Calculations

Calculations

• Arc through three points

  The center, radius, start angle and signed sweep of the arc from the first point through the second to the third.

  d = 2·(aₓ(m_y − b_y) + mₓ(b_y − a_y) + bₓ(a_y − m_y)); cₓ = Σ (x² + y²)·Δy ÷ d; r = hypot(a − c)

  Arc through three points and angle measurement arcs.js

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• Arc as Bézier pieces

  The arc as cubic Bézier segments of at most 90° each, for drawing and for the saved appearance.

  n = ⌈|sweep| ÷ (π ÷ 2) − 1e-9⌉; k = (4 ÷ 3)·tan(step ÷ 4)

  Arc through three points and angle measurement arcs.js

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• Arc length

  The length of an arc markup in points.

  length = r × |sweep|

  Arc through three points and angle measurement arcs.js

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• Angle between two legs

  The angle at a vertex between the directions to two other points, from 0° to 180°.

  θ = acos((u·w) ÷ (|u|·|w|)) × 180 ÷ π

  Arc through three points and angle measurement arcs.js

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• Angle as text

  Writes the angle with a degree sign.

  round(deg × 10ᵖ) ÷ 10ᵖ, p = 1 by default

  Arc through three points and angle measurement arcs.js

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• Snap to 45°

  Moves a point onto the nearest 45° direction from the point before it, keeping its distance.

  a = round(atan2(dy, dx) ÷ (π ÷ 4)) × (π ÷ 4); p = from + (cos a, sin a) × hypot(dx, dy)

  Markup geometry while drawing and editing geom.js

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• Square from a corner

  Makes the box from a fixed corner to the pointer square.

  s = max(|dx|, |dy|); p = a + (sign dx, sign dy) × s

  Markup geometry while drawing and editing geom.js

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• Resize by a grip

  The new box or points when a grip of a markup is dragged.

  box: dragged edges = pointer; stroke: x′ = x₀ + (x − box x₀) ÷ box width × new width

  Markup geometry while drawing and editing geom.js

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• Move a markup

  Shifts a markup's points, box and cut-outs by a distance.

  x′ = x + dx; y′ = y + dy

  Markup geometry while drawing and editing geom.js

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• Simplify a freehand stroke

  Drops the points of a pen or highlighter stroke that lie within a tolerance of the line through their neighbors.

  d = |dy·pₓ − dx·p_y + bₓ·a_y − b_y·aₓ| ÷ L

  Markup geometry while drawing and editing geom.js

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• Inverse of a transform

  The inverse of a 2-D affine transform.

  det = a·d − b·c; inverse = adjugate ÷ det

  Markup geometry while drawing and editing geom.js

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• Line-up through two pairs of points

  The move, turn and uniform scale that carries two picked points on one sheet onto two picked points on another.

  s = |b| ÷ |a|; θ = atan2(b) − atan2(a); M = [s·cos θ, s·sin θ, −s·sin θ, s·cos θ, e, f]

  Markup geometry while drawing and editing geom.js

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• Next point from a typed length and angle

  Places the next point of a line or outline at a typed real length, along a typed angle or toward the pointer.

  len = mm ÷ mmPerPt; a = (deg + page turn) × π ÷ 180; p = from + (cos a, sin a) × len

  Markup geometry while drawing and editing sketch.js

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• Cleaned pen stroke to a markup

  Turns what the stroke cleaner recognized (line, circle, arc, outline) into the points or box of a line, ellipse, arc, rectangle, polyline or polygon markup.

  circle → box (c − r, c + r); arc → points at a₀, a₀ + sweep ÷ 2, a₁ on the circle

  Markup geometry while drawing and editing pen-clean.js

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• Finished freehand stroke

  What is stored when a pen or highlighter stroke is finished.

  tolerance = 0.6 ÷ zoom (points)

  Markup geometry while drawing and editing markupcraft.js

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• Align the old sheet in a comparison

  The fit of the old sheet onto the new one from two points picked on each, and the scale and turn reported for it.

  scale = hypot(M₀, M₁); turn = atan2(M₁, M₀) × 180 ÷ π

  Markup geometry while drawing and editing markupcraft.js

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• Line up a picture by two points

  The new box of a picture markup so two points on it land on two points on the sheet.

  k = √(rw·rh ÷ (W·H)) × fit scale; box = center ± (w·k ÷ 2, h·k ÷ 2)

  Markup geometry while drawing and editing markupcraft.js

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• Length measurement label

  The dimension line, its end ticks and the length text of a measure markup.

  len = hypot(bₓ − aₓ, b_y − a_y); label = len × mmPerPt, formatted

  Markup outlines and measurement labels shapes.js

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• Area measurement label

  The area and perimeter text of an area markup, placed at the outline's centroid.

  Markup outlines and measurement labels shapes.js

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• Angle measurement label

  The angle text of an angle markup and where it sits.

  label = angle at the vertex, 1 decimal; position = vertex + (r + 1.2 × font size) along the bisector

  Markup outlines and measurement labels shapes.js

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• Count total

  The running total shown beside the first mark of a count group.

  n = number of marks

  Markup outlines and measurement labels shapes.js

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• Centroid of an outline

  The area centroid of a closed outline, used to place the label of an area or a named zone.

  k = xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ; cₓ = Σ (xᵢ + xᵢ₊₁)·k ÷ (3·Σ k)

  Markup outlines and measurement labels shapes.js

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• Circle-to-Bézier constant

  The control-point factor for drawing a quarter circle as one cubic Bézier.

