Calculations

Calculations

• Pieces back from the boolean library

  Reads a piece's corners from manifold-3d and returns them to world coordinates.

  world = org + 32-bit local corner

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  32-bit in part

• Cutting by each plane

  Splits every cell by the next plane and keeps the parts that have volume.

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  32-bit in part

• Outline of a cut through the model

  The line segments where a plane crosses the model's triangles, in the plane's own two axes.

  sᵢ = n·pᵢ − d; k = sᵢ ÷ (sᵢ − sⱼ); point = pᵢ + (pⱼ − pᵢ) × k

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Sampling a cut section

  For a grid of points over the cut: inside the model or not, and the distance to the outline.

  step = max(largest extent ÷ samples, 0.2); inside = odd number of outline crossings to the left; distance = √(min squared distance to an outline segment)

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Touching area of a joint

  The area where two pieces meet on a cut.

  area += samples in the region × step²

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Room a connector needs

  The least distance from a connector's axis to any outside surface, and how deep it goes each side.

  peg: d ÷ 2 + c + w; key: (W ÷ 2)·√2 + c + w; dovetail: w₁ ÷ 2 + c + w with w₁ = W + 2H·tan(angle); dowel, magnet: d ÷ 2 + c + w

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Round hole with its flats on the radius

  Sizes a faceted hole so its flat sides, not its corners, are at the asked radius.

  corner radius = r ÷ cos(π ÷ segments)

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Connector and hole solids

  The peg, key, dovetail, dowel pin or magnet pocket and the matching hole, in the joint's own frame.

  hole radius = r + c; hole depth = L + c; chamfer = min(0.5, r ÷ 3); dovetail half width at z: w₀ + (w₁ − w₀)·z ÷ H + growth

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Does a connector fit here

  Checks along a connector's depth that it stays inside the model with a wall all round and clear of the other cuts.

  at steps of 0.75 mm along the depth: inside, nearest outside surface ≥ lateral, distance to every other cut ≥ lateral + deepest hole

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Spreading connectors along a joint

  Picks up to k connector spots spread along the long axis of the touching area.

  θ = ½·atan2(2·Σuv, Σu² − Σv²) about the mean; samples sorted by their position along that axis and cut into k slices

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Length of a dovetail

  How far a dovetail runs each way from its spot, and whether each end is open.

  open end: (steps + ½) × step + wall + 1; stopped end: max(0, steps × step − lateral)

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Engraved joint number

  The joint's number as a solid: each stroke of the stroke font widened to a rectangle, joined and extruded.

  stroke width = max(0.6, 0.14 × h); rectangle = segment grown by half the width on all sides

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Corners back onto slanting cut planes

  Moves each corner that lies within 2e-4 mm of a slanting cut plane, or of a connector or number depth parallel to it, exactly onto that plane.

  s = n·p − d; if 0 < |s| < 2 × 10⁻⁴: p ← p − n·s

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  32-bit in part

• Building and measuring each piece

  Adds the male connectors, cuts the holes and numbers, and returns each piece with its size, volume and print check.

  piece = (cut part ∪ male parts) − holes − numbers; size = box of its corners; volume from the library

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  32-bit in part

• Laying a piece on the bed

  Turns a piece in 90° steps so it fits the bed and moves it to stand on z = 0 at x, y ≥ 0.

  new[k] = ±old[q(k)]; then new[k] −= min over the piece

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Piece list on the assembly sheet

  One line per piece with its size and volume, then the joints and the pins.

  size shown = round(mm × 10) ÷ 10; volume shown = round(mm³ ÷ 1000 × 10) ÷ 10 cm³; pin length = dowel length − clearance

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  32-bit in part

• Normal of a slanting cut

  The normal of a cut across one axis, turned by an angle about another axis.

  n = axis·cos r + (about × axis)·sin r, r = degrees × π ÷ 180

  Splitting a model into printable pieces print-split-ui.js

  • ModelCraft
  • PartCraft
  64-bit

• Cut places typed by the person

  Turns the typed distances from the low end of the model into cut planes.

  point = middle of the box, with axis coordinate lo + v; d = n·point

  Splitting a model into printable pieces print-split-ui.js

  • ModelCraft
  • PartCraft
  64-bit

• Connector spot picked by hand

  Where the ray under a click crosses the cut plane between two pieces.

  t = (d − n·o) ÷ (n·dir); p = o + dir × t

  Splitting a model into printable pieces print-split-ui.js

  • ModelCraft
  • PartCraft
  64-bit

• Welding corners

  Joins corners that lie within the tolerance of each other and drops triangles that collapse.

  join if |Δx| ≤ tol and |Δy| ≤ tol and |Δz| ≤ tol; drop if two corners are the same or |n|² ≤ (tol² × 0.01)²

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Volume, area, overhang and bed contact

  Each shell's volume, the total surface area, the area that needs support and the area lying on the bed.

  area = Σ |n| ÷ 2; V_shell = Σ a·(b × c) ÷ 6; overhang if n_z ÷ |n| < −cos((90 − angle)·π ÷ 180)

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• A shell inside another shell

  Decides whether an inside-out shell is a hollow inside another shell, which is correct, or a real fault.

  ray from one corner along [0.000123, 0.000457, 1]; inside if it crosses the other shells an odd number of times

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Ray and triangle

  The distance along a ray to a triangle, or −1 when it misses.

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

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Finding triangle pairs to test

  Finds pairs of triangles whose boxes overlap and counts those that cross, within one shell and between shells.

  eps = diagonal × 10⁻⁹; boxes grown by eps; pair tested with tolerance 10 × eps

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Do two triangles cross

  Whether two triangles pass through each other, more than touching.

  signed distances of each triangle's corners to the other's plane, set to 0 within eps; overlap of the two intervals on the planes' common line must exceed eps

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Triangles in the same plane

  Whether two triangles in one plane overlap.

  2D orientation tests on the projection that drops the normal's largest axis; crossing edges, or one centroid inside the other

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Total volume and bed fit

  The volume reported for a mesh without problems, and the warnings for a tiny model or one larger than the bed.

  V = Σ shell volumes (hollows are negative); fits if sizeₖ ≤ bedₖ + 10⁻⁹, as given or with x and y swapped

  Print check: weld, volume, area and intersections print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Binary STL

  Writes each triangle's unit normal and three corners.

  normal = n ÷ |n| from the winding; every number written with setFloat32

  STL and 3MF files for the printer print3d.js

  • ModelCraft
  • PartCraft
  32-bit in part

• A coordinate as 3MF text

  Formats one coordinate for the 3MF file.

  0 if |v| < 10⁻¹²; else v rounded to 10 significant digits

  STL and 3MF files for the printer print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• 3MF vertex list

  Writes every corner of the mesh as a vertex element.

  STL and 3MF files for the printer print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Render buffers of a definition

  The triangles of a definition's shown faces as position, normal and texture-coordinate buffers.

  Scene handed to the renderer render-mc.js

  • ModelCraft
  32-bit, display only

• Height from a baked displacement map

  The displacement height at a texture coordinate.

  bilinear blend of the four texels round (u, v), wrapping at the edges, read at texel centers

  Scene handed to the renderer render-mc.js

  • ModelCraft
  32-bit, display only

• Displaced mesh point

  Moves a mesh point along its averaged normal by the material's height times its scale, for the render.

  p′ = p + n̂ × height(u, v) × scale

  Scene handed to the renderer render-mc.js

  • ModelCraft
  32-bit, display only

• Flat area per color

  The total area of level, upward faces of each color, largest first.

  s = ∛|m₀·(m₅·m₁₀ − m₆·m₉)|; area = Σ face area × s²; shown = round(area × 100) ÷ 100

  Areas of flat upward faces by color render-mc.js

  • ModelCraft
  64-bit

• Sunrise, sunset and solar noon

  The clock hours of sunrise, sunset and solar noon and the sun's highest altitude on a day.

  hᵢ = min(0.05·i, 23.99), i = 0 … 480; rise = first hᵢ with altitude > 0 after one ≤ 0; noon = hᵢ of the highest altitude

  Time of day, date and exposure from plain words scene-say.js

  • ModelCraft
  64-bit

• Clock hour for a named time

  The hour that stands for dawn, morning, noon, afternoon, golden hour, dusk or night on a given day.

  dawn = rise + 0.3; morning = rise + 0.45·(noon − rise); afternoon = noon + 0.55·(set − noon); dusk = set − 0.1; night = set + 2; hour = round to 0.01

