Calculations

Calculations

• Pixel to coordinates

  The coordinates of a pixel position in a picture's own system.

  x = g₀ + u·g₁ + v·g₂; y = g₃ + u·g₄ + v·g₅

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
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• Numbers in a map request

  How the corners of the requested box are written into a WMS or map-server export address.

  |v| ≥ 10,000: v.toFixed(3); otherwise v.toPrecision(12)

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Ortho: the point along the nearer axis

  The point a click stands for when ortho is on and there is a base point.

  u = c·Δx + s·Δy; v = −s·Δx + c·Δy; point = base + u·(c, s) if |u| ≥ |v|, else base + v·(−s, c)

  Cursor point and status readouts cadcraft.js

  • CadCraft
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• Coordinates, distance and angle in the status line

  The cursor's coordinates and, during a command, the distance and angle from the base point.

  Δ = √(Δx² + Δy²); angle = atan2(Δy, Δx) − UCS angle, normalized to 0…360°

  Cursor point and status readouts cadcraft.js

  • CadCraft
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• Length text in the model and on a sheet

  A length written for the status line, the tracking labels and the cursor box.

  Cursor point and status readouts cadcraft.js

  • CadCraft
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• Distance and angle beside the cursor

  The live distance or angle shown in the box by the cursor while a command waits for one.

  d = √(Δx² + Δy²); angle = atan2(Δy, Δx), normalized to 0…360°

  Cursor point and status readouts cadcraft.js

  • CadCraft
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• Size of the plot area on paper

  The width and height of the chosen area at the chosen scale, shown in the Plot dialog.

  w = (maxX − minX) × s, shown as round(w × 10) ÷ 10

  Cursor point and status readouts cadcraft.js

  • CadCraft
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• Number typed in Quick select

  The number a property is compared with in the Quick select dialog.

  Cursor point and status readouts cadcraft.js

  • CadCraft
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• Turn sense of an angle constraint

  Whether the angle between two constrained lines is measured counter-clockwise or clockwise after a mirror or a trim.

  sign = +1 if atan2(u × v, u · v) ≥ 0, else −1

  Constraints kept through mirror and trim ckeep.js

  • CadCraft
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• Which piece holds the original's ends

  The piece of a trimmed line or arc that still has the original's start, and the one that still has its end.

  start piece: t₀ ≤ 10⁻⁹; end piece: t₁ ≥ 1 − 10⁻⁹

  Constraints kept through mirror and trim ckeep.js

  • CadCraft
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• Numbers in prompts and messages

  A number written for a prompt's default or a message in the command history.

  |n| < 10⁻¹⁰ → 0; otherwise n.toFixed(4) without trailing zeros

  Typed points, distances and angles commands.js

  • CadCraft
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• Typed distance, number and angle

  The value a prompt takes when a distance, a number or an angle is typed.

  angle = degrees × π ÷ 180 (+ UCS angle for an absolute angle)

  Typed points, distances and angles commands.js

  • CadCraft
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• Distance or angle shown by two clicks

  The distance or the angle a prompt takes when it is shown by picking two points.

  d = √(Δx² + Δy²); angle = atan2(Δy, Δx)

  Typed points, distances and angles commands.js

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• Typed point

  The point for typed coordinates: absolute, relative to the last point, polar, or a distance along the cursor direction.

  x,y → UCS to world; @x,y → last + R(ucs)·(x, y); d<a → from + d·(cos, sin)(a·π ÷ 180 + UCS angle); d → base + d·(cos, sin)(atan2(cursor − base))

  Typed points, distances and angles commands.js

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• Calculation on the command line

  The result CAL prints for an expression such as 12'6"+3/4" or 2*450mm.

  raw = value.toPrecision(12)

  Typed points, distances and angles commands.js

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• Circle through three points

  The center and radius of the circle through three picked points.

  d = 2·(aₓ(b_y − c_y) + bₓ(c_y − a_y) + cₓ(a_y − b_y)); centerₓ = (|a|²(b_y − c_y) + |b|²(c_y − a_y) + |c|²(a_y − b_y)) ÷ d; r = |a − center|

