Calculations

Calculations

• Solve for a parameter

  The value of the named parameter for which the expression equals the target.

  secant: c = b − f(b) × (b − a) ÷ (f(b) − f(a)); accept when |f| ≤ 1e-9 × max(1, |target|, |x₀|)

  Solving an expression for one parameter expr.js

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• Rounding of the answer

  Replaces the answer by its 13-digit rounding when the two are practically equal.

  r = x to 13 significant digits; use r when |r − x| ≤ 1e-12 × max(1, |x|)

  Solving an expression for one parameter expr.js

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• View frames

  The right, up and toward-the-viewer directions of the seven standard views.

  iso: u = (1, 1, 0) ÷ √2; v = (−1, 1, 2) ÷ √6; w = (1, −1, 1) ÷ √3

  Flat views of 3D solids flatview.js

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• Edges and triangles in the view's frame

  Every vertex of the model as (right, up, depth) in the view, and which soft edges are outlines from this side.

  q = (p · u, p · v, p · w); soft edge kept when its two faces face opposite ways

  Flat views of 3D solids flatview.js

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• Working tolerance

  The tolerance every later step uses, from the size of the projected model.

  diag = √(Δu² + Δv²); tol = diag × 1e-6

  Flat views of 3D solids flatview.js

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• Is a point hidden

  Says whether a point of an edge lies behind some triangle of the model.

  l₁, l₂ from signed areas ÷ triangle area; l₃ = 1 − l₁ − l₂; hidden when all ≥ 1e-9 and l₁ z_a + l₂ z_b + l₃ z_c > depth + tol

  Flat views of 3D solids flatview.js

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• Visible and hidden runs of an edge

  Splits an edge into stretches that are visible and stretches that are hidden.

  n = clamp(⌈L ÷ (diag ÷ 300)⌉, 1, 400) samples; each change bisected until (hi − lo) × L ≤ tol

  Flat views of 3D solids flatview.js

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• Merging pieces on one line

  Groups segments that lie on the same line and joins their overlapping spans.

  key = round(atan2(dy, dx) × 1e6) and round(offset ÷ (20 × tol)); spans joined when the gap ≤ tol

  Flat views of 3D solids flatview.js

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• Merged lines back to segments

  Writes each merged span as a segment, and removes from hidden lines what a visible line covers.

  point = offset × (−dy, dx) + t × (dx, dy)

  Flat views of 3D solids flatview.js

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• Scale from text

  Reads a scale typed as 1:50, 2:1, 1/20 or 0.5.

  a:b → a ÷ b; accepted when 0 < scale < 1e6

  Flat views of 3D solids flatview.js

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• Paper units per meter and automatic scale

  How many drawing or paper units one meter of the model is, and the largest standard scale that fits the sheet.

  inch sheet: 1 ÷ 0.0254; mm sheet: 1000; model: 1 ÷ meters per unit; fit: size × units per meter × scale ≤ 0.45 × paper size

  Flat views of 3D solids flatview.js

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• View lines in drawing units

  The 2D lines of the view, centered on the block's base point and scaled.

  x = (u − u_center) × f; y = (v − v_center) × f; f = units per meter × scale

  Flat views of 3D solids flatview.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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• 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

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• 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

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• 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

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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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