Calculations

Calculations

• Extend to a boundary

  An object with the end nearest the pick moved out to the closest boundary it reaches.

  Offset, extend, break and lengthen geom.js

  • CadCraft
  64-bit

• Break between two points

  An object with the part between two picked points removed, or split at one point.

  Offset, extend, break and lengthen geom.js

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  64-bit

• Lengthen by an amount

  An object made longer or shorter by a length, or an arc by an angle, at the end nearest the pick.

  line: moved end = fixed end ± u·(L + δ); arc: new sweep = sweep + δ ÷ r; polyline arc end: bulge = ± tan(new sweep ÷ 4)

  Offset, extend, break and lengthen geom.js

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  64-bit

• Fillet of two lines

  The arc of a given radius tangent to two lines and the lines cut back to it.

  half = acos(ua·ub) ÷ 2; t = r ÷ tan(half); center = corner + bisector × r ÷ sin(half)

  Fillet and chamfer geom.js

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  64-bit

• Tangent points and center at a corner

  The two tangent points and the center of a fillet at the corner of two straight pieces.

  half = acos(ua·ub) ÷ 2; t = r ÷ tan(half); center = corner + bisector × r ÷ sin(half)

  Fillet and chamfer geom.js

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  64-bit

• Fillet arc between two tangent points

  The arc entity of the fillet, the short way round from one tangent point to the other.

  Fillet and chamfer geom.js

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• Cutting an object back to the corner

  Moves one end of a line, arc or polyline segment to the tangent or chamfer point.

  arc end angle = atan2(T − center); polyline arc bulge = ± tan(sweep ÷ 4)

  Fillet and chamfer geom.js

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  64-bit

• Fillet of two picked objects

  The fillet arc and trimmed ends for two lines, a line and an arc or circle, two arcs, two segments of one polyline, or two parallel lines.

  with an arc: center = intersection of the two pieces offset by r; parallel lines: half circle of radius gap ÷ 2

  Fillet and chamfer geom.js

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  64-bit

• Chamfer distances

  How far to cut back each side of a corner, from two distances or from a length and an angle.

  angle method: θ = acos(ua·ub); da = len; db = len·sin α ÷ sin(θ + α)

  Fillet and chamfer geom.js

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  64-bit

• Chamfer of two picked pieces

  The chamfer line and trimmed ends for two straight pieces.

  Fillet and chamfer geom.js

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  64-bit

• Removing old fillet arcs

  Finds the fillet arcs a polyline already has and restores the sharp corners before a new radius is applied.

  θ = 4·atan(b); fillet when direction in = chord − θ ÷ 2 and direction out = chord + θ ÷ 2

  Fillet and chamfer geom.js

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  64-bit

• Every corner of a polyline

  Fillets or chamfers each corner of a polyline where two straight segments meet.

  sweep = π − acos(ua·ub); bulge = tan(sweep ÷ 4) × turn

  Fillet and chamfer geom.js

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  64-bit

• Polygon area and perimeter

  The signed area and the perimeter of a closed run of points.

  area = ½ Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ); perimeter = Σ |pᵢ₊₁ − pᵢ|

  Area, perimeter and length geom.js

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  64-bit

• Polyline area and length

  The area enclosed by a polyline with arc segments, and its length.

  area = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) + Σ ± r²(θ − sin θ)|; θ = 4·atan|b|; r = c ÷ (2·sin(θ ÷ 2)); arc length = r·θ

  Area, perimeter and length geom.js

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  64-bit

• Circle area and circumference

  The area and circumference of a circle.

  area = π·r²; circumference = 2π·r

  Area, perimeter and length geom.js

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  64-bit

• Hatch area and perimeter

  The filled area of a hatch with holes, and the total length of its boundaries.

  area = |Σ ± |ring area||, − for a ring inside an odd number of other rings; perimeter = Σ ring perimeters

  Area, perimeter and length geom.js

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  64-bit

• Length of a line, arc, polyline or spline

  The length used by the length and lengthen commands.

  line: √(Δx² + Δy²); arc: r × ((a1 − a0) mod 2π); polyline: Σ segment lengths

  Area, perimeter and length geom.js

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  64-bit

• Joining objects

  One line, arc, circle or polyline made from pieces that lie on one line, on one circle, or meet end to end.

