Calculations

Calculations

• 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

• Overlay line clipped to the near plane

  A 3D line cut at the camera's near plane and projected to screen points for the 2D overlay.

  cut at depth 1.01 × near: a′ = a + (b − a) × (near′ − da) ÷ (db − da)

  3D view drawing m3d/view3d.js

  • CadCraft
  • ModelCraft
  32-bit, display only

• Line fit

  The best straight line through a run of center-line pixels, and how far the worst pixel lies from it.

  θ = ½ × atan2(2·Sxy, Sxx − Syy); ends = feet of the first and last points on the line through the centroid

  Tracing a scan into lines and arcs raster/trace.js

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

• Circle fit

  The best circle through a run of center-line pixels, its sweep and the angles of an arc.

  Kåsa: solve the 2 × 2 system for the center about the centroid, r = √(uc² + vc² + (Suu + Svv) ÷ n); then Gauss-Newton on d − r

  Tracing a scan into lines and arcs raster/trace.js

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

• Corner detection

  Where a chain of pixels turns sharply, so it can be cut there before fitting.

  turn = |atan2(r − q) − atan2(q − p)| over ± w steps; corner when turn ≥ 0.6 rad and largest nearby

  Tracing a scan into lines and arcs raster/trace.js

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

• Splitting a chain into lines and arcs

  Fits a line, else an arc, else splits the chain at the point farthest from its chord, and joins neighbors that fit as one.

  distance to chord = |(P − A) × (B − A)| ÷ |B − A|

  Tracing a scan into lines and arcs raster/trace.js

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

• Joining collinear lines

  Merges two lines that lie on the same straight line and nearly touch.

  Tracing a scan into lines and arcs raster/trace.js

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

• Squaring level and plumb lines

  Makes nearly horizontal and vertical lines exactly so, and gives lines of one wall the same coordinate.

  y = (y₁ + y₂) ÷ 2; shared coordinate = Σ(coordinate × length) ÷ Σ length

  Tracing a scan into lines and arcs raster/trace.js

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

• Closing corners

  Moves a line end to the point where its line meets a nearby line.

  s = ((B₁ − E) × v) ÷ (u × v); new end = E + s·u

  Tracing a scan into lines and arcs raster/trace.js

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

• Lines and arcs in image coordinates

  The traced lines and arcs with the window offset added and the arc end points worked out.

  start = (cx + r·cos a₀, cy + r·sin a₀); end = (cx + r·cos a₁, cy + r·sin a₁)

  Tracing a scan into lines and arcs raster/trace.js

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

• Crossing of two traced lines

  The point where two traced line segments cross.

  d = (a₂ − a₁) × (b₂ − b₁); t = ((b₁ − a₁) × (b₂ − b₁)) ÷ d; point = a₁ + t·(a₂ − a₁)

  Tracing a scan into lines and arcs raster/snap.js

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

• Snap points of a trace

  The ends, middles, arc centers, corners and crossings offered for snapping.

  middle = (p₁ + p₂) ÷ 2; arc middle at angle (a₀ + a₁) ÷ 2; corner = mean of the ends within 3 px

  Tracing a scan into lines and arcs raster/snap.js

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

• Black-and-white level (Otsu)

  The grey level that best separates ink from paper for the whole sheet.

  maximize w_B × w_F × (m_B − m_F)² over the 256 levels

  Scan clean-up and JPEG decoding raster/image.js

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

• Local threshold (Sauvola)

  Decides ink or paper for each pixel from the mean and spread of the pixels around it.

  ink when v ≤ mean × (1 + k × (sd ÷ 128 − 1)); sd = √(Σv² ÷ n − mean²)

  Scan clean-up and JPEG decoding raster/image.js

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

• Sheet skew angle

  The angle by which the long lines of a scanned sheet are rotated.

  score(angle) = Σ rows² + Σ columns² of the ink projected at that angle; best angle on a 0.25° then 0.02° grid

  Scan clean-up and JPEG decoding raster/image.js

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

• Rotating the image

  Turns a grey image by an angle about its middle.

  source = c + R(angle) × (pixel − c); value = bilinear mix of the 4 nearest source pixels

  Scan clean-up and JPEG decoding raster/image.js

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

• JPEG inverse cosine transform

  Turns the stored frequency coefficients of an 8 × 8 block back into pixel values.

  pixel = round(Σ_u Σ_v C(x,u)·C(y,v)·coef(u,v) + 128), clamped to 0–255

  Scan clean-up and JPEG decoding raster/jpeg.js

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

• JPEG color conversion

  Converts the decoded luminance and color-difference values to red, green and blue.

  R = Y + 1.402·Cr; G = Y − 0.344136·Cb − 0.714136·Cr; B = Y + 1.772·Cb

  Scan clean-up and JPEG decoding raster/jpeg.js

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

• Matrix product

  Combines two transforms into one.

  o[c][r] = Σₖ a[k][r] × b[c][k]

  USD transforms and placement usd/geom.js

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  • TakeoffCraft
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  64-bit

• Matrix inverse

  The transform that undoes another.

  inverse = cofactor matrix ÷ determinant

  USD transforms and placement usd/geom.js

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  • TakeoffCraft
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  64-bit

• Transforming a point

  Applies a transform to a point.

  p′ = (M × [x, y, z, 1]) ÷ w

  USD transforms and placement usd/geom.js

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

• Rotation about an axis

  The matrix of a rotation by an angle in degrees about X, Y or Z.

  a = deg × π ÷ 180; entries cos a, ± sin a

  USD transforms and placement usd/geom.js

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