  K = 4·(√2 − 1) ÷ 3 ≈ 0.5522847498

  Markup outlines and measurement labels shapes.js

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• Ellipse outline

  An ellipse markup's outline as four cubic Bézier curves inside its box.

  c = (p₀ + p₁) ÷ 2; r = (p₁ − p₀) ÷ 2; control offset = r × K

  Markup outlines and measurement labels shapes.js

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• Revision cloud outline

  The scalloped outline of a cloud markup round its corner points.

  n = max(1, round(edge ÷ arc)); step = edge ÷ n; bulge height = 0.6667 × step

  Markup outlines and measurement labels shapes.js

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• Signed area of an outline

  The signed shoelace area used to tell which way an outline runs.

  A = Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) ÷ 2

  Markup outlines and measurement labels shapes.js

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• A number in an appearance stream

  Turns a number into the text written in a PDF appearance stream.

  text = v rounded to 3 decimals, trailing zeros removed; |v| < 1e-6 → 0

  Numbers written to the saved PDF pdf-write.js

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• Appearance stream of a markup

  The drawing operators that show a markup in any PDF viewer.

  Numbers written to the saved PDF pdf-write.js

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• Private copy of a markup

  The JSON kept with each annotation so it reopens exactly: points, box, cut-outs, style and the rest.

  ar = round(rect × 1000) ÷ 1000

  Numbers written to the saved PDF pdf-write.js

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• Annotation rectangle

  The /Rect of a markup's annotation, which is also the bounding box of its appearance.

  rect = bounds ± 1 pt; each side ≥ 2 × line width + 2 pt

  Numbers written to the saved PDF pdf-write.js

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• Bounds of a display list

  The box round everything a markup draws, used for its annotation rectangle.

  Numbers written to the saved PDF pdf-write.js

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• Rectangle differences of a box or ellipse

  The /RD entry: how far the shape's outer edge sits inside the annotation rectangle on each side.

  RD = [left, top, right, bottom] gaps between rect and box ± half line width, each ≥ 0

  Numbers written to the saved PDF pdf-write.js

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• Count marks as ink rings

  Writes a count group as one Ink annotation with a small ring round each mark.

  ring: 13 points at 30° steps, radius max(3, 0.8 × font size)

  Numbers written to the saved PDF pdf-write.js

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• Arc exported as a PolyLine

  Writes an arc markup's /Vertices as points along the arc, because PDF has no arc annotation.

  n = max(8, ⌈|sweep| ÷ (π ÷ 32)⌉); vertex k = c + r × (cos, sin)(a₀ + sweep × k ÷ n)

  Numbers written to the saved PDF pdf-write.js

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• Picture placement matrix

  The matrix that paints a picture markup into its box, upright for the page's rotation.

  cm = [u·W, −v·H, origin + v·H], each number to 4 decimals

  Numbers written to the saved PDF pdf-write.js

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• Scale as a /Measure dictionary

  The sheet's scale in the form other PDF programs read: the factor from points to the unit and the display precision.

  X.C = mmPerPt ÷ 304.8 (ft), ÷ 25.4 (in), ÷ 1000 (m), or mmPerPt (mm)

  Numbers written to the saved PDF pdf-write.js

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• Scale written on the page

  Stores a sheet's calibration on its page in the saved PDF.

  Numbers written to the saved PDF pdf-write.js

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• Measured text in /Contents

  The text other PDF programs show for a measurement: the length, the area and perimeter, the angle or the count.

  Numbers written to the saved PDF pdf-write.js

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• Flattened markup

  Paints a markup into the page itself instead of saving it as an annotation.

  Numbers written to the saved PDF pdf-write.js

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• Outline points of a PolyLine or Polygon

  The points of a polyline, polygon, cloud or area markup read from an annotation.

  Annotations and scales read from a PDF pdf-read.js

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• Stroke points of an Ink annotation

  The points of pen and highlighter strokes, and of count rings, read from an annotation.

  Annotations and scales read from a PDF pdf-read.js

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• End points of a Line annotation

  The two end points of a line, arrow or length measurement read from an annotation.

  Annotations and scales read from a PDF pdf-read.js

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• Box of a Square or Circle annotation

  The box of a rectangle, ellipse or cloud rectangle markup from the annotation's /Rect.

  box = rect inset by RD and by half the border width on each side

  Annotations and scales read from a PDF pdf-read.js

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• Count mark from its ring

  The position of each count mark, from the ring written for it in the ink list.

  center = mean of the ring's points (the closing point dropped)

  Annotations and scales read from a PDF pdf-read.js

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• Moved by another program

  Decides whether a markup's private JSON still matches the annotation, or another program moved or resized it since.

  |ar − rect| > 0.75 pt on any of the four numbers → use the annotation's own geometry

  Annotations and scales read from a PDF pdf-read.js

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• Scale from another program's /Measure

  A sheet calibration from the measurement dictionary another PDF program wrote.

  mmPerPt = C × mm per unit (ft 304.8, in 25.4, m 1000, cm 10, mm 1)

  Annotations and scales read from a PDF pdf-read.js

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• Which scale a sheet opens with

  Picks the calibration of a page: MarkupCraft's own saved copy, else the page's measure viewport.

  Annotations and scales read from a PDF pdf-read.js

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• Text size from an appearance stream

  The size of the first visible text that another program's text box draws.

  size = |Tf size × text matrix scale × cm scale|, rounded to 0.01

  Annotations and scales read from a PDF freetext.js

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• Text size from a style string

  Reads 12pt, 16px or 1.2em from a text box's style as points.

  px × 0.75; em × 12; pt as written

  Annotations and scales read from a PDF freetext.js

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• A page's paths as segments

  Every drawn path of a page as straight segments in page coordinates.

  curve point = Bézier at t = j ÷ 8, j = 1…8; p′ = CTM × p

  Drawing comparison by vector paths vecdiff.js

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