Calculations

Calculations

• 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

• Adding a layer

  Creates a layer of 1 to 16 numbers per element.

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• Storing a value in a layer

  Writes the values of one element of a layer.

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• Blending corner values onto new faces

  The values new face corners get when faces are cut or joined, as a weighted mix of old corners.

  out = Σ wᵢ × valueᵢ

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• Layers in the mesh file

  The layer data as it is written into the file.

  Polygon mesh: attribute layers (UV, color, crease, weights) m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit in part

• Normal per corner

  A unit normal for every corner of every face.

  n = normalize(Σ N_f) over the faces of a smooth group

  Polygon mesh: shading normals m3d/mesh.js

  • CadCraft
  • ModelCraft
  32-bit, display only

• World triangles for export

  Every triangle of the model in world coordinates, with its normal.

  p_world = M · p; n = normalize((p₁ − p₀) × (p₂ − p₀))

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF vertex positions

  The vertex coordinates and normals stored in the binary part of a .glb file.

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  32-bit in part

• glTF placement matrix

  The matrix that places each group or component in the .glb file.

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• STL triangles in millimeters

  The triangles of a binary STL file.

  x_mm = x × 1000, stored as a 32-bit float

  Export to glTF and STL m3d/io.js

  • CadCraft
  • ModelCraft
  32-bit in part

• OBJ number format

  How every coordinate and normal component is written in an OBJ file.

  text = x rounded to 6 decimals

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• OBJ vertex

  A vertex of the model as it goes into the OBJ file.

  (x, y, z) → M · p → (x, z, −y)

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• OBJ face normal

  The flat normal written for a face.

  N = Σ pᵢ × pᵢ₊₁, normalized

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• OBJ vertex and normal lines

  The lines that carry the numbers.

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada number format

  How every number is written in a Collada file.

  text = x rounded to 12 significant digits

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada positions and normals

  The vertex positions and face normals of each definition in the Collada file.

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada placement matrix

  The 4×4 matrix of each placement, written row by row.

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Is a face convex

  Decides whether a face is written as one polygon or as triangles in Collada.

  every turn (b − a) × (c − b) ≥ −1e-12 × size²

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Unit factor for DXF and DWG

  The factor that turns the model's meters into the drawing unit of the export.

  inches: 1 ÷ 0.0254; mm: 1000; cm: 100; m: 1

  Export of the 3D model to DXF and DWG m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Exported point

  A vertex of the model in the export's unit.

  p′ = (M · p) × k

  Export of the 3D model to DXF and DWG m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Placement as a block insert

  The insertion point, scale, rotation and extrusion direction that stand for a placement matrix.

  scale = column lengths; n = third unit column; rot = atan2(r₀ · a_y, r₀ · a_x); insertion = (T · a_x, T · a_y, T · n)

  Export of the 3D model to DXF and DWG m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• DXF number format

  How every coordinate, scale and angle is written in the DXF text.

  text = x rounded to 15 significant digits

  Export of the 3D model to DXF and DWG m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit