Calculations

Calculations

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• Numbers written to the model file

  The vertices, placement matrices and curve points as they go into the file.

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

  • CadCraft
  • ModelCraft
  64-bit

• Vertices read from a file

  The check each vertex passes when a file is opened.

  finite and |x| < 1e7

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

  • CadCraft
  • ModelCraft
  64-bit

• Level elevations

  The list of named levels and their heights.

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

  • CadCraft
  • ModelCraft
  64-bit

• Light values as stored

  The brightness, cone angle, size and position kept for a lamp used in rendered pictures.

  lumens = round(clamp(x, 1, 200000)); angle = round(clamp(x, 5, 170)); size clamped to 0.02…20 m

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

  • CadCraft
  • ModelCraft
  64-bit

• Drawing axes

  The frame set when the drawing axes are placed by three picks.

  X = x ÷ |x|; Y = (y − X (y · X)) normalized; Z = X × Y

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

  • CadCraft
  • ModelCraft
  64-bit

• Faces from imported polygons

  Builds faces from the polygons of an imported file, sharing vertices between them.

  key = round(x × 1e5), round(y × 1e5), round(z × 1e5)

  Welding of imported geometry m3d/model.js

  • CadCraft
  • ModelCraft
  64-bit

• Edges from imported segments

  Builds plain edges from the line segments of an imported drawing.

  key = round(x × 1e5), round(y × 1e5), round(z × 1e5)

  Welding of imported geometry m3d/model.js

  • CadCraft
  • ModelCraft
  64-bit

• Smooth shading normals

  The normal at each corner of faces joined by soft edges.

  n = normalize(Σ n_f × area_f) over the joined faces that meet at the corner

  Smooth shading and texture placement m3d/model.js

  • CadCraft
  • ModelCraft
  32-bit, display only

• Fitting a texture to a face

  The map from a point on a face to its place on an imported image.

  u = U · p + u₀, v = V · p + v₀, with U · n = 0 and V · n = 0

  Smooth shading and texture placement m3d/model.js

  • CadCraft
  • ModelCraft
  32-bit, display only

• Texture coefficients as stored

  The eight numbers of a texture's placement as they are kept in the face's paint.

  text = x rounded to 7 significant digits

  Smooth shading and texture placement m3d/model.js

  • CadCraft
  • ModelCraft
  32-bit, display only

• Ray and triangle

  Where the cursor's ray meets one triangle of a face.

  det = e₁ · (d × e₂); u, v barycentric; t = e₂ · (s × e₁) ÷ det

  Snap points picked in the 3D view m3d/infer.js

  • CadCraft
  • ModelCraft
  64-bit

• Point on the face under the cursor

  The nearest face the cursor's ray meets and the 3D point on it.

  o_L = M⁻¹ · o; d_L = M⁻¹ · d; p = o + d × t

  Snap points picked in the 3D view m3d/infer.js

  • CadCraft
  • ModelCraft
  64-bit

• World plane of a placed face

  The triangles, normal and plane of a face in model coordinates, used to draw on that face.

  n = normalize(M · n_local); d = n · (M · P)

  Snap points picked in the 3D view m3d/infer.js

  • CadCraft
  • ModelCraft
  64-bit

• Center of a face

  The center point offered as a snap on the face under the cursor.

  c = Σ (a + b + d) × w ÷ 3 ÷ Σ w, w = |(b − a) × (d − a)|

  Snap points picked in the 3D view m3d/infer.js

  • CadCraft
  • ModelCraft
  64-bit

• Is a point on a face

  Whether a snapped point lies on the face under the cursor, so a shape started there takes the face's plane.

  |n · p − d| ≤ tol and distance to a triangle ≤ tol, tol = 1e-6 × max(1, |x|, |y|, |z|)

  Snap points picked in the 3D view m3d/infer.js

  • CadCraft
  • ModelCraft
  64-bit