Calculations

Calculations

• Product of two transforms

  Combines two affine transforms into one.

  p′ = p × M (row vectors); (m × n) applies m first

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Inverse of a transform

  The inverse of the matrix an object is painted with.

  det = a·d − b·c; inverse = adjugate ÷ det

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Path coordinates rewritten

  The new numbers of one path-building operator after a move, scale or turn.

  U = C × T × C⁻¹; (x′, y′) = (x, y) × U, written with at most 4 decimals

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Matrix round a moved picture or form

  The cm matrix written before a picture or form XObject that was moved, scaled or turned.

  U = C × T × C⁻¹, six entries with at most 4 decimals

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Is this a move at all

  Decides whether an edit's matrix changes anything.

  |M − identity| < 1e-9 on all six entries

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Line width in the path's own units

  Converts a line width given in page units to the units of the path's own space.

  w = page width ÷ √|a·d − b·c|

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Replacement picture fitted into the old one's box

  Scales a replacement picture to fit, centered, inside the box of the picture it replaces, keeping its own proportions.

  ia = W ÷ H; ba = box w ÷ box h; (sx, sy) = ia > ba ? (1, ba ÷ ia) : (ia ÷ ba, 1)

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Advance of a text run as read

  How far each text-showing operator moves the text position when a page's content is read, kept so a later edit can leave the rest of the line in place.

  string: Σ (width ÷ 1000 × size + char spacing + word spacing on a single-byte space) × horizontal scale; number in TJ: −v ÷ 1000 × size × horizontal scale

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Advance of a text run

  How far a string moves the text position, in unscaled text space.

  Σ (glyph width ÷ 1000 × size + char spacing + word spacing on a space) × horizontal scale

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Padding after reworded text

  The adjustment written after a reworded run so the text that follows on the line does not move.

  k = −(old advance − new advance) × 1000 ÷ (size × horizontal scale)

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Matrix of a moved text run

  The text matrix written before a text run that was moved.

  Tm′ = (Tm × C) × T × C⁻¹

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• Line position put back after a moved run

  Restores where the text line continues after a run on it was moved.

  Tm = line matrix; then advance by −(start + advance) × 1000 ÷ (size × horizontal scale)

  Edits to a page's own content base-content.js

  • MarkupCraft
  64-bit

• A move added to an object's edits

  Adds a move, scale or turn to what an object already had.

  T′ = T_old × T_new

  Edits to a page's own content base-edit.js

  • MarkupCraft
  64-bit

• A moved path point

  The new numbers of a path when one of its points or handles is dragged to a place on the page.

  u = p_page × (C × T)⁻¹; attached handles shift by u − u_old

  Edits to a page's own content base-edit.js

  • MarkupCraft
  64-bit

• Drawn-in paths written to the page

  Writes paths that were drawn into the page, and the matrix of pasted content, as content stream text.

  Edits to a page's own content base-edit.js

  • MarkupCraft
  64-bit

• A stored matrix accepted

  Checks a move matrix read from a file or autosave before it is used.

  |a·d − b·c| > 1e-9 and every |number| < 1e7

  Edits to a page's own content base-edit.js

  • MarkupCraft
  64-bit

• Box of part of a text item

  The box on the sheet round some characters of one piece of the drawing's text.

  s = width × i ÷ n; box corners = origin + u·s + v·t, t from −0.22 to 0.9 of the size

  Text positions: search hits and the OCR text layer textsearch.js

  • MarkupCraft
  64-bit

• Cloud round a search hit

  The box of the cloud markup placed round a found word.

  box = hit box grown by pad on each side

  Text positions: search hits and the OCR text layer textsearch.js

  • MarkupCraft
  64-bit

• Picture scale for reading text

  The scale at which a scanned page is drawn before its text is read.

  s = min(dpi ÷ 72, maxSide ÷ max(w, h), √(maxPixels ÷ (w·h))), at least 0.05

  Text positions: search hits and the OCR text layer ocr.js

  • MarkupCraft
  64-bit

• Matrix that places the text layer

  The matrix that puts the recognizer's text-only page onto the sheet, written into the saved PDF.

  M = viewport⁻¹ ∘ [W_px ÷ W_pt, 0, 0, −H_px ÷ H_pt, 0, H_px]

  Text positions: search hits and the OCR text layer ocr.js

  • MarkupCraft
  64-bit

• Words placed on the sheet

  The position, direction and size of each word read from a scan, for Find and Select text.

  baseline y at x by linear interpolation along the line's baseline; size = height above the baseline ÷ 0.9

  Text positions: search hits and the OCR text layer ocr.js

  • MarkupCraft
  64-bit

• Word count and mean confidence

  The number of words read on a page and their mean confidence, for the status line.

  conf = round(Σ confidence ÷ n)

  Text positions: search hits and the OCR text layer ocr.js

  • MarkupCraft
  64-bit

• Box and cylinder

  A 20 mm cube and a 20 mm cylinder, standing on z = 0 and centered on the origin.

  S = 0.02; R = S ÷ 2; box: S × S square pulled up by S; cylinder: 32-sided polygon of radius R pulled up by S

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Cone

  A cone with a 32-sided base of radius R and its apex at height S.

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Sphere

  A sphere of radius R resting on z = 0, made of 12 rings of 24 faces.

  θ = π·i ÷ 12; φ = 2π·j ÷ 24; p = [R·sin θ·cos φ, R·sin θ·sin φ, R − R·cos θ]

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Roof

  A triangular prism: a triangle across x pulled along y from −R to +R.

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Align

  Moves each selected placement so the minimum, middle or maximum of its box on one axis meets that of the whole selection.

  target = lo, hi or (lo + hi) ÷ 2; move = target − own value; m[12 + axis] += move

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Drop to ground

  Moves each selected placement up or down so the bottom of its box is at z = 0.

  m[14] += −box.min.z

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Rounding to 0.1 mm

  The rounding applied to every stored mesh coordinate, plan polygon point and item size.

  r1(v) = round(v × 10) ÷ 10

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Mesh point

  Adds one point to a part's mesh.

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Ring loft

  Trunks, pots, limbs and conifer tiers: rings of points round an axis joined into a capped solid.

  a = 2π·i ÷ seg; p = [cx + rx·cos a, cy + ry·sin a, z]

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Crown and head blobs

  Tree crowns, shrubs, rocks and heads: an icosphere scaled to an ellipsoid with a seeded bump.

  t = (1 + √5) ÷ 2; unit point p; k = 1 + bump × (2·rand − 1); point = c + p × [rx, ry, rz] × k

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Wheel

  A car wheel: a 14-sided cylinder along y.

  a = 2π·i ÷ seg − π ÷ 2; p = [x + r·cos a, y, z + r·sin a]

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Canopy plan symbol

  The plan outline of a tree: a 14-point irregular ring, a trunk circle and a cross.

  a = 2π·i ÷ 14; k = r × (0.86 + 0.18·rand); p = [k·cos a, k·sin a]

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Library item sizes

  The width, depth and height written into the library item.

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Signed polygon area

  The signed area of a plan polygon.

  area = ½ Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Room from a floor face

  The outline, holes, height and area of a room from a level face that looks up.

  floor if n_z > 0.999; plan = model point × 1000; area (m²) = (outer − Σ holes) ÷ 10⁶

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Doors and windows near the room

  Which starter doors and windows belong to the room, with their middle, width and sill height in millimeters.

  middle = box center × 1000; width = max(Δx, Δy) × 1000; near if distance to outline ≤ 400

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Placement matrix for a piece

  The matrix that lays a furniture piece on the planner's footprint.

  x axis = (fp₁ − fp₀) ÷ 1000 ÷ W; y axis = (fp₃ − fp₀) ÷ 1000 ÷ D; origin = fp₀ ÷ 1000 − x axis·min.x − y axis·min.y; z = floor − min.z

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Reading position and turn

  The x, y, z and turn a control program reads for a placement.

  x, y, z = m[12], m[13], m[14]; rotZ = atan2(m[1], m[0]) × 180 ÷ π

  Control program host: placement position and turn plc-scene-mc.js

  • ModelCraft
  64-bit

• Writing position and turn

  Applies the values a control program set to the placement's matrix when the run is kept.

  sx = √(m₀² + m₁²); sy = √(m₄² + m₅²); a = rotZ × π ÷ 180; m₀ = sx·cos a, m₁ = sx·sin a, m₄ = −sy·sin a, m₅ = sy·cos a

  Control program host: placement position and turn plc-scene-mc.js

  • ModelCraft
  64-bit

• Triangle normals, areas and boxes

  The unit normal, area and bounding box of every triangle, and the size of the whole mesh.

  n = (b − a) × (c − a); area = |n| ÷ 2; unit normal = n ÷ |n|

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Nearest face along a ray

  The distance to the first triangle a ray meets between two limits, and whether it meets it from behind.

  det = e₁·(d × e₂); u = (s·(d × e₂)) ÷ det; v = (d·(s × e₁)) ÷ det; t = (e₂·(s × e₁)) ÷ det

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Ray and box

  Whether a ray passes through a box within the current distance limits.

  t_near = (min − o) × (1 ÷ d); t_far = (max − o) × (1 ÷ d), per axis

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point to triangle distance

  The squared distance from a point to the closest point of one triangle.

  closest point q by the corner, edge and face regions; d² = |p − q|²

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Distance to the nearest face

  The distance from a point to the nearest triangle within a search radius.

  d = √(min over triangles of d²), limited to r

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Inward offset of every corner

  Moves each corner inward so that every face around it moves by the wall thickness, and rejects the result where an offset would fold.

  q = p − wall × m, with n·m = 1 for every face direction n at the corner

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Corner move from the face directions

  The vector m with n·m = 1 for each face direction at a corner.

  one direction: m = n; two: m = (n₁ + n₂) ÷ (1 + n₁·n₂); more: (NᵀN) m = Nᵀ1

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Grid step of the distance field

  Chooses the grid step and size for the distance field.

  h = min(wall ÷ 2.5, largest size ÷ 24); h ← 1.15·h while Π(⌈sizeₖ ÷ h⌉ + 5) > maxCells; band B = wall + 2h

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Signed distance field

  The distance from every grid point to the outside surface, negative where the point is deeper inside than the wall.

  d² = min over triangles of |x − closest point|², capped at B²; φ = ±√d² + wall (− inside)

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  32-bit in part