Calculations

Calculations

• Spline (NURBS) as segments

  The points that stand for a spline with control points.

  de Boor on homogeneous points (x·w, y·w, w); 12 samples per knot span

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Spline stored with fit points only

  The points that stand for a spline that carries fit points and no control points.

  centripetal Catmull–Rom, 12 samples per span

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Bulge arcs in hatch boundaries

  The points of a curved side of a hatch boundary given as a polyline bulge.

  center = chord midpoint + ((1 ÷ b − b) ÷ 2) × (chord turned 90°) ÷ 2; sweep = 4 atan b

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Block insert transform

  The transform that places a block, with its scale, rotation and array spacing.

  M = T(x, y) · R(rot) · S(s_x, s_y) · T(−base); array offset = R(rot) · (col × d_c, row × d_r)

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Hatch holes

  Which loops of a hatch are outlines and which are holes.

  area = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|; a loop inside an odd number of larger loops is a hole

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• 3D face corners

  The corners of 3D faces and solids, which keep their heights.

  (x, y) placed by the block transform; all three × k

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Text line pitch

  The distance between lines of imported multi-line text, which is drawn as edges.

  pitch = height × 1.667 × spacing factor

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Edges of imported solids

  How the edges of an imported solid are matched to decide which ones show.

  key = round(x × 1e5), round(y × 1e5), round(z × 1e5)

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• OBJ vertex

  A vertex line of an OBJ file.

  (x, y, z) → (x, −z, y)

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF accessor values

  The numbers of a glTF buffer, such as vertex positions.

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF node transform

  The matrix of a glTF node from its matrix, or from its translation, rotation and scale.

  M = T · R(q) · S

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF vertex in the model

  A glTF vertex moved by its node transforms and turned to Z up.

  p′ = M · p, then (x, y, z) → (x, −z, y)

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada numbers

  How lists of numbers in a Collada file are read.

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada unit and up axis

  The change from the file's unit and up axis to Z-up meters.

  C = R_up · S(meter); node matrix → C · M · C⁻¹; point → C · p

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada node transforms

  The matrix of a Collada node from its matrix, translate, rotate and scale elements.

  rotate: angle in radians = degrees × π ÷ 180

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada vertex in the model

  A Collada vertex turned to Z-up meters.

  p′ = C · p

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Arc of a glyph

  The points of the round parts of letters.

  n = max(2, ⌈|a₁ − a₀| ÷ 15⌉); p = (c_x + r_x cos a, c_y + r_y sin a)

  Text from drawings as stroked edges m3d/strokefont.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Width of a string

  The advance of a string in drawing units, used to align and wrap text.

  width = (Σ (glyph width + 2) − 2) × h ÷ 9 × width factor

  Text from drawings as stroked edges m3d/strokefont.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Strokes of a string

  The polylines of a string at a given height, width factor and slant.

  x = x₀ + p_x × s_x + p_y × s × tan(oblique); y = p_y × s; s = h ÷ 9

  Text from drawings as stroked edges m3d/strokefont.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Day of the year

  The day number of a date.

  N = days before the month + day (+1 after February in a leap year)

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  Exact

• Equation of time and declination

  How far sun time runs ahead of clock time, and the sun's angle above the equator, for the date.

  γ = 2π ÷ (365 or 366) × (N − 1 + (hour − 12) ÷ 24); eqtime = 229.18 × (0.000075 + 0.001868 cos γ − 0.032077 sin γ − 0.014615 cos 2γ − 0.040849 sin 2γ)

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Hour angle and altitude

  The sun's height above the horizon.

  tst = hour × 60 + eqtime + 4 × lon − 60 × tz; H = tst ÷ 4 − 180°; cos Z = sin φ sin δ + cos φ cos δ cos H; altitude = 90° − Z

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Azimuth

  The sun's compass direction, in degrees clockwise from north.

  azimuth = atan2(sin H, cos H sin φ − tan δ cos φ) + 180°, wrapped to 0…360

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Direction toward the sun

  The unit vector shadows are cast from, in model axes (north is +Y, east is +X).

  dir = (sin a cos alt, cos a cos alt, sin alt)

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Unit vector

  A vector scaled to length 1.

  n = a ÷ |a|; [0, 0, 0] when |a| ≤ 1e-15

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Polygon normal

  The unit normal of a polygon given by its corner points.

  N = Σ (pᵢ − pᵢ₊₁) × (pᵢ + pᵢ₊₁) taken per axis, then normalized

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Polygon area in 3D

  The area of a flat polygon in space.

  area = |Σ (pᵢ − pᵢ₊₁) × (pᵢ + pᵢ₊₁)| ÷ 2

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Plane through a point

  A plane as a unit normal and an offset, with the normal's sign made canonical.

  n·p = d; first component of n with |n| > 1e-9 is made positive

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Same plane test

  Whether two planes are the same plane.

  |n₁·n₂| > 1 − 1e-9 and |d₁ × sign(n₁·n₂) − d₂| ≤ 1e-6

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Closest points of a line and a segment

  The nearest points between a ray's line and an edge.

  den = (d·d)(u·u) − (d·u)²; t = (B·E − C·D) ÷ den; s = (A·E − B·D) ÷ den, clamped to [0, 1]

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Closest points of two lines

  The nearest points between two infinite lines, or nothing when they are parallel.

  den = A·C − B²; parallel when |den| < 1e-14 × A × C

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Ray and plane

  The point where a ray's line meets a plane.

  t = (d_plane − n·o) ÷ (n·dir); p = o + t × dir

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Ray and triangle

  The distance along a ray to a triangle, or nothing when it misses.

  Möller–Trumbore: det = e₁·(d × e₂); u, v, t from three scalar triple products ÷ det

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Rotation about an axis through a point

  The 4×4 matrix that turns geometry by an angle about an axis through a given point.

  M = T(about) × R(axis, θ) × T(−about), R from Rodrigues' formula

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Transform a point

  A point moved by a 4×4 matrix.

  p′ = (M × [x, y, z, 1]) ÷ w

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Inverse of a 4×4 matrix

  The matrix that undoes a transform, used to move points into a group's own coordinates.

  M⁻¹ = adj(M) ÷ det(M), with det from twelve 2×2 minors

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• One typed length to meters

  Text such as 10'-6 1/2", 3/4", 320cm or a bare number, as meters.

  m = sign × (feet × 0.3048 + (inches + num ÷ den) × 0.0254); m = sign × value × factor(suffix or model unit)

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• Meters per unit

  The factor that turns a bare number in the model's unit into meters.

  in = 0.0254; ft = 0.3048; yd = 0.9144; mi = 1609.344; mil = 0.0254 ÷ 1000

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• Arithmetic over typed lengths

  A typed expression such as 12'6"+3/4" or 2*450mm as one length in meters.

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• The value box

  What was typed as a length, a list of lengths, a point, an offset, a count or a plain number for angles and factors.

  plain number = value in meters ÷ meters per unit (for arithmetic and bare fractions)

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• Length readout

  A length in meters as text in the model's unit.

  ft-in: total = round(m ÷ 0.0254 × den) sixteenths (or the chosen fraction); others: toFixed with the unit's decimals

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• Inches as a fraction

  Inches written as a whole number and a reduced fraction.

  sx = round(inches × den); whole = ⌊sx ÷ den⌋; fraction = (sx mod den) ÷ den, reduced by the greatest common divisor

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• Area readout

  An area in square meters as text in the model's unit.

  ft²: a ÷ (0.3048 × 0.3048); in²: a ÷ (0.0254 × 0.0254); mm model: m² from 0.01 m², else mm²

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• Volume readout

  A volume in cubic meters as text in the model's unit.

  ft³: v ÷ 0.3048³; in³: v ÷ 0.0254³; mm model: m³ from 0.01 m³, else mm³

  Typed lengths and length readouts in 3D m3d/units.js

  • CadCraft
  • ModelCraft
  64-bit

• Two-point arc from its bulge

  The radius, center and sweep of an arc through two points that stands off the chord by a given bulge.

  R = (h² + (c ÷ 2)²) ÷ (2|h|); center = midpoint + side × sign(h) × (|h| − R); sweep = 4 × atan(2|h| ÷ c)

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Sweep of an arc drawn from its center

  The angle from the arc's start direction to the pointer, counter-clockwise about the plane normal.

  a = atan2((u × w)·n, u·w), plus 2π when a ≤ 0; snapped to k × π ÷ 12 when within 0.03 rad

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Rectangle width and height

  The two sides of a rectangle from its first corner to the pointer, along the plane's two axes.

  w = (q − p₀)·u; h = (q − p₀)·v

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Distance from a point to an outline segment

  How far the pointer is from one side of the outline being offset, in the outline's plane.

  t = clamp(((p − a)·(b − a)) ÷ |b − a|², 0, 1); distance = |p − a − t(b − a)|

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Offset distance and side

  The signed distance an outline is offset by: the least distance to any side, positive into a face.

  d = ± min over sides of distance(p, side); inside by an even-odd crossing count

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Point inside an outline

  Whether the pointer lies inside a face's outline, which decides if an offset goes inward or outward.

  count the sides whose ends straddle the point's height and cross to its right; inside when the count is odd

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit