Calculations

Calculations

• Is a matrix a rigid transform

  Checks that a matrix is a pure rotation and shift before it is accepted.

  |rᵢ·rⱼ − δᵢⱼ| ≤ tol; |det − 1| ≤ tol; last row 0 0 0 1 within tol

  Scan registration register.js

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• Rotation about an axis

  The 4 × 4 matrix for a turn by an angle about an axis through a center.

  R = c·I + s·[axis]ₓ + (1 − c)·axis·axisᵀ; t = center − R·center

  Scan registration register.js

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• Angle and shift of a transform in words

  The rotation angle in degrees and the length of the shift that a result is reported with.

  angle = acos((trace − 1) ÷ 2); shift = |t|

  Scan registration register.js

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• Working points of a scan

  The thinned copy of a scan's points, measured from the session origin, that registration works on.

  work = pos + cloud origin − session origin, every stride-th point

  Scan registration register.js

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  32-bit in part

• Nearest point in the fixed scans

  For a moved point, the closest point of the fixed scans within the match distance.

  d² = Δx² + Δy² + Δz², smallest over the grid cells searched ring by ring

  Scan registration register.js

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• Surface normals

  The direction square to the surface at each fixed point, from its 10 nearest neighbors.

  normal = eigenvector of the smallest eigenvalue of the neighbors' covariance

  Scan registration register.js

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  32-bit in part

• Matching and the error of a step

  Pairs each moving point with its nearest fixed point and reports the root-mean-square distance, and the distance along the normals.

  rms = √(Σ d² ÷ matched); plane rms = √(Σ ((p − q)·n)² ÷ matched)

  Scan registration register.js

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  32-bit in part

• Point-to-plane step

  The small turn and shift that reduce the distances along the normals in one iteration.

  minimize Σ ((ω × p + t + p − q)·n)²: 6 × 6 normal equations for (ω, t); R = rotation by |ω| about ω

  Scan registration register.js

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  32-bit in part

• Solving the 6 × 6 system

  Solves the linear equations of the point-to-plane step.

  Gaussian elimination with partial pivoting, then back substitution

  Scan registration register.js

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• When the iteration stops

  Decides when a phase has settled and narrows the match distance for the next step.

  stop when |previous − error| ≤ tol × max(error, 1e-12) or error < 1e-12; cut = min(start, max(2.5 × rms, 1e-6 × diagonal))

  Scan registration register.js

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• Pixels to drawing coordinates and back

  Maps a pixel of a placed picture to the drawing, a drawing point back to a pixel, and gives the size of one pixel.

  block = p + u·px + v·(h − py); drawing = M·block; pixel size = |M·u|

  Scanned drawings: size, scale, align, crop and trace scan.js

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  64-bit

• Arc through three points

  The counterclockwise arc through a start, a middle and an end point.

  d = 2(aₓ(b_y − c_y) + bₓ(c_y − a_y) + cₓ(a_y − b_y)); center from |a|², |b|², |c|² ÷ d; r = |a − center|

  Scanned drawings: size, scale, align, crop and trace scan.js

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• Traced lines and arcs as drawing objects

  Turns a trace result in pixels into lines, arcs and circles in drawing coordinates.

  circle r = r(px) × pixel size; arc through start, middle and end

  Scanned drawings: size, scale, align, crop and trace scan.js

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• First size of an attached scan

  The size of one pixel when a scan is attached.

  per pixel = 25.4 ÷ dpi when the file states it and the drawing has units, else 1; typed width ÷ pixel width

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

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• Scale from two points and a real distance

  Rescales a scan so two picked points are a stated distance apart.

  f = wanted ÷ |b − a|; insertion′ = a + f·(insertion − a); sx′ = sx·f

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

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• Align by two pairs of points

  Moves, turns and optionally scales a scan so two points on it land on two points of the drawing.

  f = |d₂ − d₁| ÷ |s₂ − s₁|; θ = atan2(d₂ − d₁) − atan2(s₂ − s₁); insertion′ = d₁ + f·R(θ)·(insertion − s₁)

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

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• Straightening a skewed scan

  Turns the picture back by the skew angle that was found or typed.