  Shapes made by the draw commands commands.js

  • CadCraft
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• Arc through three points

  The center, radius and end angles of the arc from a start point through a second point to an end point.

  center, r from circle3; counter-clockwise when (m − s) × (e − s) > 0; a₀ = atan2(s − center), a₁ = atan2(e − center)

  Shapes made by the draw commands commands.js

  • CadCraft
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• Arc segment of a polyline

  The bulge of a polyline's arc segment that leaves the last point along the previous segment's direction and ends at the picked point.

  φ = angle(p → q) − tangent at p, wrapped to −π…π; bulge = tan(φ ÷ 2)

  Shapes made by the draw commands commands.js

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• Circle from the two ends of a diameter

  The center and radius of a circle given by two opposite points on it.

  center = (a + b) ÷ 2; r = |b − a| ÷ 2

  Shapes made by the draw commands commands.js

  • CadCraft
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• Arc from center, start and end

  The radius and end angles of an arc given its center, its start point and a point toward its end.

  r = |start − center|; a₀ = atan2(start − center); a₁ = atan2(end − center)

  Shapes made by the draw commands commands.js

  • CadCraft
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• Rectangle from two corners

  The closed four-corner polyline between two picked corners.

  Shapes made by the draw commands commands.js

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• Regular polygon

  The corners of a regular polygon inscribed in or drawn round a circle.

  R = r (inscribed) or r ÷ cos(π ÷ n) (circumscribed); pᵢ = center + R·(cos, sin)(a₀ + i·2π ÷ n)

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit

• Ellipse from an axis and a half-width

  The center, major-axis vector and axis ratio of an ellipse.

  ratio = half-width ÷ L, kept in 10⁻⁶…1; if half-width d > L: major = d·(−m_y, mₓ) ÷ L, ratio = L ÷ d

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit

• Construction line at an angle

  The direction of a construction line given by a typed angle, alone or measured from a picked straight object.

  direction = (cos a, sin a); with a reference: a = atan2(u_y, uₓ) + typed angle

  Shapes made by the draw commands commands.js

  • CadCraft
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• Construction line that bisects an angle

  The direction of the line through a vertex that halves the angle between two legs.

  b = unit(u₀ + u₁), u₀, u₁ the unit vectors of the legs

  Shapes made by the draw commands commands.js

  • CadCraft
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• Construction line parallel at a distance

  The point and direction of a construction line parallel to a picked straight object.

  foot(q) = on + ((q − on)·u)·u; new point = foot(pick) + d·n, n = unit(side − foot(side))

  Shapes made by the draw commands commands.js

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• Lines of single-line text

  Where each further line of TEXT is placed below the first.

  line k at start + k × 1.667 × h, at right angles to the text direction

  Shapes made by the draw commands commands.js

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• Paragraph text box

  The box between two picked corners, measured along the text's own direction, and the width and insertion point stored from it.

  lx = d·u; ly = d·n, u = (cos rot, sin rot), n = (−sin rot, cos rot)

  Shapes made by the draw commands commands.js

  • CadCraft
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• Move or copy through a transform

  Applies one transform to the selected objects, replacing them or adding copies.

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

  • CadCraft
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• Displacement for MOVE and COPY

  The translation from a base point to a second point, or a typed displacement.

  M = T(p − base)

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

  • CadCraft
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• Rotate and scale about a base point

  The transform for ROTATE and SCALE.

  M = T(base) · R(a) · T(−base); M = T(base) · S(s) · T(−base)

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

  • CadCraft
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• Mirror across a line

  The transform that mirrors the selection across the line through two picked points.

