Calculations

Calculations

• Line and line

  The parameters at which two lines cross.

  den = d₁ × d₂; tₐ = (w × d₂) ÷ den; t_b = (w × d₁) ÷ den, w = B.a − A.a

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Line and circle

  The parameters along a line at which it crosses a circle.

  A·t² + B·t + C = 0; q = −½·(B + sign(B)·√(B² − 4AC)); t₁ = q ÷ A, t₂ = C ÷ q

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Circle and circle

  The points where two circles cross.

  a = (r₁² − r₂² + d²) ÷ 2d; h = √(r₁² − a²); points = c₁ + a·û ± h·û⊥

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Is a crossing on the bounded curve

  Decides whether a crossing lies on the drawn part of a line or arc, and snaps it to the end when it is within the tolerance.

  line: −eps ÷ L ≤ t ≤ 1 + eps ÷ L; arc: angle ≤ sweep + eps ÷ r

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Crossing points of two curves

  All crossing points of two sketch curves, sorted along the first.

  eps = 1e-9 × max(1, size of A, size of B); duplicates within 10 × eps removed

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Knot span of a parameter

  Finds which knot span a parameter falls in.

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Basis functions

  The values of the B-spline basis functions that are not zero at a parameter.

  t = N[r] ÷ (right[r+1] + left[j−r]); N[r] = saved + right[r+1] × t; saved = left[j−r] × t (Cox–de Boor)

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Basis function derivatives

  The basis functions and their derivatives up to a given order at a parameter.

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Point on a spline

  The point of a spline at a parameter.

  C(u) = Σⱼ Nⱼ(u) × Pⱼ (degree + 1 terms)

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Derivatives of a spline

  The point, tangent and higher derivatives of a spline at a parameter.

  C⁽ⁱ⁾(u) = Σⱼ Nⱼ⁽ⁱ⁾(u) × Pⱼ

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Chord-length parameters

  Gives each fit point a parameter equal to the summed straight distances up to it.

  tᵢ = tᵢ₋₁ + |Qᵢ − Qᵢ₋₁|

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Dense linear solve

  Solves the linear system that gives a fit spline its poles.

  A · X = B by Gauss–Jordan elimination with partial pivoting

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Spline through fit points

  The cubic spline that passes through the given points, open with natural or given end directions, or closed and periodic.

  rows: C(tᵢ) = Qᵢ; ends: C″ = 0 or C′ = (cos a, sin a); solve for the poles

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Area term of a spline

  The area term of a spline by Green's formula: the enclosed area when the spline is closed.

  ½ ∫ (x·y′ − y·x′) du, 3-point Gauss–Legendre per knot span (nodes ±√(3 ÷ 5), 0; weights 5/9, 8/9, 5/9)

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Curvature of a spline

  The signed curvature of a spline at a parameter.

  κ = (x′·y″ − y′·x″) ÷ |C′|³

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Knot insertion

  Adds one knot to a spline without changing its shape.

  a = (u − Uᵢ) ÷ (Uᵢ₊ₚ − Uᵢ); Qᵢ = (1 − a)·Pᵢ₋₁ + a·Pᵢ

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Exact piece of a spline

  The part of a spline between two parameters as a curve of its own, used by trim.

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Knots rescaled to start at zero

  Shifts and scales a spline's knots so that they start at 0 and span the number of its pieces.

  u′ = (u − a) × (poles − degree) ÷ L

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Sketch tolerance

  The base length tolerance of the sketch tools.

  tolOf = 1e-7 × max(1, extent of the sketch)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Crossings of a curve with the others

  Where a curve is cut by every other curve of the sketch, for trim.

  crossings within tol of the curve's own ends, or of one already found, are left out; tol = 1e-4 × max(1, extent)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Extending a line or arc

  Moves the picked end of a line or arc to the nearest curve it meets when continued.

  line: gap = (t − 1) × L or −t × L; arc: gap = (t − sweep) × r or (2π − t) × r; nearest gap > tol wins

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Offset of each curve

  The offset copy of each line, circle, arc or spline of a chain, before the corners are joined.

  line: S = a + d·n, E = b + d·n, n = (−Δy, Δx) ÷ L; circle: r − d; arc: r ∓ d, ends c + r·(cos, sin)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Corners of an offset chain

  Joins neighboring offset curves at the point where they meet when extended.

  corner = the intersection of the two full lines or circles nearest the midpoint of the gap

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Mirror about a line

  Reflects points about a sketch line, and reflects a spline's end angles.

  t = (p − a) · u; q = a + t·u; p′ = 2q − p; angle′ = 2φ − angle

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Linear pattern

  Copies geometry a number of times along a step.

  copy i: (x + Δx × i, y + Δy × i)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Circular pattern

  Copies geometry a number of times around a center.

  step = (360 ÷ count, or angle ÷ (count − 1)) × π ÷ 180; copy i: rotate about c by step × i

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Regular polygon

  The corners of a regular polygon from its center, one corner and the number of sides.

  cornerᵢ = c + r × (cos(a₀ + 2π·i ÷ n), sin(a₀ + 2π·i ÷ n))

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Straight slot

  The four corner points of a slot from its two centers and its width.

  r = w ÷ 2; corners = center ± n × r, n = unit normal of the center line

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Projecting a model edge

  Puts a straight or circular model edge onto the sketch plane as fixed geometry.

  (u, v) = ((p − o) · x, (p − o) · y)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Radius of a circle or arc

  The radius of a circle (stored) or of an arc (the distance from its center to its start point).

  r = |s − c|

  Sketch geometry and measured values sketch.js

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• Arc angles

  The center, radius, start angle and sweep of an arc from its three points.

  a₀ = atan2(s − c); sweep = atan2(e − c) − a₀, brought to (1e-12, 2π]

  Sketch geometry and measured values sketch.js

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• New arc: end point on the radius

  Places a new arc's end point on the circle through its start point.

  r = |s − c|; e = c + r × (cos a, sin a), a = atan2(e₀ − c)

  Sketch geometry and measured values sketch.js

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• Current value of a relation or dimension

  The value a new dimension starts with: a distance, a horizontal or vertical distance, a radius, a diameter, an offset or an angle, and which side it is on.

  distance = |b − a|; point to line = |signed distance|; diameter = 2r; angle° = |atan2(u × v, u · v)| × 180 ÷ π

  Sketch geometry and measured values sketch.js

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• Signed distance from a point to a line

  How far a point is from a line through two points, positive on the left.

  s = (Δx·(p_y − a_y) − Δy·(p_x − a_x)) ÷ |b − a|

  Sketch geometry and measured values sketch.js

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• Nearly parallel test

  Decides whether two lines are close enough to parallel to take a distance instead of an angle.

  |u × v| ÷ (|u| × |v|) < 1e-6

  Sketch geometry and measured values sketch.js

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• Sketch fillet

  Trims two lines at a corner and adds a tangent arc of a given radius between them.

  θ = acos(u₁ · u₂); d = r ÷ tan(θ ÷ 2); tangent points = P + uᵢ × d; center = P + bisector × r ÷ sin(θ ÷ 2)

  Sketch geometry and measured values sketch.js

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• Size of a sketch

  The largest coordinate size in a sketch, which the tolerances of the sketch code are scaled by.

  extent = max(|x|, |y| over the points; |c_x| + r, |c_y| + r over the circles)

  Sketch geometry and measured values sketch.js

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• Box of a sketch

  The bounding box of the sketch geometry.

  Sketch geometry and measured values sketch.js

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• Typed text to a number

  Reads a typed size such as 12.5, ⌀20, R5, 30° or 12 mm as a number.

  Sketch editor: typed sizes and dimension values sketchview.js

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• Value shown in the input box

  Opens the input box at a value for the person to accept or change.

  box text = fmt(value) (at most 4 decimals)

  Sketch editor: typed sizes and dimension values sketchview.js

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• Applying the typed value

  Takes the text of the input box when it closes and hands the number to whatever asked for it.

  Sketch editor: typed sizes and dimension values sketchview.js

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• Changing a dimension's value

  Stores the typed number as the dimension's value.

  0 ≤ value; angle ≤ 180; radius and diameter > 0

  Sketch editor: typed sizes and dimension values sketchview.js

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• Keeping a typed size

  Checks a size typed while drawing and keeps it for the shape.

  angle ← ((v mod 360) + 360) mod 360; sweep must be in (0, 360); other sizes > 0

  Sketch editor: typed sizes and dimension values sketchview.js

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• Point from typed sizes

  Where the typed length and angle, width and height, diameter, radius or sweep put the next point of the shape being drawn.

  line: a + (cos r, sin r) × len, r = angle × π ÷ 180; rectangle: a ± (w, h); arc end: c + R(sweep) · (s − c)

  Sketch editor: typed sizes and dimension values sketchview.js

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• Snap to horizontal or vertical

  Makes a line drawn within 3° of horizontal or vertical exactly so.

  angle° = |atan2(Δy, Δx)| × 180 ÷ π; within 3° of 0, 180 or 90: copy the start's y or x to the end

  Sketch editor: typed sizes and dimension values sketchview.js

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• Point snapped onto a curve

  The nearest point on a line, circle, arc or spline to the pointer, used when a new point snaps onto a curve.

  line: a + t·d, t = ((p − a) · d) ÷ (d · d); circle: c + (p − c) × r ÷ |p − c|

  Sketch editor: typed sizes and dimension values sketchview.js

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• Unit vector

  Scales a vector to length 1.

  a ÷ |a|

  Vectors, plane frames and transforms vec.js

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• Frame of a plane

  The origin and the x, y and normal axes given to a sketch plane.

  o = (p·n) n; x = Y × n (or world X made square to n when |n_y| > 0.9); y = n × x

  Vectors, plane frames and transforms vec.js

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• Sketch point to model space

  The 3D position of a sketch point (u, v) on its plane.

  p = o + u x + v y

  Vectors, plane frames and transforms vec.js

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• Model point to sketch coordinates

  The sketch coordinates of a 3D point on a plane.

  u = (p − o)·x; v = (p − o)·y

  Vectors, plane frames and transforms vec.js

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