Calculations

Calculations

• A routine's number as command input

  Turns a number a routine gives to a command into the text the command reads.

  text = String(+v.toPrecision(15))

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• Answers to a command's prompts

  Hands a routine's string, number, point or object to the command that is waiting for input.

  point: UCS to world, then given as a click; number: as 15-digit text

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• Block-to-world matrix of a nested pick

  The transform from a block's own coordinates to the drawing for an object picked inside nested block references.

  M = M · [rotation × scale, insertion − base]; pick in block = M⁻¹ · pick

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• TRANS: between coordinate systems

  Converts a point or a direction between the world, the user coordinate system and the display.

  display: turn by the view's twist; UCS: by the active coordinate system (ucs.js)

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• Current lineweight as a system variable

  The current lineweight in hundredths of a millimeter.

  CELWEIGHT = round(lineweight × 100)

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• Drawing extents as system variables

  EXTMIN and EXTMAX: the box around every object.

  min and max of every object's bounds

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• Date as a Julian day number

  The DATE system variable.

  Julian date = milliseconds since 1970 ÷ 86 400 000 + 2 440 587.5

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• Position of a block attribute

  The world position of an attribute of a block reference, as entget reports it.

  p = insertion + R(rot) · ((d − base) × scale)

  Drawing values handed to and from LISP routines lisp/host-cad.js

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• Angles brought into 0 to 2π

  Normalizes an arc's, text's or block reference's angle for a data list, and when a routine writes one back.

  a′ = ((a mod 2π) + 2π) mod 2π

  Drawing values handed to and from LISP routines lisp/dxflist.js

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• Lineweight in a data list

  An object's lineweight as group 370, in hundredths of a millimeter.

  370 = round(lw × 100)

  Drawing values handed to and from LISP routines lisp/dxflist.js

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• Lineweight from a data list

  Sets an object's lineweight in millimeters from group 370.

  lw = value ÷ 100

  Drawing values handed to and from LISP routines lisp/dxflist.js

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• Arc from a data list

  Makes or changes an arc from its center, radius and two angles.

  center, radius as given; angles into 0 to 2π

  Drawing values handed to and from LISP routines lisp/dxflist.js

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• Viewport scale from a data list

  Sets a layout viewport's scale from the view height a routine gives.

  scale = paper height ÷ view height

  Drawing values handed to and from LISP routines lisp/dxflist.js

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• Linetype pattern length

  The total length of a linetype's dashes and gaps in its table record.

  length = Σ |xᵢ|

  Drawing values handed to and from LISP routines lisp/dxflist.js

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• Length and angle of a line

  The Length, Angle and Delta properties of a line.

  length = √(Δx² + Δy²); angle = atan2(Δy, Δx) into 0 to 2π

  Drawing values handed to and from LISP routines lisp/vla-host.js

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• Circle and arc properties

  Diameter, area and circumference of a circle; total angle, arc length and end points of an arc.

  area = π r²; circumference = 2π r; sweep = ((a₁ − a₀) mod 2π + 2π) mod 2π; arc length = r × sweep; end = c + r (cos a, sin a)

  Drawing values handed to and from LISP routines lisp/vla-host.js

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• Ellipse radii

  The major and minor radius of an ellipse.

  major = √(m_x² + m_y²); minor = major × ratio

  Drawing values handed to and from LISP routines lisp/vla-host.js

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• Lineweight property set by a routine

  Sets an object's lineweight in millimeters from a value in hundredths of a millimeter.

  lw = value ÷ 100

  Drawing values handed to and from LISP routines lisp/vla-host.js

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• Plane of a face

  The plane a face lies in, as a unit normal and a distance from the origin.

  n = Newell normal of the outer loop; d = n · p₀

  Face planes, areas, triangles and solid volume m3d/model.js

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• Area of a face

  The area of a face, holes taken out.

  area = |Σ (pᵢ × pᵢ₊₁)| ÷ 2 for the outer loop, minus the same for each hole

  Face planes, areas, triangles and solid volume m3d/model.js

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• Triangles of a face

  The triangles a face is cut into for volume, export and ray tests.

  Face planes, areas, triangles and solid volume m3d/model.js

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• Surface area and volume of a shape

  Whether a group or component is closed, its total face area and, when closed, its volume.

  V = Σ a · (b × c) ÷ 6 over all triangles; A = Σ face areas

  Face planes, areas, triangles and solid volume m3d/model.js

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• Vertex at a point

  The vertex a drawn point becomes: an existing one, or a new one that splits the edge it lands on.

  s = (p − A) · u ÷ (u · u); on the edge when 0 < s < 1 and |A + s·u − p| ≤ EPS

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Stops along a new edge

  Where a new edge is cut by the vertices and edges it meets.