  Time of day, date and exposure from plain words scene-say.js

  • ModelCraft
  64-bit

• The look for the choices made

  Combines time, date or season and weather into the sun's date and hour and the render's exposure.

  month, day = round(given); exposure = round((weather + time of day) × 100) ÷ 100

  Time of day, date and exposure from plain words scene-say.js

  • ModelCraft
  64-bit

• Places things can stand on

  Top-level placements and large flat faces with their box, top height, height and area.

  placement area = Δx × Δy of its box; face area from the model kernel; every number rounded to 0.001

  Placing scene trees and people scene-entourage.js

  • ModelCraft
  64-bit

• Point inside a face outline

  Whether a plan point lies inside a polygon.

  count edges with (y_a > y) ≠ (y_b > y) and x < (x_b − x_a)·(y − y_a) ÷ (y_b − y_a) + x_a; inside if odd

  Placing scene trees and people scene-entourage.js

  • ModelCraft
  64-bit

• The ground a request names

  The region to scatter over, the height things stand at, and the boxes to keep clear.

  on a placement: its box in plan, z = top of the box; on a face: its outline, z = first point's z; avoid boxes with top > z + 0.3 and bottom < z + 2.5

  Placing scene trees and people scene-entourage.js

  • ModelCraft
  64-bit

• Planting trees

  Places a number of trees over the region, each with a random turn and a size within 10% of the library's.

  minimum distance = max(0.5, 0.9 × crown width); yaw = rand × 2π; scale = 0.9 + 0.2 × rand

  Placing scene trees and people scene-entourage.js

  • ModelCraft
  64-bit

• Placing people

  Places figures over the region; along a line they face along it, half each way.

  heading = atan2(Δy, Δx) of the line; yaw = heading + π ÷ 2 (+ π for half of them) + (rand − ½) × 0.3; scale = 0.97 + 0.06 × rand

  Placing scene trees and people scene-entourage.js

  • ModelCraft
  64-bit

• Scattering points with a minimum spacing

  Seeded points inside a rectangle or along a line, none closer than the minimum distance to another.

  rectangle: x = x₀ + margin + rand × w, y likewise; line: p = a + (b − a) × i ÷ (count − 1) + normal × jitter; kept x, y = round to 0.01

  Placing scene trees and people scene-say.js

  • ModelCraft
  64-bit

• Placement matrix

  The 4×4 matrix for a turn about the vertical axis, a uniform scale and a position.

  c = cos(yaw)·scale; s = sin(yaw)·scale; columns [c, s, 0], [−s, c, 0], [0, 0, scale], [x, y, z]

  Placing scene trees and people scene-say.js

  • ModelCraft
  64-bit

• Arc and circle as straight segments

  The ground segments for one traced line, arc or circle.

  aₖ = a₀ + sweep × k ÷ n; pₖ = [cx + r·cos aₖ, cy + r·sin aₖ, 0]

  Traced sketch into model edges sketch-photo-mc.js

  • ModelCraft
  64-bit

• Picked points in millimeters

  The picked model points as the surfaces library receives them.

  v_mm = round(v × 1000 × 10⁶) ÷ 10⁶

  Points picked for surfaces surfaces-mc.js

  • ModelCraft
  64-bit

• Letter outlines from strokes

  The outline of the text: each stroke segment as a rectangle of the pen's width, each stroke point as a disc.

  r = height × weight ÷ 2; rectangle corners = a ∓ r·n̂, b ∓ r·n̂; disc corner = c + r·[cos(i·π ÷ 6), sin(i·π ÷ 6)]; line k at y − 1.6·k·height

  3D text text3d.js

  • ModelCraft
  64-bit

• Letters as faces or solids

  The component the app places: flat faces, or a closed solid from z = 0 to the depth.

  g6(v) = round(v × 10⁶) ÷ 10⁶ on every coordinate

  3D text text3d.js

  • ModelCraft
  32-bit in part

• Did another program change the points

  Decides whether a definition's native USD points still match the model's own.

  changed if any fround(own coordinate) ≠ file coordinate, or the counts differ

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  Exact

• Mesh from the native points

  Rebuilds a definition that was changed elsewhere from the file's native points and faces.

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  64-bit