Calculations

Calculations

• Grid lines an elevation shows

  The lines that run within 30° of a view's direction, and where they sit across the view.

  kept when |e · d| ÷ |e| ≥ 0.866; x = a · right

  Grid lines model-grids.js

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• Is the grid rectangular

  A yes or no that decides the grid's type in the file.

  parallel when |d₁ × d₂| < 1e-6; square when |u · v| < 1e-6

  Grid lines model-grids.js

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• Grid line ends read from the file

  A grid line's two ends in building millimeters, from its polyline and the grid's placement.

  world = M × (x × unit, y × unit, 0)

  Grid lines model-grids.js

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• Grid lines in a story's plan

  Grid line ends moved from building coordinates into one story's plan axes.

  plan = (story world)⁻¹ × point

  Grid lines model-grids.js

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• Boxes of the things that move

  The plan box of each wall, drawn floor, column and placed item.

  wall corners = end ± n × t ÷ 2

  Move, align and distribute model-modify.js

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• Align

  The move that brings each element's chosen side or middle to a common line.

  move = target − own; middle = (min + max) ÷ 2

  Move, align and distribute model-modify.js

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• Distribute

  The moves that space elements evenly between the first and the last.

  gap = (span − Σ sizes) ÷ (n − 1)

  Move, align and distribute model-modify.js

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• Where placed items stand

  The position and rotation of each placed library item in its story's plan.

  rot = atan2(M[1], M[0]) × 180 ÷ π

  Move, align and distribute model-move.js

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• Moving and turning placed items

  The new position and rotation of an item moved by an amount or turned, on its own or about a shared point.

  p′ = c + R(Δ) × (p − c); rot = ((rot mod 360) + 540) mod 360 − 180

  Move, align and distribute model-move.js

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• Rejoining wall corners after a type change

  The new corner point of two walls that met end to end, after their center lines moved.

  s = ((B.a − A.a) × v) ÷ (u × v); p = A.a + s u

  Move, align and distribute model-ops.js

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• Doors and windows when a wall end slides

  How far a wall's start moved along its own line, so that what it hosts keeps its place.

  shift = (B.a − A.a) · u_A; new offset = offset − shift

  Move, align and distribute model-ops.js

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• Fitting an opening in a wall

  The offset along the wall at which an opening of a given width fits.

  offset = max(gap, min(len − width − gap, offset))

  Doors and windows in walls model-openings.js

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• Sill of a door

  Whether a door stands on its wall's own level or on an upper level the wall passes.

  kept when |level elevation − base elevation − sill| ≤ 1 mm

  Doors and windows in walls model-openings.js

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• Offset, size and sill read back

  Where an existing door or window sits and how big it is, from the file.

  sill = M[14] + base offset; width = stored × unit

  Doors and windows in walls model-openings.js

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• Default offset and height check

  Where a new opening goes when no offset is given, and the check that it is not taller than the wall.

  offset = (len − W) ÷ 2; refused when sill + height > wall height + 1

  Doors and windows in walls model-openings.js

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• The box that cuts the wall

  The body of the opening that is subtracted from the wall.

  box: x from 0 to width, y from −t ÷ 2 − 10 to t ÷ 2 + 10, z from 0 to height

  Doors and windows in walls model-openings.js

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• Width, height and area written

  The quantities and sizes written on a door or window.

  area = width × height

  Doors and windows in walls model-openings.js

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• Shift of a wall's start along its line

  How far a wall's start moved along the wall, so hosted openings can keep their place in the world.

  shift = (a′ − a) · u

  Doors and windows in walls model-openings.js

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• Door body

  The boxes a starter door is built from: frame, leaves, glass.

  frame = min(1.5 in, W ÷ 6)

  Doors and windows in walls model-openings.js

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• Window body

  The boxes a starter window is built from: frame, rails, stiles, glass and an optional well.

  frame = min(2 in, W ÷ 6, H ÷ 6)

  Doors and windows in walls model-openings.js

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• Slope and outline kind

  Turns the pitch into a slope and chooses the method for the outline.

  s = tan(pitch)

  Roof planes over an outline model-roof.js

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• Faces of a roof over a convex outline

  Each pitched edge's face, the eave, the ridges and the area.

  keep p where (nᵢ − nⱼ) · p ≤ nᵢ · aᵢ − nⱼ · aⱼ; area = plan area × √(1 + s²)

  Roof planes over an outline model-roof.js

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• Clipping a polygon by a half-plane

  Keeps the part of a convex polygon on one side of a line.

  t = f_a ÷ (f_a − f_b); point = a + t (b − a)

  Roof planes over an outline model-roof.js

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• Where two offset edges meet

  The corner of the eave where two neighboring edges, pushed out by the overhang, cross.

  t = ((p₂ − p₁) × d₂) ÷ (d₁ × d₂)

  Roof planes over an outline model-roof.js

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• Roof over an outline with square corners

  A hip roof for every largest rectangle that fits in the outline with its overhang.

  m = min(w, h) ÷ 2; ridge from (x₀ + m, c_y) to (x₁ − m, c_y)

  Roof planes over an outline model-roof.js

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• Height of the roof over a point

  The roof's underside height above the top of the walls at a plan point.

  z = z₀ + s × (distance inside the face's eave edge)

  Roof planes over an outline model-roof.js

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• Is the outline convex

  Chooses between the two roof methods.

  convex when every corner turns the same way

  Roof planes over an outline model-roof.js

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• Plan area of a polygon

  The shoelace area used for the eave and the faces.

  A = ½ Σ (xᵢ yᵢ₊₁ − yᵢ xᵢ₊₁)

  Roof planes over an outline model-roof.js

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• Dimension chain of a wall

  The overall length of a wall along one face and the chain of lengths between its ends and its openings.

  overall = e₁ − e₀; piece = sᵢ₊₁ − sᵢ with s rounded to 0.01 mm

  Plan view: picked points, wall outlines and dimensions model-plan.js

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• Wall outline with openings taken out

  The pieces of a wall's plan outline left between the openings the cut plane passes through.

  Plan view: picked points, wall outlines and dimensions model-plan.js

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• Clipping a polygon at a position along the wall

  Keeps the part of a polygon on one side of x = k.

  t = (k − a_x) ÷ (b_x − a_x); y = a_y + t (b_y − a_y)

  Plan view: picked points, wall outlines and dimensions model-plan.js

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• Viewer coordinates to a story's plan

  The matrix from the viewer's Y-up meters to a story's plan millimeters.

  M = (story world)⁻¹ × S × C⁻¹

  Plan view: picked points, wall outlines and dimensions model-plan.js

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• Pointer position to plan millimeters

  The plan point under the pointer, which the drawing tools then snap and use.

  x = (px − w ÷ 2) ÷ scale + cx; y = (h ÷ 2 − py) ÷ scale + cy

  Plan view: picked points, wall outlines and dimensions model-planview.js

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• Where two face lines meet

  The point where two lines, each given by a point and a direction, cross.

  s = ((p₂ − p₁) × d₂) ÷ (d₁ × d₂); point = p₁ + s·d₁

  Wall joins and plan outlines model-walls.js

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• Nodes and tees

  Groups wall ends that meet and finds ends that land on another wall's side.

  Wall joins and plan outlines model-walls.js

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• Corner points at a node

  The mitre or corner point between each pair of neighboring walls at a node.

  c = left face of A ∩ right face of B; kept when |c − node| ≤ 4 × max(h_A, h_B) + tol

  Wall joins and plan outlines model-walls.js

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• Faces of curved walls

  Where a straight face meets a circular face, or two circular faces meet, nearest a given point.

  line and circle: t = −b ± √(b² − c); two circles: x = (d² + r₁² − r₂²) ÷ 2d, y = √(r₁² − x²)

  Wall joins and plan outlines model-walls.js

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• Curved wall outline

  The outline of a curved wall with each arc edge kept as a bulge.

  bulge = tan(sweep ÷ 4)

  Wall joins and plan outlines model-walls.js

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• Removing repeated and collinear points

  Drops outline points that repeat or lie on a straight run.

  collinear when |ab × bc| ≤ 1e-6 × (|ab|·|bc| + 1e-12) and ab·bc ≥ 0

  Wall joins and plan outlines model-walls.js

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• Signed outline area

  The signed area of an outline, used for the footprint and to check its direction.

  A = ½ Σ (xᵢ·yᵢ₊₁ − yᵢ·xᵢ₊₁)

  Wall joins and plan outlines model-walls.js

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• Outline validity test

  Decides whether an outline is anticlockwise and does not cross itself.

  Wall joins and plan outlines model-walls.js

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• Location line offset

  How far a wall's location line is from its center line for each justification.

  face: ± t ÷ 2; core: ± (t ÷ 2 − layer thickness up to the core face or core middle)

  Wall joins and plan outlines model-walls.js

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• Point along a snapped direction

  The point at an orthogonal or stepped angle from the last point, stopping at a wall it crosses or at a rounded length.

  θ = round(θ ÷ step) × step; l = len × cos(θ₀ − θ); p = from + l × (cos θ, sin θ)

  Drawn points: angle, length and wall-face paths model-tools.js

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• Nearest point on a wall line or face

  The point on a wall's center line or face nearest the pointer.

  s = clamp((p − a)·u, 0, L); q = a + s·u

  Drawn points: angle, length and wall-face paths model-tools.js

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• Signed area of a drawn loop

  The signed area of a closed chain of points, used to tell which side is outside.

  A = ½ Σ (xᵢ·yᵢ₊₁ − yᵢ·xᵢ₊₁)

  Drawn points: angle, length and wall-face paths model-tools.js

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• Center lines for a path drawn on a wall face

  Offsets a drawn path by half the wall thickness and mitres the corners.

  offset = ± t ÷ 2 along the left normal; corner = intersection of neighboring offset lines

  Drawn points: angle, length and wall-face paths model-tools.js

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• Where two center lines meet

  The meeting point of line a→b and line c→d, with its position along each.

  den = r × s; t = ((c − a) × s) ÷ den; u = ((c − a) × r) ÷ den; p = a + t·r

  Trim, extend and corner walls model-trim.js

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• New ends of a trimmed wall

  Which part of the wall stays and where its moved end goes.

  inside when 0.5 ÷ L < t < 1 − 0.5 ÷ L; click side by t_c = (click − a)·(b − a) ÷ L²

  Trim, extend and corner walls model-trim.js

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• Trim or extend to a wall

  A wall's new ends when it is trimmed or extended to another wall.

  allowed when −slack ≤ u ≤ 1 + slack, slack = (t_edge ÷ 2) ÷ L_edge

  Trim, extend and corner walls model-trim.js

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• Corner two walls

  Both walls' new ends at the point their center lines meet.

  Trim, extend and corner walls model-trim.js

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