Calculations

Calculations

• Trimmed curve

  The part of a line, circle, ellipse or other curve between two trims given as parameters or points.

  conic: angle = parameter × angle unit, or atan2 of the trim point in the conic's frame

  IFC curves and profiles as point lists ifc/geo-curve.js

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• B-spline curve points

  Points on a B-spline curve, rational or not, at 12 evenly spaced parameters per knot span.

  de Boor: d_j = (1 − a) d_{j−1} + a d_j, a = (u − k_i) ÷ (k_{i+p+1−r} − k_i); point = (x, y, z) ÷ w

  IFC curves and profiles as point lists ifc/geo-curve.js

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• Offset of a polyline

  A 2D polyline moved to its left by a distance, with mitered corners.

  p' = p + n × d ÷ max(0.2, n · n₁), n = normalized sum of the two edge normals

  IFC curves and profiles as point lists ifc/geo-curve.js

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• Circle and ellipse profiles

  The outline of a circular, hollow circular or elliptical profile.

  (r₁ cos a, r₂ sin a), a = 2π i ÷ 36

  IFC curves and profiles as point lists ifc/geo-profile.js

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• Fillets and edge radii

  Replaces corners of a profile outline with arcs of given radii tangent to both edges.

  d = r ÷ tan(half); center = corner + bisector × r ÷ sin(half), half = half the corner angle

  IFC curves and profiles as point lists ifc/geo-profile.js

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• I-shape outline

  The outline of a symmetric or asymmetric I-section with flange slopes, fillets and edge radii.

  inner flange face = thickness + tan(slope) × distance from the quarter point of the outstand

  IFC curves and profiles as point lists ifc/geo-profile.js

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• Closing an outline

  Drops repeated points of a profile curve and its closing point.

  IFC curves and profiles as point lists ifc/geo-profile.js

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• Straight extrusion

  The solid of a profile swept along a direction by a depth, with end caps and sides.

  top = bottom + direction × depth

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Side face of a sweep

  Adds a four-sided face as one flat face or as two triangles.

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Revolved solid

  The solid of a profile turned about an axis by an angle.

  p_k = o + R(axis, angle × k ÷ steps) · (p − o), steps = 36 per full turn

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Frames along a path

  The tangent, normal and binormal at each point of a 3D polyline, and the miter stretch at each bend.

  tangent = normalize(t_in + t_out); miter = 1 ÷ max(0.25, t_in · tangent)

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Normal carried round a bend

  Turns the section's normal by the rotation that takes the previous tangent to the next.

  angle = atan2(|a × b|, a · b); v' = R(a × b, angle) v, made square to b

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Swept disk

  A round bar or tube along a path, such as rebar, with an optional inner radius and bend radius.

  section = 36-sided polygon of the radius; swept on the path's frames

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Bends of a swept path

  Rounds the corners of a 3D polyline with arcs of a given radius.

  d = min(r ÷ tan(half), 0.999 × each leg); center = corner + bisector × r' ÷ sin(half)

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Sphere

  The mesh of an IfcSphere.

  (r sin θ cos φ, r sin θ sin φ, r cos θ), θ = π i ÷ 12, φ = 2π j ÷ 24

  IFC swept solids, primitives and faceted shapes ifc/geo-solid.js

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• Outer bound of a face

  Picks the largest loop of a face as its outer bound when the file marks none.

  size = |Σ ((yₐ − y_b)(zₐ + z_b), (zₐ − z_b)(xₐ + x_b), (xₐ − x_b)(yₐ + y_b))| (twice the area)

  IFC swept solids, primitives and faceted shapes ifc/geo-brep.js

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• Repeated points of a loop

  Drops consecutive points of a face loop that coincide, and a closing point equal to the first.

  IFC swept solids, primitives and faceted shapes ifc/geo-brep.js

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• Point on a clothoid

  The point at a length along a clothoid from its origin.

  x = ∫₀ˢ cos(k u²) du, y = ∫₀ˢ sin(k u²) du, k = sign(A) ÷ (2A²); Simpson's rule

  IFC alignment curves and stations ifc/geo-align.js

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• Points of a curve segment

  Samples one alignment segment from its parent curve and places it at the segment's start.

  n = max(2, min(400, ceil(|length| ÷ 2))); p_i = T · f(start + length × i ÷ n)

  IFC alignment curves and stations ifc/geo-align.js

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• Distance along the alignment

  The horizontal distance of each sampled point from the start.

  d_i = d_{i−1} + √(Δx² + Δy²)

  IFC alignment curves and stations ifc/geo-align.js

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• Height from the vertical profile

  The height of each horizontal point from the alignment's vertical profile.

  z = z_lo + (z_hi − z_lo) × (d − d_lo) ÷ (d_hi − d_lo)

  IFC alignment curves and stations ifc/geo-align.js

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• Point at a station

  The position and direction on a curve at a distance along it.

  p = p_{i−1} + (p_i − p_{i−1}) × (d − d_{i−1}) ÷ (d_i − d_{i−1})

  IFC alignment curves and stations ifc/geo-align.js

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• Placement along an alignment

  The matrix of an element placed at a station with lateral, vertical and longitudinal offsets.

  origin = p + left × lateral + (0, 0, vertical) + heading × longitudinal

  IFC alignment curves and stations ifc/geo-align.js

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• Solid between cross sections

  The solid of an IfcSectionedSolidHorizontal: cross sections placed across the alignment at their stations and joined.

  ring point = o + x × ((1 − f) a_x + f b_x) + y × ((1 − f) a_y + f b_y), f = (d − d₀) ÷ (d₁ − d₀)

  IFC alignment curves and stations ifc/geo-align.js

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• Cylinder and cone

  A point on a cylindrical or conical surface from (u, v), and (u, v) from a point.

  p = o + (r + v tan α)(x cos u + y sin u) + z v; u = atan2(d · y, d · x), v = d · z

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Swept curve by length

  The generating curve of an extruded or revolved surface as a polyline with a length parameter.

  s_i = s_{i−1} + |p_i − p_{i−1}|; point at s by linear interpolation

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• B-spline basis functions

  The non-zero basis functions of a B-spline at a parameter.

