Calculations

Calculations

• 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

• Numbers written into the macro

  A number as text in the macro.

  v.toPrecision(15)

  Sketch exchange macro and import swsketch.js

  • CadCraft
  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

  • CadCraft
  64-bit

• 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

  • CadCraft
  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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 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

• Ray angles

  The directions offered at every multiple of the angle step.

  aᵢ = start + i × step × (π ÷ 180), i = 0 … round(360 ÷ step) − 1

  Polar and object snap tracking tracking.js

  • CadCraft
  64-bit

• Pointer projected onto a ray

  The point of a ray nearest the pointer, and how far the pointer is from it.

  t = (q − p)·u; point = p + u·t, u = (cos a, sin a)

  Polar and object snap tracking tracking.js

  • CadCraft
  64-bit

• Crossing of two tracking rays

  The point where rays from two acquired points cross.

  den = a × b; t = ((B − A) × b) ÷ den; point = A + a·t

  Polar and object snap tracking tracking.js

  • CadCraft
  64-bit

• User coordinates to world

  A point or direction typed in the user coordinate system, in world coordinates.

  x = u.x + pₓ·cos − p_y·sin; y = u.y + pₓ·sin + p_y·cos

  User coordinate systems ucs.js

  • CadCraft
  64-bit

• World coordinates to user

  A world point in the user coordinate system, as shown in coordinates and prompts.

  d = p − origin; x = dₓ·cos + d_y·sin; y = −dₓ·sin + d_y·cos

  User coordinate systems ucs.js

  • CadCraft
  64-bit

• Is a coordinate system the world

  Normalizes a coordinate system and stores it as the world when it is the same.

  world when |x| < 1e-12, |y| < 1e-12 and angle within 1e-12 of 0 or 2π

  User coordinate systems ucs.js

  • CadCraft
  64-bit

• Coordinate system from an origin and an X-axis point

  The coordinate system whose X axis runs from one picked point through another.

  angle = atan2(p_y − o_y, pₓ − oₓ)

  User coordinate systems ucs.js

  • CadCraft
  64-bit

• Coordinate system from a polyline segment

  The coordinate system at the start of the polyline segment picked, with X along it.

  angle = atan2 of (point at 0.001 of the segment − its start)

  User coordinate systems ucs.js

  • CadCraft
  64-bit

• Pointer position to drawing coordinates

  The drawing point under a screen position, in the active pane.

  u = (sx − w ÷ 2) ÷ zoom; v = (h ÷ 2 − sy) ÷ zoom; x = cx + cos(tw)·u − sin(tw)·v; y = cy + sin(tw)·u + cos(tw)·v

  Picked points, object snaps and viewport views view.js

  • CadCraft
  64-bit

• Object snap under the pointer

  The best snap point (endpoint, midpoint, center, intersection and so on) within the aperture of the pointer.

  r = px ÷ zoom; keep candidates with |c − p| ≤ r; lowest priority number wins, then the smallest distance

  Picked points, object snaps and viewport views view.js

  • CadCraft
  64-bit

• Viewport view after pan or zoom inside it

  The center, scale and twist an entered sheet viewport shows on screen, returned for the caller to store in the viewport when they differ from the stored ones.

  M = P⁻¹ · T; s = √(M₀² + M₁²); tw = atan2(M₁, M₀); center = M⁻¹ · (vp.x, vp.y)

  Picked points, object snaps and viewport views view.js

  • CadCraft
  64-bit

• View to matrix

  The drawing-to-pixel matrix of a view for a pane of a given size.

  [z·c, z·s, z·s, −z·c, w ÷ 2 − z·(c·cx + s·cy), h ÷ 2 + z·(−s·cx + c·cy)] with c = cos(tw), s = sin(tw), z = zoom

  Picked points, object snaps and viewport views view.js

  • CadCraft
  64-bit

• Matrix to view

  Reads the center, zoom and turn back out of a drawing-to-pixel matrix.

  zoom = √(A₀² + A₁²); tw = atan2(A₁, A₀); center = A⁻¹ · (w ÷ 2, h ÷ 2)

  Picked points, object snaps and viewport views view.js

  • CadCraft
  64-bit

• Inverse of a 2 × 3 matrix

  The inverse of a drawing-to-pixel matrix.

  det = m₀·m₃ − m₁·m₂; inverse = adjugate ÷ det

  Picked points, object snaps and viewport views view.js

  • CadCraft
  64-bit