Calculations

Calculations

• Angle brought into one turn

  An angle reduced to the range −π to π.

  r = atan2(sin a, cos a); |r| < 1e-12 → 0

  Picked points, object snaps and viewport views view.js

  • CadCraft
  64-bit

• Drawing coordinates to screen position

  Where a drawing point is on screen, in CSS pixels of the active pane.

  sx = (x − cx)·zoom + w ÷ 2; sy = h ÷ 2 − (y − cy)·zoom (with a turn: the difference is rotated first)

  2D drawing on screen view.js

  • CadCraft
  32-bit, display only

• Origin of the cached paths

  The point the cached drawing paths are stored relative to.

  origin = middle of the extents of the visible objects; path point = point − origin

  2D drawing on screen view.js

  • CadCraft
  32-bit, display only

• Grid spacing

  The spacing of the background grid: 1, 2 or 5 times a power of ten, at least 14 pixels apart.

  target = 14 ÷ zoom; p = 10^⌊log₁₀ target⌋; step = first of p, 2p, 5p, 10p that is ≥ target; major = 5 × step

  2D drawing on screen view.js

  • CadCraft
  32-bit, display only

• Pixels per paper millimeter

  How many CSS pixels one millimeter of the sheet takes, used to draw lineweights at their paper width.

  √(T₀² + T₁²) ÷ (25.4 when the paper is in inches, else 1)

  2D drawing on screen view.js

  • CadCraft
  32-bit, display only

• Band of a wide polyline segment

  The filled outline of one polyline segment that has a start and end width.

  h(t) = (w₀ + (w₁ − w₀)·t) ÷ 2; left = p + n·h; right = p − n·h; arc: k = max(2, ⌈sweep ÷ 10°⌉) steps

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit

• Corner between two wide segments

  The wedge that fills the outside gap where one wide segment meets the next.

  cross = tₐ × t_b; s = ((P₂ − P₁) × t_b) ÷ cross; X = P₁ + s·tₐ; used when s ≥ 0 and |X − corner| ≤ 10 × max(hₐ, h_b)

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit

• Text background mask

  The rectangle painted behind a text: its box grown by a margin and turned with the text.

  m = max(0, scale − 1) × height; corner = (x, y) + R(rot) · (x₀ − m, y₀ − m) …

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit

• Ring direction

  Turns a ring clockwise so that overlapping rings add up when filled.

  a = Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ); a > 0 → reverse the points

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit

• Ellipse as line pieces

  The line pieces an ellipse is drawn and plotted with.

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit

• Linetype scale of an object

  The factor that sizes the dashes of an object's linetype.

  scale = LTSCALE × object scale × plot style table scale (non-ISO linetypes only)

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit

• Point in polygon

  Whether a point lies inside the clip polygon.

  count edges with (aᵧ > pᵧ) ≠ (bᵧ > pᵧ) and pₓ < (bₓ − aₓ)·(pᵧ − aᵧ) ÷ (bᵧ − aᵧ) + aₓ; odd → inside

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Where a line crosses the clip edges

  The positions along a line segment where it crosses the edges of the clip polygon.

  den = d × e; t = ((p − a) × e) ÷ den; u = ((p − a) × d) ÷ den; kept when EPS < t < 1 − EPS and −EPS ≤ u ≤ 1 + EPS

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Line cut to the clip

  The pieces of a line that lie inside the clip polygon.

  piece ends = a + (b − a)·t₀ and a + (b − a)·t₁; kept when the middle point is inside

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Arc cut to the clip

  The sub-arcs of an arc that lie inside the clip polygon.

  A·s² + B·s + C = 0 on each clip edge; s = (−B ± √(B² − 4AC)) ÷ 2A; angle = atan2(y − cy, x − cx) − a₀

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Fill cut to the clip

  A filled ring cut to a convex clip polygon.

  per clip edge: keep points with side ≥ −EPS; crossing = p + (q − p)·((a − p) × e) ÷ ((q − p) × e)

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Is the clip polygon convex

  Decides whether fills can be cut to this clip polygon.

  z = (b − a) × (c − b) at every corner; all signs equal → convex

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Clip polygon in drawing coordinates

  The clip polygon of a reference moved from the block's coordinates into the drawing.

