Calculations

Calculations

• 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

• Distance from a point to a triangle

  The distance used to decide whether a point is inside a face's outline or on it.

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

  • CadCraft
  • ModelCraft
  64-bit

• Midpoint of an edge

  The middle of an edge, offered as a snap point.

  m = (a + b) × 0.5

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

  • CadCraft
  • ModelCraft
  64-bit

• Point on an edge

  The point of an edge nearest the cursor, offered when no end or midpoint is in reach.

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

  • CadCraft
  • ModelCraft
  64-bit

• Point along an axis from the start point

  The point on a red, green or blue line (or one parallel or square to an edge) through the start point.

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

  • CadCraft
  • ModelCraft
  64-bit

• Where an axis line meets an edge

  The point where the line from the start point along an axis meets the edge or guide under the cursor.

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

  • CadCraft
  • ModelCraft
  64-bit

• Point on a locked direction

  The point kept on a line when a direction is locked with Shift or an arrow key.

  p = from + d × ((q − from) · d) ÷ (d · d)

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

  • CadCraft
  • ModelCraft
  64-bit

• Distance along a line under the cursor

  The point on a line nearest the cursor and how far along it lies, used by push/pull and moves along an axis.

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

  • CadCraft
  • ModelCraft
  64-bit

• Newell normal of a mesh face

  The normal of a face with any number of corners; its length is twice the face's area.

  N = Σ ((y_a − y_b)(z_a + z_b), (z_a − z_b)(x_a + x_b), (x_a − x_b)(y_a + y_b))

  Polygon mesh: normals, areas, triangles and volume m3d/mesh.js

  • CadCraft
  • ModelCraft
  64-bit

• Unit normal of a mesh face

  The direction a mesh face points.

  n = N ÷ |N|

  Polygon mesh: normals, areas, triangles and volume m3d/mesh.js

  • CadCraft
  • ModelCraft
  64-bit

• Area of a mesh face

  The area of one mesh face.

  area = |N| ÷ 2

  Polygon mesh: normals, areas, triangles and volume m3d/mesh.js

  • CadCraft
  • ModelCraft
  64-bit

• Center of a mesh face

  The point used as a face's middle.

  c = Σ pᵢ ÷ n

  Polygon mesh: normals, areas, triangles and volume m3d/mesh.js

  • CadCraft
  • ModelCraft
  64-bit

• Is a mesh face flat

  Whether every corner of a face lies in one plane, which decides if it stays one face or becomes triangles.

  |(p − c) · n| ≤ tol, tol = 1e-6 m by default

  Polygon mesh: normals, areas, triangles and volume m3d/mesh.js

  • CadCraft
  • ModelCraft
  64-bit

• Triangles of a mesh

  The triangles every face is cut into for area, volume, ray tests and drawing.

  Polygon mesh: normals, areas, triangles and volume m3d/mesh.js

  • CadCraft
  • ModelCraft
  64-bit

• Area and volume of a mesh

  The surface area of a mesh and, when it is closed, the volume it encloses.

  A = Σ |u × v| ÷ 2; V = Σ a · (b × c) ÷ 6

  Polygon mesh: normals, areas, triangles and volume m3d/mesh.js

  • CadCraft
  • ModelCraft
  64-bit