Calculations

Calculations

• Scale from two points and a real distance

  Rescales a scan so two picked points are a stated distance apart.

  f = wanted ÷ |b − a|; insertion′ = a + f·(insertion − a); sx′ = sx·f

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

  • CadCraft
  64-bit

• Align by two pairs of points

  Moves, turns and optionally scales a scan so two points on it land on two points of the drawing.

  f = |d₂ − d₁| ÷ |s₂ − s₁|; θ = atan2(d₂ − d₁) − atan2(s₂ − s₁); insertion′ = d₁ + f·R(θ)·(insertion − s₁)

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

  • CadCraft
  64-bit

• Straightening a skewed scan

  Turns the picture back by the skew angle that was found or typed.

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

  • CadCraft
  64-bit

• Crop window in pixels

  The whole-pixel window of a crop and the new corner of the cropped picture.

  x₀ = ⌊min⌋, y₀ = ⌊min⌋, x₁ = ⌈max⌉, y₁ = ⌈max⌉; corner = p + u·x₀ + v·(h − y₁)

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

  • CadCraft
  64-bit

• Trace tolerance

  The tolerance of a trace, typed in drawing units and used in pixels.

  tolerance (px) = typed ÷ pixel size, held to 0.5 … 20

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

  • CadCraft
  64-bit

• Outline moved inward

  The outline offset inward by a distance, with mitered corners.

  corner′ = corner + bisector × d ÷ cos(half angle)

  Tapered, revolved and swept solids shapes3d.js

  • CadCraft
  64-bit

• Signed volume and face direction

  Works out whether the faces point outward and refuses a solid with no volume.

  V = Σ p₀ · (pᵢ × pᵢ₊₁) ÷ 6 over fan triangles

  Tapered, revolved and swept solids shapes3d.js

  • CadCraft
  64-bit

• Tapered extrusion

  A solid whose top is the outline inset by the height times the tangent of the taper angle.

  inset = |h| × tan(angle); vertices relative to the outline's minimum x and y

  Tapered, revolved and swept solids shapes3d.js

  • CadCraft
  64-bit

• Revolution about an axis

  A solid made by turning the outline about an axis in steps.

  n = max(2, ⌈48 × |angle| ÷ 2π⌉); p′ = v·cos θ + (u × v)·sin θ + u(u·v)(1 − cos θ)

  Tapered, revolved and swept solids shapes3d.js

  • CadCraft
  64-bit

• Sweep along a path

  A solid made by standing the outline across a path, with mitered corners.

  centroid = Σ (pᵢ + pⱼ)wᵢ ÷ 3Σwᵢ, wᵢ = xᵢyⱼ − xⱼyᵢ; frame normal = bisector ÷ cos(half angle)

  Tapered, revolved and swept solids shapes3d.js

  • CadCraft
  64-bit

• Repeated path points

  Drops path points that repeat the one before.

  |pᵢ − pᵢ₋₁| ≤ 1e-9 → dropped

  Tapered, revolved and swept solids shapes3d.js

  • CadCraft
  64-bit

• Pen moves

  Moves the pen by a displacement of the shape.

  x′ = x + dx × mul ÷ div

  SHX font text: strokes and width shx.js

  • CadCraft
  64-bit

• Arcs as chords

  Draws an arc of a shape as short straight pieces.

  n = max(2, ⌈|sweep| ÷ 7.5°⌉); pointₖ = c + r(cos aₖ, sin aₖ), aₖ = a₀ + sweep × k ÷ n

  SHX font text: strokes and width shx.js

  • CadCraft
  64-bit

• Bulge arcs

  An arc given by its chord and a bulge value.

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

  SHX font text: strokes and width shx.js

  • CadCraft
  64-bit

• Octant and fractional arcs

  Arcs given by a radius, a starting octant and a span of octants, with optional fractions of an octant.

  a₀ = (s₀ + so ÷ 256) × π ÷ 4; center = pen − r(cos a₀, sin a₀)

  SHX font text: strokes and width shx.js

  • CadCraft
  64-bit

• Text strokes and width

  The strokes of a text for capitals of a given height, and the text's width.

  k = h ÷ capital height of the font; width = Σ glyph advance × k

  SHX font text: strokes and width shx.js

  • CadCraft
  64-bit

• Width of a text in an SHX font

  The width the text layout uses for a text whose style names a loaded .shx font.

  width = advance × width factor

  SHX font text: strokes and width shx-text.js

  • CadCraft
  64-bit

• Strokes placed in the drawing

  Each stroke point of a text taken to drawing coordinates with alignment, width factor, slant and rotation.

  lx = x₀ + (gx + gy·tan(oblique)) × wf; X = x + lx·cos − gy·sin; Y = y + lx·sin + gy·cos

  SHX font text: strokes and width shx-text.js

  • CadCraft
  64-bit

• Circle through a stroke

  The circle that best fits the stroke's points, with its root-mean-square error.

  u = x − x̄, v = y − ȳ; solve [Σu² Σuv; Σuv Σv²]·(u_c, v_c) = ½(Σu³ + Σuv², Σv³ + Σvu²); r² = u_c² + v_c² + (Σu² + Σv²) ÷ n

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• Turn of a stroke about a center

  How far the stroke goes round the fitted center, signed.

  Σ (aᵢ − aᵢ₋₁) with each step brought into (−π, π]

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• Squaring a direction

  Makes a direction level or plumb when it is within a few degrees of it.

  q = round(a ÷ (π ÷ 2)) × (π ÷ 2); used when |a − q| ≤ deg × π ÷ 180

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• Where two sides meet

  The corner where two straight sides cross.

  det = l × m; t = ((m₀ − l₀) × m) ÷ det; point = l₀ + l·t

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• Distance from a point to a segment

  How far a pen point is from the straight line between two others.

  t = ((p − a)·(b − a)) ÷ |b − a|², held to [0, 1]; distance = |p − (a + t(b − a))|

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• Straight sides and rectangles

  Cuts a stroke at its corners, squares each side, and rebuilds the corners where the sides cross.

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• A stroke as a line

  Makes a nearly straight stroke a line, exactly level or plumb when it is close to it.

  straight when max distance ≤ max(eps, 3.5 % of the gap) and gap > 90 % of the length

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• A stroke as a circle or an arc

  Makes a round stroke a circle, or an arc from the stroke's first point to its last.

  accepted when rms ÷ r < 0.07 and r < 3 × box diagonal; circle when |turn| > 0.85 × 2π

  Pen sketch cleanup sketch.js

  • CadCraft
  • MarkupCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • 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

  • CadCraft
  64-bit