  Cox-de Boor: N_r = saved + right × N_r ÷ (right + left)

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Point on a B-spline surface

  The point of a B-spline surface, rational or not, at (u, v).

  p = Σ N_i(u) N_j(v) (w P)_ij ÷ Σ N_i(u) N_j(v) w_ij

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Parameters of a point on a B-spline surface

  Finds (u, v) of a 3D point on a B-spline surface.

  start at the nearest of 25 × 25 samples; Newton: solve (Sᵤ·Sᵤ, Sᵤ·Sᵥ; Sᵤ·Sᵥ, Sᵥ·Sᵥ) Δ = (r·Sᵤ, r·Sᵥ)

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Boundary points put back on the surface

  Moves a boundary point onto the surface when the file's edge curve leaves it.

  tolerance = max(10⁻⁹, 0.001 × diagonal of the loops' box)

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Spacing of interior points

  The spacing of the points added inside a face, from how fast the surface normal turns.

  h = clamp(0.2 ÷ turn, extent ÷ 96, extent ÷ 4), turn = largest change of the normal per unit over a 9 × 9 grid

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Scale of the parameter plane

  Scale factors that make distances in (u, v) roughly equal to distances on the surface.

  s_u = mean |S(u + e, v) − S(u, v)| ÷ e, e = 10⁻⁴ (1 + |u|)

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Circumcircle of a triangle

  The center and squared radius of a triangle's circumcircle, for the Delaunay test.

  D = 2 (a_x (b_y − c_y) + b_x (c_y − a_y) + c_x (a_y − b_y)); center = Σ |p|² × (difference) ÷ D

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Delaunay triangulation

  Triangulates the face's points in the scaled parameter plane.

  a point is inside a circumcircle when (x − c_x)² + (y − c_y)² < r² (1 + 10⁻¹²)

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• Orientation and crossing tests

  Whether three points turn left or right, and whether two segments cross, for recovering the boundary edges.

  orient = (b_x − a_x)(c_y − a_y) − (b_y − a_y)(c_x − a_x)

  IFC faces on curved surfaces ifc/geo-nurbs.js

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• SI value of the file's units

  The factor that turns the file's length, area, volume and mass values into meters, square meters, cubic meters and kilograms.

  SI unit: factor = prefix; conversion-based unit: the file's factor; gram: × 0.001

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Quantities of an element

  The lengths, areas, volumes and weights an element states in its quantity sets, in SI units.

  SI value = stated value × unit factor

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Length of an alignment

  The length of an alignment curve from the lengths of its segments.

  L = Σ |SegmentLength|

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Rounding of totals

  Rounds a total length, area or volume before it is listed or written to a quantities file.

  ROUND(v) = v to 12 significant digits

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Totals per class

  The count and the total length, area and volume of each class and predefined type.

  total = Σ of the first stated quantity of that kind per element

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Numbers in a written file

  Formats a coordinate, length or quantity for a new IFC 4.3 file.

  R(v, d) = v with d decimals, trailing zeros removed; d = 9, or 12 for directions

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Alignment segments written

  The straight segments of a written alignment: each one's start, direction and length.

  length = √(Δx² + Δy²); direction = (Δx, Δy) ÷ length

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Area of a written profile

  The cross-section area of an element in a written file, from its rectangle, circle or polygon.

  circle: π r²; polygon: |Σ (xₐ y_b − x_b yₐ)| ÷ 2; rectangle: w × d

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Quantities written for an element

  The length, width, height, footprint area and volume written to an element's quantity set.

  Length = longer side; Width = shorter side; GrossVolume = area × height

  IFC 4.3 infrastructure quantities and files ifc/ifc43.js

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• Position and direction of a port

  Where a port is, in millimeters, and the way it faces.

  p = M · (0, 0, 0) × mm; z = normalize(M · (0, 0, 1) direction)

  Piping in an IFC file as PCF components ifc/pipes.js

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• Ends and bore from the extruded body

  The two ends of a pipe segment and its bore from its extruded solid.

  a = T · 0; b = T · (direction × depth); bore = 2 × radius × mm

  Piping in an IFC file as PCF components ifc/pipes.js

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• Turn angle of a fitting

  The angle a two-port fitting turns the pipe through.

  turn = 180 − acos(p · q) × 180 ÷ π, rounded to 2 decimals

  Piping in an IFC file as PCF components ifc/pipes.js

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• Center point of a bend

  The corner where the axes of a bend's two ports meet.

  s = (b e − c d) ÷ (a c − b²), t = (a e − b d) ÷ (a c − b²); center = midpoint of the two closest points

  Piping in an IFC file as PCF components ifc/pipes.js

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• Size in millimeters

  The width and height of the copied area, on paper or at the drawing's real size.

  paper: mm = pixels × ptPerPx × (25.4 ÷ 72); real: mm = pixels × mmPerPx

  Size of a copied area areaclip.js

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• Render size in pixels

  The pixel width and height for an area given in points at a resolution, lowered so neither side passes the cap.

  d = dpi × min(1, maxPx ÷ (longer side × dpi ÷ 72)); w = round(wPt × d ÷ 72); h = round(hPt × d ÷ 72)

  Size of a copied area areaclip.js

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