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

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• Crop window in pixels

  The whole-pixel window of a crop and the new corner of the cropped picture.

  x₀ = ⌊min⌋, y₀ = ⌊min⌋, x₁ = ⌈max⌉, y₁ = ⌈max⌉; corner = p + u·x₀ + v·(h − y₁)

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

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• Trace tolerance

  The tolerance of a trace, typed in drawing units and used in pixels.

  tolerance (px) = typed ÷ pixel size, held to 0.5 … 20

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

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• Outline moved inward

  The outline offset inward by a distance, with mitered corners.

  corner′ = corner + bisector × d ÷ cos(half angle)

  Tapered, revolved and swept solids shapes3d.js

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• Signed volume and face direction

  Works out whether the faces point outward and refuses a solid with no volume.

  V = Σ p₀ · (pᵢ × pᵢ₊₁) ÷ 6 over fan triangles

  Tapered, revolved and swept solids shapes3d.js

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• Tapered extrusion

  A solid whose top is the outline inset by the height times the tangent of the taper angle.

  inset = |h| × tan(angle); vertices relative to the outline's minimum x and y

  Tapered, revolved and swept solids shapes3d.js

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• Revolution about an axis

  A solid made by turning the outline about an axis in steps.

  n = max(2, ⌈48 × |angle| ÷ 2π⌉); p′ = v·cos θ + (u × v)·sin θ + u(u·v)(1 − cos θ)

  Tapered, revolved and swept solids shapes3d.js

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• Sweep along a path

  A solid made by standing the outline across a path, with mitered corners.

  centroid = Σ (pᵢ + pⱼ)wᵢ ÷ 3Σwᵢ, wᵢ = xᵢyⱼ − xⱼyᵢ; frame normal = bisector ÷ cos(half angle)

  Tapered, revolved and swept solids shapes3d.js

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• Repeated path points

  Drops path points that repeat the one before.

  |pᵢ − pᵢ₋₁| ≤ 1e-9 → dropped

  Tapered, revolved and swept solids shapes3d.js

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  64-bit

• Pen moves

  Moves the pen by a displacement of the shape.

  x′ = x + dx × mul ÷ div

  SHX font text: strokes and width shx.js

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• Arcs as chords

  Draws an arc of a shape as short straight pieces.

  n = max(2, ⌈|sweep| ÷ 7.5°⌉); pointₖ = c + r(cos aₖ, sin aₖ), aₖ = a₀ + sweep × k ÷ n

  SHX font text: strokes and width shx.js

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  64-bit

• Bulge arcs

  An arc given by its chord and a bulge value.

  θ = 4·atan(b); r = chord ÷ (2·sin(|θ| ÷ 2)); center = chord middle ± r·cos(θ ÷ 2) across the chord

  SHX font text: strokes and width shx.js

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  64-bit

• Octant and fractional arcs

  Arcs given by a radius, a starting octant and a span of octants, with optional fractions of an octant.

  a₀ = (s₀ + so ÷ 256) × π ÷ 4; center = pen − r(cos a₀, sin a₀)

  SHX font text: strokes and width shx.js

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• Text strokes and width

  The strokes of a text for capitals of a given height, and the text's width.

  k = h ÷ capital height of the font; width = Σ glyph advance × k

  SHX font text: strokes and width shx.js

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• Width of a text in an SHX font

  The width the text layout uses for a text whose style names a loaded .shx font.

  width = advance × width factor

  SHX font text: strokes and width shx-text.js

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• Strokes placed in the drawing

  Each stroke point of a text taken to drawing coordinates with alignment, width factor, slant and rotation.

  lx = x₀ + (gx + gy·tan(oblique)) × wf; X = x + lx·cos − gy·sin; Y = y + lx·sin + gy·cos

  SHX font text: strokes and width shx-text.js

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• Circle through a stroke

  The circle that best fits the stroke's points, with its root-mean-square error.

  u = x − x̄, v = y − ȳ; solve [Σu² Σuv; Σuv Σv²]·(u_c, v_c) = ½(Σu³ + Σuv², Σv³ + Σvu²); r² = u_c² + v_c² + (Σu² + Σv²) ÷ n

  Pen sketch cleanup sketch.js

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• Turn of a stroke about a center

  How far the stroke goes round the fitted center, signed.

