Calculations

Calculations

• Angle constraint: legs and sign

  Which direction along each line the angle is measured between, and whether it turns counterclockwise.

  parallel when |u × v| < 1e-9 × |u| × |v|; sign = sign of atan2(u′ × v′, u′ · v′)

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Tangent circles: inside or outside

  Decides whether two circles made tangent touch from inside or from outside.

  internal when |c_b − c_a| < max(r_a, r_b) − 1e-12

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Crossing point of two lines

  Where two infinite lines meet, used as the corner of an angle constraint.

  den = d₁ × d₂; t = ((l₂ − l₁) × d₂) ÷ den; x = l₁ + d₁ × t

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Foot of a perpendicular

  The point of a line nearest to a given point.

  t = ((p − l₁) · u) ÷ (u · u); foot = l₁ + t × u

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Mirror gap

  How far a point is from being the mirror image of another across a line.

  f = foot of a on the line; gap = 2f − a − b

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Value rounded to 6 decimals

  Rounds a constraint's value to a millionth of a unit for its label and its parameter text.

  Math.round(v × 1e6) ÷ 1e6

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Constraint value as a parameter

  Creates the parameter for a dimensional constraint read from a file that lacks one.

  expr = formatNumber(round6(value))

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Label value

  The number shown in a dimensional constraint's label.

  text = formatNumber(round6(v)), with ° for angles

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Did solving collapse an object

  Refuses a solution that shrinks a line, a polyline segment or a radius to almost nothing.

  collapsed when new < 1e-6 × old and old > 1e-9

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Which fields an edit moved

  Finds the coordinates an edit changed, so the solver holds them where the edit put them.

  moved when |before − after| > 1e-12 × max(1, |before|)

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• Fixed values moved on paste

  Shifts the stored position of a fix constraint by the paste offset.

  x′ = x + dx; y′ = y + dy

  Constraint values and signs gcons.js

  • CadCraft
  64-bit

• 8-byte base-16 real

  Converts the stream format's excess-64 base-16 real (units, magnification, angle) to a number.

  value = sign × (m ÷ 2⁵⁶) × 16^(e − 64), m = the 7 mantissa bytes as one integer

  GDSII layout read and placement gds.js

  • CadCraft
  64-bit

• Reference transform

  The matrix that places one cell reference: reflect, magnify, rotate, move, and its product with the parent's matrix.

  r = angle × π ÷ 180; M = [mag·cos r, mag·sin r, −f·mag·sin r, f·mag·cos r, x, y], f = −1 when reflected

  GDSII layout read and placement gds.js

  • CadCraft
  64-bit

• Flattened coordinates, array steps and path width

  Places every element of every referenced cell and array and converts its points and width to user units.

  array step = (corner − origin) ÷ count; point = (M × p) × userUnit; width = w × √(M₀² + M₁²) × userUnit

  GDSII layout read and placement gds.js

  • CadCraft
  64-bit

• GEODATA fields from the drawing's location

  The design point, reference point, units and north vector written to the file.

  rot = degrees × π ÷ 180; north = (round₉(sin rot), round₉(cos rot))

  Drawing location written to DWG and DXF geodwg.js

  • CadCraft
  64-bit

• Location from another program's GEODATA

  Turns a file's GEODATA into the drawing's coordinate system, placement point and rotation.

  rot = round₉(atan2(north.x, north.y) × 180 ÷ π)

  Drawing location written to DWG and DXF geodwg.js

  • CadCraft
  64-bit

• Fingerprint of the written location

  Decides whether the GEODATA in a file is still the one CadCraft wrote or another program moved the location since.

  Drawing location written to DWG and DXF geodwg.js

  • CadCraft
  64-bit

• Arc of a polyline segment

  The center, radius and angles of a polyline's arc segment from its two vertices and its bulge.

  θ = 4·atan(b); r = |c ÷ (2·sin(θ ÷ 2))|; h = r·cos(θ ÷ 2); center = chord midpoint ± h along the chord's normal

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Angle on an arc

  Whether an angle lies on the counter-clockwise sweep of an arc.

  d = (a − a0) mod 2π; on the arc when d ≤ sweep + tol or d ≥ 2π − tol

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Arc angle range

  Puts an arc's two stored angles in counter-clockwise order with the end after the start.

  a0, a1 reduced to [0, 2π); a1 += 2π when a1 ≤ a0 + 1e-9

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Polyline segments

  The straight and arc pieces of a polyline, one per pair of vertices.

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Point at a parameter

  The point a fraction t of the way along a line or arc.

  line: p₁ + (p₂ − p₁)·t; arc: c + r·(cos a, sin a), a = a0 + (a1 − a0)·t

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Nearest parameter

  The parameter of the point on a line or arc closest to a given point.

  line: t = (q − p₁)·(p₂ − p₁) ÷ |p₂ − p₁|²; arc: t = ((atan2(q − c) − a0) mod 2π) ÷ sweep

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Length of a piece

  The length of a straight segment or an arc.

  line: √(Δx² + Δy²); arc: r × (a1 − a0)

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Intersections on both pieces

  The points where two pieces meet, limited to the pieces themselves unless one is to be treated as unbounded.

