_qq − a_pp) ÷ 2a_pq; t = sign(θ) ÷ (|θ| + √(θ² + 1))

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Combining transforms

  The product of two 4 × 4 transforms, used to add each iteration's step to the running transform.

  o = a · b, each entry a sum of four products

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Inverse of a rigid transform

  The transform that undoes a rotation and shift.

  R⁻¹ = Rᵀ; t′ = −Rᵀ·t

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Transform in shifted coordinates

  Rewrites a transform for coordinates measured from the working origin, and back.

  t_local = t + R·o − o; back: t = t_local − R·o + o

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Is a matrix a rigid transform

  Checks that a matrix is a pure rotation and shift before it is accepted.

  |rᵢ·rⱼ − δᵢⱼ| ≤ tol; |det − 1| ≤ tol; last row 0 0 0 1 within tol

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Rotation about an axis

  The 4 × 4 matrix for a turn by an angle about an axis through a center.

  R = c·I + s·[axis]ₓ + (1 − c)·axis·axisᵀ; t = center − R·center

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Angle and shift of a transform in words

  The rotation angle in degrees and the length of the shift that a result is reported with.

  angle = acos((trace − 1) ÷ 2); shift = |t|

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Working points of a scan

  The thinned copy of a scan's points, measured from the session origin, that registration works on.

  work = pos + cloud origin − session origin, every stride-th point

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Nearest point in the fixed scans

  For a moved point, the closest point of the fixed scans within the match distance.

  d² = Δx² + Δy² + Δz², smallest over the grid cells searched ring by ring

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Surface normals

  The direction square to the surface at each fixed point, from its 10 nearest neighbors.

  normal = eigenvector of the smallest eigenvalue of the neighbors' covariance

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Matching and the error of a step

  Pairs each moving point with its nearest fixed point and reports the root-mean-square distance, and the distance along the normals.

  rms = √(Σ d² ÷ matched); plane rms = √(Σ ((p − q)·n)² ÷ matched)

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Point-to-plane step

  The small turn and shift that reduce the distances along the normals in one iteration.

  minimize Σ ((ω × p + t + p − q)·n)²: 6 × 6 normal equations for (ω, t); R = rotation by |ω| about ω

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Solving the 6 × 6 system

  Solves the linear equations of the point-to-plane step.

  Gaussian elimination with partial pivoting, then back substitution

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• When the iteration stops

  Decides when a phase has settled and narrows the match distance for the next step.

  stop when |previous − error| ≤ tol × max(error, 1e-12) or error < 1e-12; cut = min(start, max(2.5 × rms, 1e-6 × diagonal))

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Label position and text height

  The point where a remnant's number is written and the height of that text.

  d = chamfer distance in cells (steps 1 and 1.4142); x = (i + 0.5) × cell; y = (j + 0.5) × cell; h = max(cell, min(0.6 × d × cell, height ÷ 4))

  Where a remnant's number is written addons/apparel/lib/nest/nest-stock.js

  • CadCraft
  • PartCraft
  32-bit in part

• Gray value of each engraving cell

  The average gray of the picture's pixels under each cell of the engraving grid.

  gray = Σ w × pixel ÷ Σ w, w = overlap area of pixel and cell

  Burn levels of an engraved picture laser/laser-image.js

  • CadCraft
  • PartCraft
  32-bit in part

• Burn level of each cell

  Turns the gray values into burn levels from 0 to 255 by the chosen method.

  darkness = 1 − (gray ÷ 255)^gamma; threshold: darkness > t; gray steps: round(round(darkness × (n − 1)) ÷ (n − 1) × 255)

  Burn levels of an engraved picture laser/laser-image.js

  • CadCraft
  • PartCraft
  32-bit in part

• Adding a layer

  Creates a layer of 1 to 16 numbers per element.

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• Storing a value in a layer

  Writes the values of one element of a layer.

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• Blending corner values onto new faces

  The values new face corners get when faces are cut or joined, as a weighted mix of old corners.

  out = Σ wᵢ × valueᵢ

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• Layers in the mesh file

  The layer data as it is written into the file.

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• World triangles for export

  Every triangle of the model in world coordinates, with its normal.

  p_world = M · p; n = normalize((p₁ − p₀) × (p₂ − p₀))

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF vertex positions

  The vertex coordinates and normals stored in the binary part of a .glb file.

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  32-bit in part

• glTF placement matrix

  The matrix that places each group or component in the .glb file.

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• STL triangles in millimeters

  The triangles of a binary STL file.

  x_mm = x × 1000, stored as a 32-bit float

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  32-bit in part