  T = 1e-9 × max(width, height, 1); pieces join when they agree within 1000 × T

  Join and stretch geom.js

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  64-bit

• Bulge that keeps an arc's height

  The new bulge of an arc whose chord changes length, so that its height stays the same.

  b′ = b·c₀ ÷ c₁

  Join and stretch geom.js

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  64-bit

• Stretching an object

  An object with the points inside the stretch window moved and the rest left in place.

  Join and stretch geom.js

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  64-bit

• Normal of a grid line

  The unit direction a positive distance moves the grid line in: to the left of its start-to-end direction.

  n = (−Δy, Δx) ÷ L

  Moving a grid line with what hangs on it gridmove.js

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  64-bit

• What moves, and by how much

  The new geometry of every object the grid line carries with it.

  tol = max(0.0005 × L, 1e-6); reach = 0.15 × L; move = n × distance

  Moving a grid line with what hangs on it gridmove.js

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  64-bit

• Chaining pieces into paths

  Joins line and arc pieces that follow one another into open or closed paths.

  arc → bulge = tan((a1 − a0) ÷ 4); full circle → two half arcs with bulge 1

  Drawing to laser paths laser-cad.js

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  64-bit

• A path through a block transform

  A path placed by its block reference, keeping arcs where the transform allows.

  Drawing to laser paths laser-cad.js

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  64-bit

• Drawing units to millimeters

  The one scale factor that puts the drawing into millimeters.

  k = millimeters per drawing unit, or 1 when the drawing has no units

  Drawing to laser paths laser-cad.js

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  64-bit

• Picture placement

  Where a picture lies on the laser bed and how large it is.

  w = u_x × picture width; h = v_y × picture height, u and v the picture's axes through the transform

  Drawing to laser paths laser-cad.js

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  64-bit

• Landing and text start

  Where the landing ends and where the text starts.

  end = last + dir·(landing length, 0); text = end + dir·(gap, 0)

  Multileader geometry leader.js

  • CadCraft
  64-bit

• Leader line, arrowhead and text

  The polyline, the solid arrowhead and the text a multileader is drawn and exploded as.

  u = (tip − next) ÷ |tip − next|; base = tip − u·a; corners = base ± (−u_y, u_x)·a ÷ 6; text top = landing y + h ÷ 2

  Multileader geometry leader.js

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  64-bit

• abs, round, ceil and floor

  The four functions an expression may call.

  round(x) = floor(x + 0.5) for x ≥ 0, −floor(−x + 0.5) for x < 0; ceil(x) = ⌈x − 1e-9·max(1, |x|)⌉; floor(x) = ⌊x + 1e-9·max(1, |x|)⌋

  Arithmetic on typed lengths lencalc.js

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  64-bit

• Adding and subtracting

  Sums and differences of lengths, left to right.

  Arithmetic on typed lengths lencalc.js

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  • MarkupCraft
  • TakeoffCraft
  • CivilCraft
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  64-bit

• Multiplying and dividing

  Products and quotients of a length and plain numbers, left to right.

  Arithmetic on typed lengths lencalc.js

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  • ModelCraft
  • MarkupCraft
  • TakeoffCraft
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  64-bit

• The answer

  The final value and whether it is a length or a plain number.

  Arithmetic on typed lengths lencalc.js

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  64-bit

• Millimeters per unit

  The table of unit factors.

  in 25.4; ft 304.8; yd 914.4; mi 1 609 344; mil 0.0254; µm 0.001

  Lengths and units for Library items libitem.js

  • CadCraft
  • BimCraft
  Exact

• Length from text

  A typed or stored length in millimeters.

  with a unit: n × factor; feet and inches: (feet × 12 + inches + a ÷ b) × 25.4; bare number: n × mm per unit

  Lengths and units for Library items libitem.js

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  • BimCraft
  64-bit

• Drawing units

  The millimeters per drawing unit of a DWG or DXF file.