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Bounds of a piece

  The smallest box round a line or arc, used for the drawing's extents.

  arc: its two ends and each of the points at 0, π ÷ 2, π, 3π ÷ 2 that is on the sweep

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Tangent points from a point

  The points on a circle where a line from a given point touches it.

  a = atan2(p − c); b = acos(r ÷ d); touch points = c + r·(cos(a ± b), sin(a ± b))

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Lines tangent to two circles

  The outer and inner lines that touch two circles, as pairs of touch points.

  k = (r₁ − s·r₂) ÷ d, s = +1 outer, −1 inner; g = acos(k); n = (cos(base ± g), sin(base ± g)); points = c₁ + r₁·n and c₂ + s·r₂·n

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Object snap points

  The end, middle, center, quadrant, perpendicular, tangent and nearest points of an object that a new point can be placed on.

  middle of a line = (p₁ + p₂) ÷ 2; arc points = c + r·(cos a, sin a)

  Line and arc primitives geom.js

  • CadCraft
  64-bit

• Composing two transforms

  The single transform that applies n first and then m.

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Mirror across a line

  The transform that reflects across the line through two points.

  a = atan2(q − p); M = [cos 2a, sin 2a, sin 2a, −cos 2a] about p

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Scale and rotation of a transform

  The uniform scale factor, rotation and handedness of a matrix, or nothing when it shears or stretches.

  sx = √(a² + b²); sy = √(c² + d²); rotation = atan2(b, a); mirrored when a·d − b·c < 0

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Inverse transform

  The transform that undoes a given one.

  det = a·d − b·c; inverse = [d, −b, −c, a, c·f − d·e, b·e − a·f] ÷ det

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Block reference matrix

  The transform from a block's own coordinates to where a reference places it.

  M = T(x, y) · R(rotation) · S(sx, sy) · T(−base)

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Viewport matrix

  The transform from model coordinates to paper coordinates for a layout viewport, at its scale and twist.

  no twist: [s, 0, 0, s, x − cx·s, y − cy·s]; twisted: c = s·cos tw, n = s·sin tw, [c, n, −n, c, x − (c·cx − n·cy), y − (n·cx + c·cy)]

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Viewport view center

  Converts a viewport's view center between the file's display coordinates and the model point shown at the viewport's center.

  x = tx + cos tw·dx + sin tw·dy; y = ty − sin tw·dx + cos tw·dy

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Transforming an object

  A copy of a drawing object put through a move, rotation, scale or mirror.

  point: M·p; arc: angles = atan2(M·end − M·center); radius, text height, block scale × s

  Move, rotate, scale and mirror geom.js

  • CadCraft
  64-bit

• Cut parameters

  Where along an object each cutting edge crosses it.

  t = segment index + parameter of the intersection; t = round(t × 10⁹) ÷ 10⁹

  Trim geom.js

  • CadCraft
  64-bit

• Piece between two parameters

  The part of a line, arc, open polyline or construction line between two parameters, as a new object.

  line: p₁ + (p₂ − p₁)·t; arc: a0 + sweep·t; polyline arc piece: bulge = tan(atan(b)·(t_b − t_a))

  Trim geom.js

  • CadCraft
  64-bit

• Trim at a pick

  The pieces that stay when the part of an object round the picked point is cut away.

  circle: remaining arc from angle t_hi × 2π to t_lo × 2π

  Trim geom.js

  • CadCraft
  64-bit

• Splitting into pieces

  An object divided at all its cutting edges into pieces that can be removed one by one.

  Trim geom.js

  • CadCraft
  64-bit

• Offset of a line, circle or arc

  A copy of the object a set distance to one side.

  line: p ± d·n, n = (−Δy, Δx) ÷ L; circle, arc: r ± d

  Offset, extend, break and lengthen geom.js

  • CadCraft
  64-bit

• Offset of a polyline

  A polyline offset to one side, with its corners rebuilt.

  Offset, extend, break and lengthen geom.js

  • CadCraft
  64-bit

• Extend to a boundary

  An object with the end nearest the pick moved out to the closest boundary it reaches.

  Offset, extend, break and lengthen geom.js

  • CadCraft
  64-bit

• Break between two points

  An object with the part between two picked points removed, or split at one point.

  Offset, extend, break and lengthen geom.js

  • CadCraft
  64-bit

• Lengthen by an amount

  An object made longer or shorter by a length, or an arc by an angle, at the end nearest the pick.

  line: moved end = fixed end ± u·(L + δ); arc: new sweep = sweep + δ ÷ r; polyline arc end: bulge = ± tan(new sweep ÷ 4)

  Offset, extend, break and lengthen geom.js

  • CadCraft
  64-bit

• Fillet of two lines

  The arc of a given radius tangent to two lines and the lines cut back to it.

  half = acos(ua·ub) ÷ 2; t = r ÷ tan(half); center = corner + bisector × r ÷ sin(half)

  Fillet and chamfer geom.js

  • CadCraft
  64-bit

• Tangent points and center at a corner

  The two tangent points and the center of a fillet at the corner of two straight pieces.

  half = acos(ua·ub) ÷ 2; t = r ÷ tan(half); center = corner + bisector × r ÷ sin(half)

  Fillet and chamfer geom.js

  • CadCraft
  64-bit

• Fillet arc between two tangent points

  The arc entity of the fillet, the short way round from one tangent point to the other.

  Fillet and chamfer geom.js

  • CadCraft
  64-bit

• Cutting an object back to the corner

  Moves one end of a line, arc or polyline segment to the tangent or chamfer point.

  arc end angle = atan2(T − center); polyline arc bulge = ± tan(sweep ÷ 4)

  Fillet and chamfer geom.js

  • CadCraft
  64-bit