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Crossing of two edges

  The point where two edges cross inside both of them.

  n = u × w; s = ((C − A) × w) · n ÷ (n · n); t = ((C − A) × u) · n ÷ (n · n)

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Planes a new edge may close a face on

  The planes tried when a new edge might complete a face.

  n = u × (C − A), kept when |n| > 1e-9 × |u| × |C − A|

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Faces of a plane from its edges

  Which closed loops of edges on a plane are faces, which are holes, and which side each face shows.

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Loops of the planar edge graph

  The closed loops formed by the edges on a plane, and the signed area of each.

  edges round a vertex sorted by atan2(Δy, Δx); area = ½ Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Point inside a loop

  Whether a 2D point is inside a loop, used to match holes and old faces.

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Test point inside a region

  A point well inside a region, used to ask which old faces cover it.

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Welding coincident vertices

  Merges vertices at the same place and removes edges and loops that collapse.

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Removing a vertex on a straight run

  Joins two edges that continue each other in a straight line into one.

  merged when u₁ · u₂ ≤ −1 + 1e-9

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Merging coplanar faces

  Removes an edge between two faces that lie in one plane and face the same way.

  n₁ · n₂ ≥ 1 − 1e-9 and |d₁ − d₂| ≤ EPS

  Drawing an edge: welding, splitting and re-facing m3d/model.js

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• Push or pull a face

  Moves a face along its normal by a typed or dragged distance, or extrudes it into side faces.

  p′ = p + n × distance

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Depth to the face behind

  How far a face can be pushed in before it meets the first face behind it.

  smallest ray parameter t > EPS along −n

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Moving vertices by a matrix

  The new place of each selected vertex under a move, rotate, scale or flip.

  p′ = M · p

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Is a face still flat

  Whether every corner of a face lies in the face's plane after an edit.

  |n · p − d| ≤ 10 × EPS for every corner

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Stretch to a new size

  Moves every vertex beyond a cut plane by a distance, as used by typed dimensions and sizes.

  p′ = p + d × δ for every vertex with p · d ≥ cut − 10 × EPS

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Profile ring at a path corner

  Where each corner of a swept profile lands on the next miter plane of the path.

  q′ = q + d × (n · (o − q)) ÷ (n · d)

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Where a line crosses a face

  The stretches of a line that lie inside a face, holes left out.

  c = P + (Q − P) × s_P ÷ (s_P − s_Q); t = (c − x₀) · u

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Line where two faces cross

  The line shared by the planes of two faces, clipped to both.

  dir = n_a × n_b; x₀ = ((n_b·d_a − n_a·d_b) × dir) ÷ |dir|²

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Mirror matrix

  The matrix that flips geometry through a plane.

  M = T(c) · (I − 2·n·nᵀ) · T(−c)

  Push/pull, move, stretch, follow me and intersect m3d/model.js

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• Corners of a circle or polygon

  The corner points of a circle or regular polygon with a given number of sides.

  pᵢ = c + u·r·cos aᵢ + v·r·sin aᵢ, aᵢ = (i ÷ sides) × 2π

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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• Corners of an arc

  The corner points of an arc from its center, radius, sweep and segment count.

  pᵢ = c + u·r·cos aᵢ + v·r·sin aᵢ, aᵢ = sweep × i ÷ segs

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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• What a curve is

  The length, radius, center, normal and sweep reported for an arc, circle, polygon or smooth curve.

  length = Σ |segment|; center = circle through three corners; sweep = |Σ atan2((u₁ × u₂) · n, u₁ · u₂)|

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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• Give a curve a new radius

  Moves the corners of an arc, circle or polygon to a typed radius about its center.

  p′ = c + (p − c) × r ÷ r_old

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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• Equal division points

  Where the parts of an edge or curve divided into n equal parts meet.

  straight edge: a + (b − a) × (i + 1) ÷ n; arc: new corners at equal angles; other curve: equal lengths along its segments

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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• Dividing an edge or curve

  Adds the corners that divide an edge or curve into n equal parts.

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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• Points of a smooth curve

  The points of the smooth curve drawn through clicked points.

  centripetal Catmull–Rom: knot step = √|pᵢ₊₁ − pᵢ|, evaluated at k ÷ segs of each span

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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• Offset of a 2D outline

  The corners of an outline moved sideways by a distance.

  each segment line moved d along its left unit normal; corner = intersection of neighboring lines

  Arcs, circles, smooth curves, divide and offset m3d/model.js

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