  M = m · insert matrix; point = M · p

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Clip read from a file

  Turns the clip boundary stored in a DWG or DXF file into the polygon kept on the reference.

  point = I · (T · p), with T the clip transform and I the inverse block transform (3 × 4 each)

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Objects cut to the clip when binding

  The objects of a block cut to the clip polygon: whole when inside, as line and arc pieces when crossing.

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• New clip boundary into block coordinates

  Stores a boundary picked in the drawing as a polygon in the block's own coordinates.

  d = m₀·m₃ − m₁·m₂; inv = adjugate ÷ d; stored point = inv · picked point

  Clipping block references to a boundary xclip.js

  • CadCraft
  64-bit

• Meters to drawing units

  A 3D point in meters as a point in drawing units.

  x = pₓ ÷ k; y = pᵧ ÷ k; z = p_z ÷ k

  Points and heights picked in the 3D view ws3d-core.js

  • CadCraft
  64-bit

• Point on a level plane

  Where the ray under the pointer meets the plane at height z.

  plane: n = (0, 0, 1), d = z × k; point = ray hit ÷ k

  Points and heights picked in the 3D view ws3d-core.js

  • CadCraft
  64-bit

• Height following the cursor

  A height above a base outline, taken from where the pointer's ray passes nearest the vertical line through the outline's middle.

  c = Σ ring points ÷ n; height = s ÷ k, with s the distance along the vertical line from (cₓ·k, cᵧ·k, z₀·k)

  Points and heights picked in the 3D view ws3d-core.js

  • CadCraft
  64-bit

• 2D drawing as snap edges

  The visible lines and arcs of the 2D drawing as edges on the ground, in meters, for the 3D snaps.

  edge = (x·k, y·k, 0); arc: m = max(2, ⌈sweep ÷ 7.5°⌉) chords

  Points and heights picked in the 3D view ws3d-core.js

  • CadCraft
  64-bit

• Height shown beside the cursor

  The text of the height tip while a height is being picked.

  |n| < 1e-9 → 0; else n to 4 decimals, trailing zeros removed

  Points and heights picked in the 3D view ws3d-core.js

  • CadCraft
  64-bit

• Kept solids as screen meshes

  The vertex positions and edge lines of the kept 3D solids as drawn on the 3D tab.

  position = (P − corner) × k, stored as 32-bit

  3D tab drawing of the 2D lines and kept solids ws3d.js

  • CadCraft
  32-bit, display only

• 2D lines as screen lines

  The 2D drawing's line pieces as drawn on the ground of the 3D tab.

  position = (x − min x, y − min y, 0), stored as 32-bit

  3D tab drawing of the 2D lines and kept solids ws3d.js

  • CadCraft
  32-bit, display only

• Unit vector

  A direction scaled to length 1, or nothing when it has no length.

  l = √(x² + y² + z²); l > 1e-300 → (x, y, z) ÷ l

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Plane

  A point of a plane from its (u, v), and back.

  P = O + U·u + V·v; u = (p − O)·U; v = (p − O)·V

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Cone and cylinder

  A point of a cone or cylinder (circular or elliptical base) from its angle and its height along the axis, and back.

  t = ± sin ÷ cos; q = 1 + h·t ÷ R; P = C + X·R·q·cos u + Y·R_m·q·sin u + N·h; u = atan2((d·Y) ÷ R_m, (d·X) ÷ R); apex at h = −R ÷ t

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Sphere

  A point of a sphere from longitude and latitude, and back.