  M = T(a) · [cos 2θ, sin 2θ; sin 2θ, −cos 2θ] · T(−a), θ = atan2(b − a)

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

  • CadCraft
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• Copies of a rectangular or polar array

  The copies an array makes, in rows and columns or round a center.

  rectangular: T(k·dx, r·dy); polar: step = fill ÷ items (full circle) or fill ÷ (items − 1); copy i = rotation by step × i about the center

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

  • CadCraft
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• Which points a stretch moves

  Whether a point lies inside one of the crossing windows that define a stretch.

  minX − 10⁻⁹ ≤ x ≤ maxX + 10⁻⁹ and minY − 10⁻⁹ ≤ y ≤ maxY + 10⁻⁹

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

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• Length change for LENGTHEN

  How much an object's end moves for the delta, percent, total and dynamic options.

  percent: Δ = L × (p ÷ 100 − 1); total: Δ = total − L; total angle: Δθ = angle − sweep; dynamic: Δ = ((cursor − A)·(B − A)) ÷ |B − A| − |B − A|

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

  • CadCraft
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• Nearest position on a line or arc

  The position (0 at the start, 1 at the end) of the point on a line or arc nearest a given point.

  line: t = clamp(((q − p₁)·(p₂ − p₁)) ÷ |p₂ − p₁|², 0, 1); arc: t = (angle from the start) ÷ sweep

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

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• Point on an object nearest a pick

  Where a break point or a pick lands on a line, arc, circle or polyline.

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

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• Offset distance through a point

  The distance OFFSET uses: the typed one, or the distance from the object to a picked point.

  through: d = min over the primitives of |p − nearest point on the primitive|

  Move, rotate, scale, mirror, array, stretch and lengthen commands.js

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• Distance, Δx, Δy and angle between two points

  What DIST prints for two picked points.

  d = √(Δx² + Δy²); angle = atan2(Δy, Δx), normalized to 0…360°

  Distance, area, angle and point inquiries commands.js

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• Area and perimeter of picked points

  The area and perimeter of the outline through points picked one by one.

  area = |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| ÷ 2; perimeter = Σ |pᵢ₊₁ − pᵢ|

  Distance, area, angle and point inquiries commands.js

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• Area and perimeter of an object

  What AREA prints for a picked circle, polyline, ellipse or hatch.

  Distance, area, angle and point inquiries commands.js

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• Running area total

  The total AREA keeps while areas are added and subtracted.

  total = Σ ± areaᵢ

  Distance, area, angle and point inquiries commands.js

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• Angle at a vertex, of an arc, or between two lines

  The angle MEASUREGEOM prints.

  angle = |atan2(b − v) − atan2(a − v)|, folded to 0…180°; arc: its sweep, with 0 read as 360°

  Distance, area, angle and point inquiries commands.js

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• Coordinates of a point

  What ID prints for a picked point.

  Distance, area, angle and point inquiries commands.js

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• One object in a line of text

  The description LIST prints and assistants read for an object: its points, length, radius and angles.

  line length = √(Δx² + Δy²); angles in degrees = radians × 180 ÷ π

  Distance, area, angle and point inquiries commands.js

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• Units of exported points

  The factor between drawing units and the units of the point file EXPORTPOINTS writes.

  factor = mm per unit ÷ 1000 (meters), ÷ 304.8 (feet), or 1 (drawing units)

  Distance, area, angle and point inquiries commands.js

  • CadCraft
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• Two origins of a linear dimension

  The two points a linear or aligned dimension measures between: picked, or the ends of a picked object.

  arc end = center + r·(cos a, sin a); circle: (cx − r, cy) and (cx + r, cy)

  Where dimensions are attached commands.js

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• Horizontal or vertical

  Whether a linear dimension measures along the X or the Y axis, from where its line is placed.

  Where dimensions are attached commands.js

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• Point on the circle for a radius or diameter

  The point on an arc or circle that a radius or diameter dimension is attached to.

  p₁ = center + r·(cos, sin)(atan2(cursor − center))

  Where dimensions are attached commands.js

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• Are two lines parallel

  Whether two picked lines have an angle between them that can be dimensioned.

  parallel when |e × f| < 10⁻¹² × |e| × |f|

  Where dimensions are attached commands.js

  • CadCraft
  64-bit