Calculations

Calculations

• Transforming an object

  A copy of a drawing object put through a move, rotation, scale or mirror.

  point: M·p; arc: angles = atan2(M·end − M·center); radius, text height, block scale × s

  Move, rotate, scale and mirror geom.js

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• Cut parameters

  Where along an object each cutting edge crosses it.

  t = segment index + parameter of the intersection; t = round(t × 10⁹) ÷ 10⁹

  Trim geom.js

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• Piece between two parameters

  The part of a line, arc, open polyline or construction line between two parameters, as a new object.

  line: p₁ + (p₂ − p₁)·t; arc: a0 + sweep·t; polyline arc piece: bulge = tan(atan(b)·(t_b − t_a))

  Trim geom.js

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• Trim at a pick

  The pieces that stay when the part of an object round the picked point is cut away.

  circle: remaining arc from angle t_hi × 2π to t_lo × 2π

  Trim geom.js

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• Splitting into pieces

  An object divided at all its cutting edges into pieces that can be removed one by one.

  Trim geom.js

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• Offset of a line, circle or arc

  A copy of the object a set distance to one side.

  line: p ± d·n, n = (−Δy, Δx) ÷ L; circle, arc: r ± d

  Offset, extend, break and lengthen geom.js

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• Offset of a polyline

  A polyline offset to one side, with its corners rebuilt.

  Offset, extend, break and lengthen geom.js

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• 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

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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• Chamfer of two picked pieces

  The chamfer line and trimmed ends for two straight pieces.

  Fillet and chamfer geom.js

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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• Points along an ellipse

  The points that stand in for an ellipse or elliptical arc.

  n = max(8, ⌈steps × sweep ÷ 2π⌉); t = t0 + (t1 − t0)·i ÷ n; p = c + R(rot)·(a·cos t, a·ratio·sin t)

  Ellipses and curves as short segments geom.js

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• Ellipse area and perimeter

  The area and perimeter of an ellipse, or the area and length of an elliptical arc.

  full: area = π·a·b; perimeter ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))), h = ((a − b) ÷ (a + b))²; arc: 720 chords per turn

  Ellipses and curves as short segments geom.js

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• A piece through a transform

  A line or arc piece of a block put through the block reference's transform.

  uniform: center M·c, radius r·s, start angle = atan2(M·start − M·c); otherwise n = max(8, ⌈48 × sweep ÷ 2π⌉) chords

  Ellipses and curves as short segments geom.js

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• Outline of an object as points

  The outline a revision cloud follows, with arcs and curves in short pieces.

  Ellipses and curves as short segments geom.js

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• Even spacing along a path

  Points spaced evenly along a run of points, about a given step apart.

  n = max(1 or 3, round(L ÷ step)); target k = L·k ÷ n; point = linear interpolation on the segment that holds the target

  Ellipses and curves as short segments geom.js

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• Revision cloud

  A polyline of equal arcs along a path, bulging outward.

  bulge = ± tan(2π ÷ 3 ÷ 4) = ± tan 30°; outward side from the sign of Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Ellipses and curves as short segments geom.js

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• Exploding a spline

  Replaces a spline with a polyline along the curve.

  Ellipses and curves as short segments geom.js

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• 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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• 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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• 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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• 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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• 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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• Solver tolerance

  The size below which an equation counts as met.

  tol = 1e-9 × max(1, width, height, |min x|, |max x|, |min y|, |max y|)

  2D constraint solver gsolve.js

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• Constraint equations

  The residual of each constraint: how far the current geometry is from meeting it.

  coincident: P₀ − P₁; distance: |P₁ − P₀| − d; parallel: (u × v) ÷ (|u|·|v|); tangent: distance(center, line) − r; angle: atan2(u × v, u·v) − θ·π ÷ 180

  2D constraint solver gsolve.js

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• Circle of a polyline arc segment

  The center and radius of a polyline's arc segment as the solver sees it.

  θ = 4·atan(b); r = |c ÷ (2·sin(θ ÷ 2))|; h = ± r·cos(θ ÷ 2); center = chord midpoint + normal × h

  2D constraint solver gsolve.js

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• Derivatives

  How each equation changes with each coordinate it depends on.

  ∂f/∂x ≈ (f(x + h) − f(x − h)) ÷ 2h, h = 1e-6 × max(1, |x|)

  2D constraint solver gsolve.js

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• Cholesky solve

  Solves the damped normal equations for one step.

  A = L·Lᵀ; L·y = b; Lᵀ·z = y

  2D constraint solver gsolve.js

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• Newton iteration

  Repeats damped least-norm steps until every equation is met.

  (J·Jᵀ + μ·max(scale, 1e-12)·I)·y = −r; dx = Jᵀ·y; x ← x + α·dx, α = 1, ½, ¼ …

  2D constraint solver gsolve.js

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• Writing the solution back

  The changed objects after a solve, and the remaining residual.

  residual = √(Σ rᵢ²)

  2D constraint solver gsolve.js

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• Radius sign and arc angles

  Puts solved values back into an object in its stored form.

  2D constraint solver gsolve.js

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• Degrees of freedom and redundant constraints

  How many independent constraints there are, which ones add nothing, which conflict and which objects are fully fixed.

  v = row ÷ |row|; v ← v − (v·b)·b for each kept row b, twice; redundant when |v| < 1e-7; degrees of freedom = variables − rank

  2D constraint solver gsolve.js

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• Arcs as straight pieces

  The points that stand in for an arc or circle in a hatch boundary.

  n = max(4, ⌈|sweep| ÷ (2π ÷ 128)⌉); point = c + r·(cos a, sin a)

  Hatch boundaries and pattern lines hatch.js

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• A closed object as a boundary

  The boundary loop of a circle, a closed polyline or a full ellipse picked as an object.

  Hatch boundaries and pattern lines hatch.js

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• Boundary tolerance

  The distance within which boundary pieces count as touching.

  tol = 1e-9 × max(width, height, 1)

  Hatch boundaries and pattern lines hatch.js

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• Where two boundary pieces cross

  The parameters at which two segments cross or touch, so each can be split there.

  u = ((c − a) × q) ÷ (r × q); v = ((c − a) × r) ÷ (r × q)

  Hatch boundaries and pattern lines hatch.js

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• Merging boundary corners

  The corner points of the boundary, with points that nearly coincide made one.

  snap = 1000 × tol; split point = start + (end − start)·t

  Hatch boundaries and pattern lines hatch.js

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