Calculations

Calculations

• 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

• Push/pull distance

  The distance a face is pushed, in the open group's own scale, and the distance reported back.

  kernel distance = d × k; reported d = returned distance ÷ k; k = |normal taken into the group's axes|

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Move and copy

  The translation applied to the selection, in the open group's own coordinates.

  local delta = M⁻¹ × delta (direction only); transform = T(local delta)

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Rows and rings of copies

  The transform of each copy when a copy is repeated (3x) or divided (/3).

  copy k: T(k × delta) or R(k × angle); divided: k ÷ n of the delta or angle

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Rotation angle

  The angle the selection is turned by, from the reference direction to the pointer.

  a = atan2((ref × v)·n, ref·v); snapped to k × π ÷ 12 when within 2.5°

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Uniform scale factor

  The factor the selection is scaled by about a fixed point, from a reference point to the pointer.

  f = max(0.001, ((p − base)·u) ÷ (u·u)), u = ref − base

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Stretch by a grip

  The matrix that scales the selection by separate factors along the group's three axes about an anchor.

  M = T(a) × diag(fx, fy, fz) × T(−a)

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Guide line offset from an edge

  The perpendicular offset of a new guide line from the edge or guide it was measured from.

  w = w₀ − dir × (w₀·dir), w₀ = pointer − start

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Guide direction at an angle

  The direction of a guide line at a given angle from the protractor's zero direction.

  d = normalize(u × cos a + (n × u) × sin a)

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Dimension line offset

  Where the dimension line sits: the pointer's offset square to the measured line.

  off = q − ab × (q·ab), q = pointer on the plane − first point, ab = unit direction of the measured line

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Terrain brush weights

  How much each terrain point under the round brush is raised or lowered.

  w = 0.5 × (1 + cos(π × d ÷ R)) for d < R; new z = z + w × Δz

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Selection bounding box

  The box around the selection in world coordinates; its middle is the center for turning and scaling from the keyboard.

  min and max per axis over the transformed points; center = (min + max) × 0.5

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Frame of a placed component

  The three axis directions a component gets when it is laid on a face.

  z = face normal toward the eye; x = red axis less its part along z (or blue × z); y = z × x

  3D drawing and editing tools m3d/tools.js

  • CadCraft
  • ModelCraft
  64-bit

• Mesh buffers for the graphics card

  The vertex positions, normals and colors of a definition's faces as the arrays the graphics card draws.

  3D view drawing m3d/view3d.js

  • CadCraft
  • ModelCraft
  32-bit, display only

• Box of a placement as drawn

  The world-space box of one placed definition, used to frame the view and place the lights.

  3D view drawing m3d/view3d.js

  • CadCraft
  • ModelCraft
  32-bit, display only