  Σ (aᵢ − aᵢ₋₁) with each step brought into (−π, π]

  Pen sketch cleanup sketch.js

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• Squaring a direction

  Makes a direction level or plumb when it is within a few degrees of it.

  q = round(a ÷ (π ÷ 2)) × (π ÷ 2); used when |a − q| ≤ deg × π ÷ 180

  Pen sketch cleanup sketch.js

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• Where two sides meet

  The corner where two straight sides cross.

  det = l × m; t = ((m₀ − l₀) × m) ÷ det; point = l₀ + l·t

  Pen sketch cleanup sketch.js

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• Distance from a point to a segment

  How far a pen point is from the straight line between two others.

  t = ((p − a)·(b − a)) ÷ |b − a|², held to [0, 1]; distance = |p − (a + t(b − a))|

  Pen sketch cleanup sketch.js

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• Straight sides and rectangles

  Cuts a stroke at its corners, squares each side, and rebuilds the corners where the sides cross.

  Pen sketch cleanup sketch.js

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• A stroke as a line

  Makes a nearly straight stroke a line, exactly level or plumb when it is close to it.

  straight when max distance ≤ max(eps, 3.5 % of the gap) and gap > 90 % of the length

  Pen sketch cleanup sketch.js

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• A stroke as a circle or an arc

  Makes a round stroke a circle, or an arc from the stroke's first point to its last.

  accepted when rms ÷ r < 0.07 and r < 3 × box diagonal; circle when |turn| > 0.85 × 2π

  Pen sketch cleanup sketch.js

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• Joining loose ends

  Pulls each loose end onto the nearest corner of another stroke or of the drawing.

  nearest corner with distance ≤ tol; tol = 6 % of the sketch's box diagonal unless given

  Pen sketch cleanup sketch.js

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• Point on a spline

  The curve's point at a parameter.

  de Boor: dⱼ = (1 − a)·dⱼ₋₁ + a·dⱼ, a = (u − kᵢ) ÷ (kᵢ₊ₚ₋ᵣ₊₁ − kᵢ); point = (Σ w·P) ÷ Σ w

  Splines: curve points, fitting and editing spline.js

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• Clamped knots

  The knot list for a spline that starts and ends on its first and last control points.

  p + 1 zeros, 1 … n − p − 1, p + 1 values of n − p

  Splines: curve points, fitting and editing spline.js

  • CadCraft
  Exact

• Spline through fit points

  Builds the spline that passes through the points picked.

  two points: control points at 0, ⅓, ⅔ and 1 of the segment; more: clamped cubic with chord-length parameters

  Splines: curve points, fitting and editing spline.js

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• Spline from control points

  Builds a spline with a given control polygon, clamped or closed.

  closed: control points wrapped by the degree, knots 0, 1, 2, …

  Splines: curve points, fitting and editing spline.js

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• A spline as points

  The curve as a list of points for drawing and for anything that works from points.

  pieces per span = max(4·max(1, p − 1), ⌈√(polygon length ÷ tol)⌉), within a budget of 4000

  Splines: curve points, fitting and editing spline.js

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• A spline as a polyline

  Points for converting a spline to a polyline, a set number of segments per knot span.

  u = a + (b − a) × i ÷ n, n = 1 … 99 (10 unless given)

  Splines: curve points, fitting and editing spline.js

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• Adding a control point

  The same curve with one more control point at a parameter.

  Qᵢ = (1 − a)·Pᵢ₋₁ + a·Pᵢ, a = (u − kᵢ) ÷ (kᵢ₊ₚ − kᵢ), in weighted form

  Splines: curve points, fitting and editing spline.js

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• Reversing a spline

  The same curve run the other way.

  knot′ = (first + last) − knot, in reverse order

  Splines: curve points, fitting and editing spline.js

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• Control-point spline to fit points

  Samples points on a spline and builds a fit-point spline through them.

  n points evenly spaced in the parameter (12 unless given, at least 3)

  Splines: curve points, fitting and editing spline.js

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  64-bit