Calculations

Calculations

• Number text for the command line

  Writes a value taken from the pointer as text.

  text = value rounded to 10 decimals, trailing zeros dropped

  Dynamic input: points from typed distance, angle, X and Y dynin.js

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• Unit factors

  What a number with a unit is multiplied by to give millimeters or degrees.

  mm 1; cm 10; m 1000; in 25.4; ft 304.8; deg 1; rad 180 ÷ π

  Parameter expressions expr.js

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• A number with a unit

  Reads a number and converts it by the unit written right after it.

  v = Number(text) × factor

  Parameter expressions expr.js

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• Sine, cosine and tangent in degrees

  sin, cos and tan of an angle given in degrees.

  sin(x) = clean(Math.sin(x × π ÷ 180)); tan refused when |cos| < 1e-12

  Parameter expressions expr.js

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• Trig result rounded to 15 digits

  Rounds a sine, cosine or tangent to 15 significant digits and turns a value below 1e-15 into 0.

  |x| < 1e-15 → 0; otherwise Number(x.toPrecision(15))

  Parameter expressions expr.js

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• Inverse trigonometry and square root

  asin, acos, atan, atan2 in degrees, and sqrt.

  asin(x) = Math.asin(clamp(x, −1, 1)) ÷ (π ÷ 180); sqrt(x) = √max(0, x)

  Parameter expressions expr.js

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• Power, exponential and logarithms

  pow, exp, ln and log.

  pow(a, b) = aᵇ; ln and log need x > 0

  Parameter expressions expr.js

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• ceil, floor and int

  Rounds up, down or toward zero to a whole number.

  ceil(x) = ⌈x − ε⌉; floor(x) = ⌊x + ε⌋; int(x) = trunc(x + sign(x) × ε); ε = 1e-9 × max(1, |x|)

  Parameter expressions expr.js

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• round(x, digits)

  Rounds to a number of decimal places from 0 to 10, halves away from zero.

  y = |x| × 10ᵈ; r = ⌊y + 0.5 + 1e-9 × max(1, y)⌋; result = ± r ÷ 10ᵈ

  Parameter expressions expr.js

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• Comparisons

  The six comparison operators, giving 1 or 0.

  tol = 1e-9 × max(1, |a|, |b|); a == b when |a − b| ≤ tol; a < b when a < b − tol

  Parameter expressions expr.js

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• Arithmetic operators

  Addition, subtraction, multiplication, division, remainder and power.

  a % b = a − b × ⌊a ÷ b⌋; division and remainder refuse b = 0

  Parameter expressions expr.js

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• Number as text

  The text a number is written as when it is put back into an expression or a message.

  Number(v.toPrecision(12)) as text

  Parameter expressions expr.js

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• What a dimensional constraint measures

  The current horizontal, vertical or direct distance, radius, diameter or angle of the constrained geometry.

  hdist = s × (x₁ − x₀); dist = |p₁ − p₀| or s × signed distance from the line; diameter = 2r; angle = s × atan2(u × v, u · v) × 180 ÷ π; arc angle = (a₁ − a₀) mod 2π in degrees

  Constraint values and signs gcons.js

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• Signed distance from a line

  How far a point or a circle's center lies from an infinite line, positive on the left of its direction.

  d = (u_x × (p_y − l_y) − u_y × (p_x − l_x)) ÷ |u|

  Constraint values and signs gcons.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Label value

  The number shown in a dimensional constraint's label.

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

  Constraint values and signs gcons.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Polyline segments

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

  Line and arc primitives geom.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Composing two transforms

  The single transform that applies n first and then m.

  Move, rotate, scale and mirror geom.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Offset of a polyline

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

  Offset, extend, break and lengthen geom.js

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