Calculations

Calculations

• Knot span

  Which knot interval a B-spline parameter falls in.

  STEP curves and surfaces: points from parameters step/geom.js

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  Exact

• B-spline basis functions

  The p + 1 basis functions that are not zero at a parameter.

  N_{i,0} = 1; N_{i,j} built from (t − u) ÷ (u_b − u_a) terms in the Cox–de Boor triangular table

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
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  64-bit

• B-spline curve point

  The point on a B-spline or rational B-spline curve at a parameter.

  C(t) = Σ Nᵢ wᵢ Pᵢ ÷ Σ Nᵢ wᵢ

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
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  64-bit

• Is a B-spline curve closed

  Decides whether a B-spline curve's two ends meet, so it can be treated as periodic.

  |C(t₀) − C(t₁)| < 1e-9 × max(1, max |Pᵢ|)

  STEP curves and surfaces: points from parameters step/geom.js

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• B-spline surface point

  The point on a B-spline or rational B-spline surface at (u, v).

  S(u, v) = Σ Σ Nᵢ(u) Nⱼ(v) wᵢⱼ Pᵢⱼ ÷ Σ Σ Nᵢ(u) Nⱼ(v) wᵢⱼ

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• Seams and poles of a B-spline surface

  Decides which direction of a B-spline surface closes on itself and which boundary rows collapse to a point.

  two points are the same when |a − b| < 1e-7 × max(1, |a|)

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• Cylinder

  The point at (angle, height) on a cylinder, and the angle and height of a point.

  S(u, v) = o + r cos u · x + r sin u · y + v · z

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
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  64-bit

• Cone

  The point at (angle, distance along the axis) on a cone, and the inverse.

  r(v) = R + v tan α; v = ((ρ − R) tan α + l_z) ÷ (tan² α + 1)

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• Sphere

  The point at (longitude, latitude) on a sphere, and the inverse.

  S(u, v) = o + r cos v cos u · x + r cos v sin u · y + r sin v · z

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• Torus

  The point at two angles on a torus, and the inverse.

  c = R + r cos v; S(u, v) = o + c cos u · x + c sin u · y + r sin v · z

  STEP curves and surfaces: points from parameters step/geom.js

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• Surface of revolution

  The point of a profile curve turned about an axis by an angle, and the inverse.

  S(u, v) = o + h z + rr cos u + (z × rr) sin u, h = (C(v) − o)·z, rr = (C(v) − o) − h z

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• Angle of a point on an ellipse

  The ellipse parameter of the point on the ellipse nearest a given point.

  f(t) = (E(t) − q)·E′(t) = 0, solved by t ← t − f ÷ f′

  STEP curves and surfaces: parameters from points step/geom.js

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• Parameter of a point on a B-spline curve

  The curve parameter nearest a given point.

  STEP curves and surfaces: parameters from points step/geom.js

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  64-bit

• Parameters of a point on a B-spline surface

  The (u, v) of the surface point nearest a given point.

  [S_u·S_u S_u·S_v; S_u·S_v S_v·S_v] d = −[r·S_u; r·S_v], r = S(u, v) − q

  STEP curves and surfaces: parameters from points step/geom.js

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  64-bit

• Surface of linear extrusion

  The point of a curve swept along a vector, and the (u, v) of a point.

  S(u, v) = C(u) + v · vec; v = (q − C(u))·d ÷ |vec|, u = project(q − v · vec)

  STEP curves and surfaces: parameters from points step/geom.js

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• Offset surface

  The point of a surface moved a distance along its normal.

  S_off(u, v) = S(u, v) + d · n, n = (ΔS_u × ΔS_v) ÷ |…|

  STEP curves and surfaces: parameters from points step/geom.js

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  64-bit

• Length and angle units

  The file's length unit in millimeters and its plane-angle unit in radians.

  mm = prefix × 1000 for a meter; a conversion-based unit = its value × its base unit

  STEP import: units, points and assembly placement step/read.js

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• Point coordinates

  A CARTESIAN_POINT in millimeters.

  (x, y, z) × unit in mm

  STEP import: units, points and assembly placement step/read.js

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• Radii and cone angles

  The radius of a cylinder, cone, sphere or torus in millimeters and a cone's half angle in radians.

  r × unit in mm; α × angle unit, accepted when 0 < |α| < π ÷ 2

  STEP import: units, points and assembly placement step/read.js

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  64-bit

• Knot vectors

  The full knot vector of a B-spline from the file's knots and multiplicities, or the one a uniform, quasi-uniform or Bézier form implies.

  STEP import: units, points and assembly placement step/read.js

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  Exact

• Model size and chord tolerance

  The size of the model and the distance a triangle may stray from a curved face.

  size = diagonal of the box around every point; tol = max(4e-4 × size, 1e-4 mm)

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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• Closing a loop of edges

  Drops the last point of an edge loop when it repeats the first.

  |first − last| ≤ 0.01 × tol

  STEP import: units, points and assembly placement step/read.js

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• Placement matrices

  A placement as a 4 × 4 matrix, the product of two and the inverse of one.

  M⁻¹ = [Rᵀ, −Rᵀ t]

  STEP import: units, points and assembly placement step/read.js

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• A component placed in its assembly

  The transform that carries a component's coordinates into the assembly that uses it.

