Calculations

Calculations

• Vertex at a breakline crossing

  Places a new vertex where two breaklines cross and gives it an elevation.

  x = x_a + (x_b − x_a) t; z from the first breakline

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• Side of a breakline a vertex is on

  Decides whether a vertex lies left of, right of, or on the line of a breakline being forced in.

  on the line when |orient| ÷ L < 1e-7 × (1 + L)

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• Whether a diagonal can be flipped

  Tests that the two triangles on an edge form a strictly convex quadrilateral.

  orient(c, d, a) and orient(c, d, b) have opposite signs

  Ground surface triangulation and contours tin.js

  • BimCraft
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  Exact

• Whether an edge crosses a breakline

  Tests whether two segments properly cross during the flips that force a breakline in.

  each segment's ends lie on opposite sides of the other

  Ground surface triangulation and contours tin.js

  • BimCraft
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  Exact

• Long outer triangles removed

  Removes triangles on the outside whose outer edge is longer than a limit, working inward.

  remove when outer edge length > maxEdge

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• Contour lines

  The contour lines of the surface at every multiple of an interval.

  level = base + k × interval; t = (L − zᵢ) ÷ (zⱼ − zᵢ); point = pᵢ + (pⱼ − pᵢ) t

  Ground surface triangulation and contours tin.js

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  64-bit

• Triangle clipped at an elevation

  Keeps the part of a triangle above or below a level, for the elevation bands.

  t = f_p ÷ (f_p − f_q); new point = p + (q − p) t at z = L

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• Convex hull

  The convex hull of a set of points, counter-clockwise.

  monotone chain; pop while cross(o, a, b) ≤ 0

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• Polygon area

  The signed area of a polygon.

  A = ½ Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• Point inside a polygon

  Tests whether a point is inside a polygon.

  count crossings of a ray to the right; odd = inside

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• Polygon clipped by a convex polygon

  Clips a polygon to a convex polygon, used to keep a boundary inside the data's hull.

  inside when cross(b − a, p − a) ≥ −1e-9; t = cross(a − p, e) ÷ cross(d, e)

  Ground surface triangulation and contours tin.js

  • BimCraft
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  64-bit

• SI value of a file unit

  Meters per file length unit and radians per file angle unit.

  SI unit: 10^prefix; conversion-based unit: factor × factor of its base unit; by name: degree π ÷ 180, inch 0.0254, foot 0.3048

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• Axis placement as a matrix

  The matrix of an IfcAxis2Placement3D, 2D or IfcAxis1Placement.

  columns = (x, z × x, z, origin), x = reference direction projected square to z

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• First projection axis

  The X axis of a placement: the reference direction projected into the plane normal to Z.

  x = normalize(v − z (v · z))

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• Cartesian transformation operator

  The matrix of an IfcCartesianTransformationOperator, uniform or not, in 2D or 3D.

  columns = (x s₁, y s₂, z s₃, origin)

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• World matrix of a placement

  The position of an element in the model from its chain of relative placements.

  world = parent world × local

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• Grid placement

  The point where two grid axes meet, with their offsets, as a placement.

  t = ((pB − pA) × dB) ÷ (dA × dB); point = pA + dA t

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• Map conversion

  The matrix that takes model coordinates to map coordinates, with its rotation and scale.

  map = (E, N, H) + R(θ) · S · model; θ = atan2(ordinate, abscissa)

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• Matrix product

  The product of two 4 × 4 matrices.

  IFC units and placements ifc/geo-math.js

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  64-bit

• Point through a matrix

  Transforms a point by a matrix.

  p' = M · (x, y, z, 1)

  IFC units and placements ifc/geo-math.js

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  64-bit

• Matrix inverse

  The inverse of a 4 × 4 matrix, or null when it has none.

  M⁻¹ = adjugate ÷ determinant

  IFC units and placements ifc/geo-math.js

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  64-bit

• Rotation about an axis

  The matrix that turns by an angle about an axis through the origin.

  R = c I + s [a]× + (1 − c) a aᵀ

  IFC units and placements ifc/geo-math.js

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  64-bit

• Polygon normal

  The normal of a 3D polygon, with a length of twice its area.

  n = Σ ((yₐ − y_b)(zₐ + z_b), (zₐ − z_b)(xₐ + x_b), (xₐ − x_b)(yₐ + y_b))

  IFC units and placements ifc/geo-math.js

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  64-bit

• Signed area of a 2D polygon

  The area of a profile outline, positive when counter-clockwise; used to orient outlines and holes.

  A = Σ (xₐ y_b − x_b yₐ) ÷ 2

  IFC units and placements ifc/geo-math.js

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  64-bit

• Positions and normals handed to the app

  Packs the mesh into the vertex and index arrays the apps read.

  vertices = Float32Array of (x, y, z, nx, ny, nz) per vertex

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

  • BimCraft
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  32-bit in part

• Planar face

  Adds one planar face with holes as triangles, all with the face's normal.

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

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  64-bit

• Mesh moved by a matrix

  Adds another mesh, transformed, with its normals turned to match.

  p' = M p; n' = normalize((M⁻¹)ᵀ n); a mirror swaps the winding

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

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  64-bit

• Signed volume

  The volume enclosed by the mesh, positive when the triangles face outward.

  V = Σ a · (b × c) ÷ 6

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

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  64-bit

• Shared vertices

  Merges vertices that fall in the same grid cell, for boolean operations.

  key = (round(x ÷ eps), round(y ÷ eps), round(z ÷ eps))

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

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  64-bit

• Segments of an arc

  How many chords replace an arc of a given sweep.

  n = max(2, ceil(|sweep| ÷ 2π × 36))

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

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  Exact

• Arc through three points

  The circular arc of an indexed poly-curve from its start, middle and end points.

  center = a + ((|ab|² (ac × n)) + (|ac|² (n × ab))) ÷ (2 |n|²), n = ab × ac

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

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  64-bit

• Circle and ellipse points

  Points on a circle or ellipse between two parameters.

  p = M · (r₁ cos θ, r₂ sin θ, 0), θ = a₀ + sweep × i ÷ n

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

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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit