Calculations

Calculations

• Joining loose ends

  Pulls each loose end onto the nearest corner of another stroke or of the drawing.

  nearest corner with distance ≤ tol; tol = 6 % of the sketch's box diagonal unless given

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• Point on a spline

  The curve's point at a parameter.

  de Boor: dⱼ = (1 − a)·dⱼ₋₁ + a·dⱼ, a = (u − kᵢ) ÷ (kᵢ₊ₚ₋ᵣ₊₁ − kᵢ); point = (Σ w·P) ÷ Σ w

  Splines: curve points, fitting and editing spline.js

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• Clamped knots

  The knot list for a spline that starts and ends on its first and last control points.

  p + 1 zeros, 1 … n − p − 1, p + 1 values of n − p

  Splines: curve points, fitting and editing spline.js

  • CadCraft
  Exact

• Spline through fit points

  Builds the spline that passes through the points picked.

  two points: control points at 0, ⅓, ⅔ and 1 of the segment; more: clamped cubic with chord-length parameters

  Splines: curve points, fitting and editing spline.js

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• Spline from control points

  Builds a spline with a given control polygon, clamped or closed.

  closed: control points wrapped by the degree, knots 0, 1, 2, …

  Splines: curve points, fitting and editing spline.js

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• A spline as points

  The curve as a list of points for drawing and for anything that works from points.

  pieces per span = max(4·max(1, p − 1), ⌈√(polygon length ÷ tol)⌉), within a budget of 4000

  Splines: curve points, fitting and editing spline.js

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• A spline as a polyline

  Points for converting a spline to a polyline, a set number of segments per knot span.

  u = a + (b − a) × i ÷ n, n = 1 … 99 (10 unless given)

  Splines: curve points, fitting and editing spline.js

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• Adding a control point

  The same curve with one more control point at a parameter.

  Qᵢ = (1 − a)·Pᵢ₋₁ + a·Pᵢ, a = (u − kᵢ) ÷ (kᵢ₊ₚ − kᵢ), in weighted form

  Splines: curve points, fitting and editing spline.js

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• Reversing a spline

  The same curve run the other way.

  knot′ = (first + last) − knot, in reverse order

  Splines: curve points, fitting and editing spline.js

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• Control-point spline to fit points

  Samples points on a spline and builds a fit-point spline through them.

  n points evenly spaced in the parameter (12 unless given, at least 3)

  Splines: curve points, fitting and editing spline.js

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  64-bit

• Parameter nearest a picked point

  Where along the curve a new control point is added.

  nearest of 401 samples evenly spaced in the parameter

  Splines: curve points, fitting and editing spline-cmd.js

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• Arc as a polyline segment

  Turns an arc into a two-point polyline with a bulge.

  ends = c + r(cos a, sin a); bulge = tan((a₁ − a₀) ÷ 4)

  Splines: curve points, fitting and editing spline-cmd.js

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• Curve-fitted polyline

  A polyline redrawn as a smooth curve through or along its vertices, kept as many short segments.

  8 segments per knot span unless given

  Splines: curve points, fitting and editing spline-cmd.js

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  64-bit

• Mass per meter of a section

  The kilograms per meter shown for a steel section.

  kg/m = published mass, else area (mm²) × 1e-6 × 7850

  Standard sections: member solids, mass and parts list stdlib.js

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• Member solid

  The faces of a straight member along two points, or of a column of a given height.

  outline (mm) ÷ 1000; L = |b − a| × k; point = eₓ·u + e_y·(v + lift) + e_z·s

  Standard sections: member solids, mass and parts list stdlib.js

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• Parts list quantities

  Groups placed parts and adds up how many there are of each.

  qty = Σ CC-QTY (1 when absent)

  Standard sections: member solids, mass and parts list stdlib.js

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• Total mass in the parts list

  The mass column of the parts list and its CSV.

  mass = round(mass each × qty × 100) ÷ 100

  Standard sections: member solids, mass and parts list stdlib.js

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• A level written as text

  Reads a level such as 104.37, +104,37 or EL 104.37 from a text.

  value = Number(whole digits . decimal digits) × sign

  Survey levels from text and the survey check survey.js

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  • BimCraft
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• Levels into project units

  A level point's x, y and z in project units.

  x·k, y·k, z·k

  Survey levels from text and the survey check survey.js

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• Extent and distance from the origin

  How far the data spans and how far it lies from zero, in meters.

  span = |(maxX − minX, maxY − minY)| × m per unit; far = max |coordinate| × m per unit

  Survey levels from text and the survey check survey.js

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  • BimCraft
  • CivilCraft
  64-bit

• Median

  The middle value of a list of levels.

  odd count: the middle value; even: mean of the two middle values

  Survey levels from text and the survey check survey.js

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  • BimCraft
  • CivilCraft
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• Level outliers

  Counts levels that are far from the rest, and levels of exactly 0 that look like missing heights.

  MAD = median(|z − median|) × 1.4826; limit = max(6 × MAD, 5 m); outlier when |z − median| > limit

  Survey levels from text and the survey check survey.js

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  • BimCraft
  • CivilCraft
  64-bit

• Duplicate points

  Counts points that sit on top of another point, with the same or a different level.

  cell = 1 mm; key = (round(x ÷ cell), round(y ÷ cell)); same level when |Δz| ≤ cell

  Survey levels from text and the survey check survey.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Drawing units to meters

  A coordinate or length as the text the macro passes, in meters.

  m = round₁₂(round₉(v) × mm per unit ÷ 1000)

  Sketch exchange macro and import swsketch.js

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  64-bit

• Numbers written into the macro

  A number as text in the macro.

  v.toPrecision(15)

  Sketch exchange macro and import swsketch.js

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  64-bit

• Millimeters per drawing unit

  The size of the drawing's unit in millimeters.

  inch 25.4; foot 304.8; yard 914.4; mil 0.0254; microinch 0.0000254

  Sketch exchange macro and import swsketch.js

  • CadCraft
  Exact

• Arc ends on export

  The start and end points of an arc, which the macro needs beside its center.

  start = c + r(cos a₀, sin a₀); end = c + r(cos a₁, sin a₁)

  Sketch exchange macro and import swsketch.js

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• Center of a polyline arc segment

  The center of the arc between two polyline vertices with a bulge.

  θ = 4·atan(b); r = |chord ÷ (2·sin(θ ÷ 2))|; center = middle ± r·cos(θ ÷ 2) across the chord

  Sketch exchange macro and import swsketch.js

  • CadCraft
  64-bit

• Dimension values on export

  The value given to each dimension in the macro.

  angle: degrees × π ÷ 180; length: M(value)

  Sketch exchange macro and import swsketch.js

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  64-bit

• Arcs on import

  Builds an arc from the center, start and end points of an imported sketch.

  r = |start − center|; a₀ = atan2(start − center); a₁ = atan2(end − center), each in [0, 2π)

  Sketch exchange macro and import swsketch.js

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• Dimension values on import

  The value each imported dimension gets, and the parameter that drives it.

  parameter = round(value × 1e9) ÷ 1e9

  Sketch exchange macro and import swsketch.js

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• The line or circle under a pick

  Reads the picked object as an endless line with a unit direction, or as a full circle.

  direction = (p₂ − p₁) ÷ |p₂ − p₁|

  Circles tangent to two or three objects tancircle.js

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• Objects offset by the radius

  A line moved to both sides by r, and a circle grown and shrunk by r.

  line: p ± n·r; circle: R + r and |R − r|

  Circles tangent to two or three objects tancircle.js

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• Where two offset objects cross

  The crossing points of two lines, a line and a circle, or two circles.

  line–circle: t = −B ± √(B² − C), C = |f|² − r²; circles: l = (r₁² − r₂² + d²) ÷ 2d, h = √(r₁² − l²)

  Circles tangent to two or three objects tancircle.js

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• Is a circle tangent to an object

  Checks a candidate circle against a line or a circle.

  line: | |distance to line| − r | ≤ tol; circle: |d − (R + r)| ≤ tol or |d − |R − r|| ≤ tol

  Circles tangent to two or three objects tancircle.js

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• Tangent, tangent, radius

  The circle of radius r tangent to two picked objects, nearest the picks.

  center = crossing of a offset by r and b offset by r; tol = 1e-7 × max(1, size of a, b and r)

  Circles tangent to two or three objects tancircle.js

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  64-bit

• Newton solve for three tangents

  Solves the three tangency equations for a circle's center and radius from one starting guess.

  F(x, y, r) = 0; J by forward differences with h = 1e-7 × max(1, |x|, |y|, |r|); step = −J⁻¹F by Cramer's rule

  Circles tangent to two or three objects tancircle.js

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• Tangent, tangent, tangent

  The circle tangent to three picked objects, nearest the picks.

  8 sign choices × 18 starting guesses round the picks; kept when tangent to all three within 10 × tol

  Circles tangent to two or three objects tancircle.js

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• Touch point and choice by the picks

  Where a circle touches an object, and which candidate circle touches nearest the picked points.

  line: foot of the center on the line; circle: point of o on the line of centers

  Circles tangent to two or three objects tancircle.js

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• Height unit factor

  The factor that turns a file's heights into the project's unit.

  factor = meters per file unit ÷ meters per project unit

  Elevation grids into survey points and a grid surface terrain.js

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• Cell center in the file's coordinates

  The x and y of a raster cell's center.

  x = g₀ + (col + 0.5)·g₁ + (row + 0.5)·g₂; y = g₃ + (col + 0.5)·g₄ + (row + 0.5)·g₅

  Elevation grids into survey points and a grid surface terrain.js

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  • BimCraft
  • CivilCraft
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• Cells inside a box

  The range of columns and rows whose centers can fall inside a picked box.

  (col, row) = G⁻¹·(x − g₀, y − g₃) − 0.5; c₀ = ⌈min − 1e-9⌉, c₁ = ⌊max + 1e-9⌋ + 1

  Elevation grids into survey points and a grid surface terrain.js

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  • BimCraft
  • CivilCraft
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• A box through a coordinate conversion

  The box round a rectangle after its outline is converted to another coordinate system.

  17 points along each side, converted, then minimum and maximum

  Elevation grids into survey points and a grid surface terrain.js

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  • BimCraft
  • CivilCraft
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• Survey points from a grid

  One survey point per grid node that has a height.

  (x, y) = conversion(cell center); z = file height × unit factor

  Elevation grids into survey points and a grid surface terrain.js

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  • BimCraft
  • CivilCraft
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• Grid surface

  A triangulated surface from the grid: two triangles per cell.

  cell corners a (SW), b (SE), c (NE), d (NW): triangles a–b–c and a–c–d

  Elevation grids into survey points and a grid surface terrain.js

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  • BimCraft
  • CivilCraft
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• 1-2-5 step

  Rounds a step up to 1, 2 or 5 times a power of ten.

  p = 10^⌊log₁₀ x⌋; m = x ÷ p; 1, 2, 5 or 10 × p

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Band of a wide polyline segment

  The filled outline of one polyline segment that has a start and end width.

  h(t) = (w₀ + (w₁ − w₀)·t) ÷ 2; left = p + n·h; right = p − n·h; arc: k = max(2, ⌈sweep ÷ 10°⌉) steps

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

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  64-bit

• Corner between two wide segments

  The wedge that fills the outside gap where one wide segment meets the next.

  cross = tₐ × t_b; s = ((P₂ − P₁) × t_b) ÷ cross; X = P₁ + s·tₐ; used when s ≥ 0 and |X − corner| ≤ 10 × max(hₐ, h_b)

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit

• Text background mask

  The rectangle painted behind a text: its box grown by a margin and turned with the text.

  m = max(0, scale − 1) × height; corner = (x, y) + R(rot) · (x₀ − m, y₀ − m) …

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

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  64-bit

• Ring direction

  Turns a ring clockwise so that overlapping rings add up when filled.

  a = Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ); a > 0 → reverse the points

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

  • CadCraft
  64-bit