Calculations

Calculations

• 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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• Do two outline segments cross

  The test that refuses an offset whose loops cross themselves or each other.

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

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• Offset of a face's loops

  The offset outline of a face, back in 3D.

  p = u·x + v·y + n·h, h = p₀ · n

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

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• Offset of a chain of edges

  The offset of a connected run of edges lying in one plane.

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

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• Vertex a dimension end attaches to

  Finds the vertex at a picked point so the dimension follows the geometry.

  |M · q − p| ≤ 100 × EPS

  Dimensions and typed sizes in 3D m3d/model.js

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• Where a dimension end is now

  The current world position of a dimension's end.

  p = M · q

  Dimensions and typed sizes in 3D m3d/model.js

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• Radius and center of a dimensioned curve

  The world center, radius and normal a radius or diameter dimension reads.

  C = M · c; r = |C − M · p₀|

  Dimensions and typed sizes in 3D m3d/model.js

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• Typing a new value into a dimension

  Resizes the model so a dimension shows the typed length, radius or diameter.

  radius: r_local′ = r_local × want ÷ r_world; length: see resize between two ends

  Dimensions and typed sizes in 3D m3d/model.js

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• Resize between two ends

  Moves one or both ends so the distance between them is the typed length.

  δ = L − |p_b − p_a|, applied along the unit direction from a to b

  Dimensions and typed sizes in 3D m3d/model.js

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• Moving the end of a dimension

  Carries the move into the group that holds the moving end and stretches or moves the geometry there.

  d_local = M⁻¹ · d (direction); s = |d_local|; local move = δ × s

  Dimensions and typed sizes in 3D m3d/model.js

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• Typing a new length for an edge

  Sets an edge to a typed length by moving one of its ends.

  Dimensions and typed sizes in 3D m3d/model.js

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• Typing a new size for a group

  Sets a group's or component's size along one of its own axes.

  δ = size − (max − min) on that axis

  Dimensions and typed sizes in 3D m3d/model.js

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• Bounding box

  The smallest axis-aligned box round a group and everything placed in it.

  min and max of M · p over every vertex

  Dimensions and typed sizes in 3D m3d/model.js

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• Length of a linear dimension

  The length a dimension between two points shows, and where its line is drawn.

  L = |p_b − p_a|

  Dimensions and typed sizes in 3D m3d/infer.js

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• Value of a radius or diameter dimension

  The radius or diameter a dimension on an arc or circle shows.

  L = r, or 2 × r for a diameter

  Dimensions and typed sizes in 3D m3d/infer.js

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• Value of an angle dimension

  The angle a dimension at a corner shows.

  θ = atan2(v · (n × u), v · u); the dimension shows θ or π − θ

  Dimensions and typed sizes in 3D m3d/infer.js

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• Angle label

  The text of an angle dimension, in degrees.

  degrees = round(θ × 180 ÷ π × 10) ÷ 10

  Dimensions and typed sizes in 3D m3d/infer.js

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• World matrix of a nested placement

  The matrix that takes a group's own coordinates to the model's, through every group it sits in.

  M_world = M₁ · M₂ · … · Mₖ

  Groups, placements and the model file m3d/model.js

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• Editing context of an open group

  The world matrix of the group being edited and its inverse, used to take picked points into the group.

  M = M₁ · … · Mₖ; M⁻¹ by cofactors

  Groups, placements and the model file m3d/model.js

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• Making a group

  Moves the selected geometry into a new group whose origin is the lowest corner of its box.

  p_group = p − origin; placement = translation(origin)

  Groups, placements and the model file m3d/model.js

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• Exploding a group

  Puts a group's geometry back into its parent at the group's place.

  p_parent = M · p

  Groups, placements and the model file m3d/model.js

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• Flattening a group into a mesh

  Gathers a group and everything placed in it into one set of points before it becomes a mesh.

  key = round(x × 1e8), round(y × 1e8), round(z × 1e8)

  Groups, placements and the model file m3d/model.js

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