  Lengths and units for Library items libitem.js

  • CadCraft
  • BimCraft
  Exact

• Pipe size in eighths of an inch

  A decimal inch size written as a fraction, such as 1 1/2".

  eighths = round((n − ⌊n⌋) × 8)

  Lengths and units for Library items libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Size label on the preview

  The overall width and depth printed on an item's preview picture.

  millimeters: round(mm); inches: round(mm ÷ 25.4 × 8) eighths

  Lengths and units for Library items libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Millimeters to drawing units

  The factor that puts an item's millimeters into the drawing's units.

  k = 1 ÷ (mm per drawing unit), or 1 when the drawing has no units

  Library item placed as a block libplace.js

  • CadCraft
  64-bit

• Plan entities in drawing units

  Each line, circle, arc, polyline, ellipse, text, point and hatch of the item as a drawing object.

  length × k; angle in degrees × π ÷ 180, reduced to [0, 2π)

  Library item placed as a block libplace.js

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  64-bit

• Size attributes

  The width, depth, height and weight written as attributes on the block.

  Library item placed as a block libplace.js

  • CadCraft
  64-bit

• Count and quantity of each item

  How many times each Library item is placed and the total of their quantities, for the credits and schedule.

  placed = number of references; quantity = Σ parseFloat(quantity attribute), 1 when not above 0

  Library item placed as a block libplace.js

  • CadCraft
  64-bit

• Pattern in drawing units

  The dash and gap lengths of a linetype for this drawing.

  metric drawing: inch pattern × 25.4

  Linetype dashes linetypes.js

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  64-bit

• Dashes along lines and arcs

  The pieces of each line or arc that are drawn.

  piece from fraction s ÷ length to (s + step) ÷ length; arc length = r × sweep

  Linetype dashes linetypes.js

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  64-bit

• Local placement record

  The scale, rotation and point pair that tie drawing coordinates to grid coordinates.

  scale = m per drawing unit ÷ m per grid unit; from = (x, y); to = (e, n)

  Drawing placed on a map grid maplayers.js

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  • BimCraft
  64-bit

• Axes of an object's own coordinate system

  The X, Y and Z axes of the plane an object was drawn on, from its extrusion direction.

  Z = N ÷ |N|; X = (Wy × Z or Wz × Z) ÷ length; Y = Z × X

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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  64-bit

• Is the plane the drawing's own

  Whether an extrusion direction is straight up, so that nothing needs mapping.

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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  64-bit

• Plane to plan matrix

  The transform from an object's own (x, y) at its elevation to the plan.

  M = [X.x, X.y, Y.x, Y.y, Z.x·elevation, Z.y·elevation]

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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• Mapped circle, arc or ellipse

  The circle, arc or ellipse that has exactly the points of the original after the map.

  t* = ½·atan2(2 u·v, |u|² − |v|²); U = u·cos t* + v·sin t*; V = −u·sin t* + v·cos t*; a = |U|, b = |V|

  Objects on a tilted or flipped plane mapped to the plan ocs.js

  • CadCraft
  64-bit

• Polyline arcs as points

  A polyline's arc segments as points, for a map that does not keep shapes.

  k = max(4, ⌈64 × |sweep| ÷ 2π⌉); point = c + r·(cos a, sin a)

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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  64-bit

• Mapping each kind of object

  A drawing object in its own coordinate system put through the plane's matrix.

  text: rotation = atan2(M·d); height × |M·d × M·up| ÷ |M·d|; block: scales and rotation read from the mapped axes

  Objects on a tilted or flipped plane mapped to the plan ocs.js

  • CadCraft
  64-bit