  T = M(A₂) × M(A₁)⁻¹

  STEP import: units, points and assembly placement step/read.js

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• Placed vertex positions

  Every vertex of a solid moved to its place and stored for the app.

  p′ = R p + t

  STEP import: units, points and assembly placement step/read.js

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• Faceted faces

  Triangles for a flat face given as loops of points.

  n = Σ ((yᵢ − yⱼ)(zᵢ + zⱼ), (zᵢ − zⱼ)(xᵢ + xⱼ), (xᵢ − xⱼ)(yᵢ + yⱼ))

  STEP import: units, points and assembly placement step/read.js

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• Points along an edge

  The points of an edge from its start vertex to its end vertex along its curve.

  STEP import: triangles for curved faces step/tess.js

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• How finely a curve is divided

  The parameters at which an edge curve is sampled.

  circle step = min(π ÷ 24, 2 acos(1 − tol ÷ r)); else halve while the middle is more than tol from the chord or the turn exceeds π ÷ 24

  STEP import: triangles for curved faces step/tess.js

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• Turn and crossing tests

  Whether three points turn left, right or run straight, and whether two segments cross.

  x = (a − o) × (b − o); left when x > 1e-9 × |ab| × |bc|, right when x < −1e-9 × |ab| × |bc|

  STEP import: triangles for curved faces step/tess.js

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• Holes joined to the outer ring

  One ring made from a face's outer loop and its holes by bridges, ready for ear clipping.

  STEP import: triangles for curved faces step/tess.js

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• Ear clipping

  Triangles of a simple polygon in the plane.

  STEP import: triangles for curved faces step/tess.js

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• A loop in the surface's (u, v)

  The parameters of each boundary point, with periodic directions unwrapped and poles split.

  uᵢ ← uᵢ + round((uⱼ − uᵢ) ÷ period) × period

  STEP import: triangles for curved faces step/tess.js

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• Scaled layout and first triangles

  The (u, v) points scaled to lengths, the loops given their turning sense, and the first triangulation.

  M = (u × s_u, v × s_v); loop sense = sign of Σ (xᵢ yⱼ − xⱼ yᵢ)

  STEP import: triangles for curved faces step/tess.js

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• Apex of a cone

  The parameter at which a cone's radius reaches zero, for a face that closes at the apex.

  R = |S(0, 0) − S(π, 0)| ÷ 2; k = |S(0, 1) − S(π, 1)| ÷ 2 − R; v_apex = −R ÷ k

  STEP import: triangles for curved faces step/tess.js

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• Interior grid spacing

  How far apart interior points are placed on each kind of surface.

  angle step = min(π ÷ 24, 2 acos(1 − tol ÷ r))

  STEP import: triangles for curved faces step/tess.js

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• Interior points inserted

  Grid points added inside a face and joined into the triangulation.

  STEP import: triangles for curved faces step/tess.js

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• In-circle test

  Whether a fourth point lies inside the circle through three others.

  det of rows (a − d, |a − d|²), (b − d, |b − d|²), (c − d, |c − d|²)

  STEP import: triangles for curved faces step/tess.js

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• Edge flips

  Inside edges flipped toward a Delaunay triangulation in the scaled (u, v).

  STEP import: triangles for curved faces step/tess.js

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• Refinement to the chord tolerance

  Inside edges split where the surface is farther than the tolerance from the triangles.

  split when |S(mid uv) − (P_a + P_b) ÷ 2| > tol, or when a triangle's center is more than tol from its plane

  STEP import: triangles for curved faces step/tess.js

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• Facing and self-check

  The 3D position of every point, the triangles turned to face outward, and a check of how far the mesh is from the surface.

  worst = max |S(center uv) − center of triangle| over up to 64 triangles; noted when worst > 25 × tol

  STEP import: triangles for curved faces step/tess.js

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• Distance between two points

  The length a headset measurement reports.

  d = √(Δx² + Δy² + Δz²)

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

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• Snap to a corner

  The corner of the hit triangle nearest the hit, when it is within the snap radius.

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

  • BimCraft
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  64-bit

• Model to headset transform

  The matrix that places the model in the headset's space.

  M = T(anchor) · R_y(yaw) · S(scale) · T(−at)

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

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• Inverse of a rig or pose

  The inverse of a rigid motion with one uniform scale.

  M⁻¹ = [Rᵀ ÷ s², −(Rᵀ ÷ s²) t], s² = squared length of the first column

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

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• Section plane

  The plane of a section cut through what was pointed at.

  n = normal toward the eye; p = hit − n × depth; d = n·p

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

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• Issue camera

  The viewpoint, direction and up vector saved with an issue, in the app's coordinates.

  to app = M_up⁻¹ · M_rig⁻¹; eye = to app × head position

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

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• The point being pointed at

  The model point, surface normal and triangle under a controller's ray.

  Headset review: measuring, section cuts and issue cameras xr/xr-session.js

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  32-bit in part

• A measurement in the headset

  The length shown on the label between two picked points.

  d = |q − p|, shown through the app's length format

  Headset review: measuring, section cuts and issue cameras xr/xr-session.js

  • BimCraft
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  32-bit in part

• Where an issue is placed

  The point and camera handed to the app with a new issue.

  Headset review: measuring, section cuts and issue cameras xr/xr-session.js

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  32-bit in part

• Two-point fit

  The turn and move that carry two model points onto two site points, with the remaining error.

  yaw = atan2(Δw) − atan2(Δm); t = mid(w) − R mid(m); residual = max(|R m₁ + t − w₁|, |R m₂ + t − w₂|); scale off = |Δw| ÷ |Δm| − 1

  AR on site: lining the model up with the building xr/xr-core.js

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