  P = C + r·cos v·(X·cos u + Y·sin u) + Z·r·sin v; u = atan2(y, x); v = atan2(z, √(x² + y²))

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Torus

  A point of a torus from its two angles, and back.

  w = A + B·cos v; P = C + X·w·cos u + Y·w·sin u + N·B·sin v; u = atan2(y, x); v = atan2(z, √(x² + y²) − A)

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Spline surface and its inverse

  A point of a B-spline surface from (u, v), its normal, and the (u, v) nearest to a given point.

  first guess: nearest of 25 × 25 samples; then Gauss–Newton on |p − S(u, v)|² with partial derivatives by central differences of 1e-6 of the range

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Knot span

  Which knot interval a parameter falls in.

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• De Boor evaluation

  A point of a B-spline at a parameter, in homogeneous coordinates.

  a = (t − Uᵢ) ÷ (Uᵢ₊ₚ₋ᵣ₊₁ − Uᵢ); d_j = (1 − a)·d_j₋₁ + a·d_j

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Spline curve point

  A point of a rational B-spline curve.

  P = (X, Y, Z) ÷ W from de Boor

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Spline surface point

  A point of a rational B-spline surface.

  de Boor along v on each of the p + 1 rows, then de Boor along u on the results

  ACIS solids turned into triangles acis/geom.js

  • CadCraft
  • ModelCraft
  64-bit

• Model size and tolerances

  The size of the whole solid and the tolerances every later test uses.

  size = max(|hi − lo|, 2·rad, 1e-9); tol = max(10 × resabs, 1e-5 × size); weld = 1e-9 × size

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Circular and elliptical edges

  The points along an edge that is part of a circle or an ellipse.

  step = min(7.5°, 2·acos(1 − min(1, 0.0002·size ÷ R))); n = min(4096, ⌈|t₁ − t₀| ÷ max(step, 0.001)⌉)

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Spline edges

  The points along an edge that follows a B-spline curve.

  N = min(4000, max(64, 24 × spans)) samples; end parameters by nearest sample, then 60 steps of a three-way search; n = ⌈|t₁ − t₀| ÷ (b − a) × N ÷ 4⌉ pieces, 2 to 4096

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Merging vertices

  Gives points that coincide one shared vertex, so a closed solid stays closed.

  key = round(x ÷ weld), round(y ÷ weld), round(z ÷ weld)

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Does the surface fit its edges

  Checks that every edge point of a face lies on the face's surface before the face is used.

  |S(inv(p)) − p| ≤ tol for every loop point

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Which way the triangles face

  Turns a face's triangles so they point the way the face's outward normal does.

  n = (t₁ − t₀) × (t₂ − t₀); vote by the sign of n · normal at the triangle's center

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Flat faces

  Triangles of a flat face with its holes.

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Outer loops and holes

  Sorts the loops of a face into outlines and the holes inside them.

  A = |Σ (uᵢ·vᵢ₊₁ − uᵢ₊₁·vᵢ)| ÷ 2; depth = number of larger loops that contain the middle of the loop's first edge

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Loop unwound on a curved surface

  The (u, v) of a loop's points with the angles unwound along the loop, and how many times it goes round the surface.

  du wrapped to ±half a turn and added up; windings = round(total ÷ period); pole when cos v · r ≤ tol (sphere) or the cone radius ≤ tol

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
  • ModelCraft
  64-bit

• Step sizes on a curved face

  How finely a curved face is divided.

  angles: 7.5°; spline: range ÷ 16 each way; along a cone's axis: max(3 × 7.5° in radians × R, 1e-12)

  ACIS solids turned into triangles acis/tess.js

  • CadCraft
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  64-bit

• Extra points inside a curved face

  The points added inside a curved face so that its triangles follow the surface.

  one point per grid cell at (i + 0.5 + 0.3·jitter)·cell; kept when inside the face and at least 0.35 × the smaller cell side from every boundary edge

  ACIS solids turned into triangles acis/tess.js

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• Fixed offset of a grid point

  A repeatable offset between −0.5 and 0.5 for each grid cell.

  s = sin(12.9898·i + 78.233·j) × 43758.5453; s − ⌊s⌋ − 0.5

  ACIS solids turned into triangles acis/tess.js

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  • ModelCraft
  64-bit