Calculations

Calculations

• Canvas position to time and value

  The time and value under a canvas point.

  t = t₀ + (x − pad) ÷ (w − 2 pad) × (t₁ − t₀); v = v₀ + (h − pad − y) ÷ (h − 2 pad) × (v₁ − v₀)

  Keys dragged in the graph editor anim/graph.js

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• A key dragged to a new time and value

  Moves a key in time and sets one number of its value.

  time = max(0, round(t × 1000) ÷ 1000); value = v

  Keys dragged in the graph editor anim/graph.js

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• A handle dragged

  The handle's new offset from its key in time and value.

  Δt = t − key time (≤ 0 for the left handle, ≥ 0 for the right); Δv = v − key value

  Keys dragged in the graph editor anim/graph.js

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• 4×4 matrix product

  The product of two column-major 4×4 matrices.

  Bone frames and pose matrices anim/rig.js

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• 4×4 matrix inverse

  The inverse of a 4×4 matrix by cofactors.

  M⁻¹ = adjugate(M) ÷ det(M)

  Bone frames and pose matrices anim/rig.js

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• A bone's rest frame

  The matrix from a bone's own frame to the mesh: Y from head to tail, Z as near the mesh's +Z as it can be, turned by the roll.

  y = (tail − head) made unit; z = (up − y (up · y)) made unit; x = y × z; roll turns x and z about y

  Bone frames and pose matrices anim/rig.js

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• Posed bone and skin matrices

  Each bone's posed frame and the matrix that moves rest points to the pose.

  W = W_parent × R_parent⁻¹ × R × local; skin = W × R⁻¹

  Bone frames and pose matrices anim/rig.js

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• Skin matrices as dual quaternions

  Each bone's turn and move as a unit dual quaternion, for dual-quaternion skinning.

  dual part = ½ × (t, 0) × q

  Bone frames and pose matrices anim/rig.js

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• A bone's pose from a drag

  The location or rotation a bone gets when it is dragged in the view.

  turn: angle swept round the bone's head on screen, about the view axis carried into the bone's frame; move: pixels × pixel size, along the camera's right and up, carried into the bone's frame

  Bone frames and pose matrices anim/rig-view.js

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• Axes and unit

  The matrix that takes the file's coordinates to Z up and meters.

  k = UnitScaleFactor × 0.01; columns = ± k along the model axis each file axis maps to

  FBX import: axes, units and transforms io/fbx-import.js

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• Node matrices into the model's axes

  Carries a node's matrix from the file's axes and unit into the model's.

  M′ = A × M × A⁻¹

  FBX import: axes, units and transforms io/fbx-import.js

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• Euler angles to a matrix

  The rotation matrix of three angles in degrees in the file's rotation order.

  XYZ order: R = Rz × Ry × Rx, angles × π ÷ 180

  FBX import: axes, units and transforms io/fbx-import.js

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• An object's own matrix

  Builds a model's local matrix from its translation, offsets, pivots, pre-, own and post-rotation, and scaling.

  M = T × Roff × Rp × Rpre × R × Rpost⁻¹ × Rp⁻¹ × Soff × Sp × S × Sp⁻¹

  FBX import: axes, units and transforms io/fbx-import.js

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• Mesh points into the model

  Carries each control point through the axis and unit matrix.

  p′ = A × p

  FBX import: axes, units and transforms io/fbx-import.js

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• Lamps

  A lamp's position, direction and brightness from an FBX light.

  lumens = Intensity × 8; direction = the node's −Y, made unit

  FBX import: axes, units and transforms io/fbx-import.js

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• Cameras

  A saved view's eye, target and field of view from an FBX camera.

  target = eye + 10 × forward; parallel view height = OrthoZoom × 10

  FBX import: axes, units and transforms io/fbx-import.js

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• Reading a vertex

  Parses the three coordinates of a 'v' line.

  OBJ import and export io/obj.js

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• Points into Z up

  Copies an object's points into the model's coordinates.

  (x, y, z) → (x, −z, y)

  OBJ import and export io/obj.js

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• Sharp edges

  Marks an edge sharp where the file's normals break across it, where smoothing groups differ, or where the faces bend.

  with normals: sharp unless n₁ · n₂ > 0.9995 at both ends; without normals or groups: sharp when the face normals' dot product < 0.99996

  OBJ import and export io/obj.js

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• Number format on export

  Prints a number for the exported file.

  text = the number rounded to 6 decimals, trailing zeros removed

  OBJ import and export io/obj.js

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• Exported points

  Carries a mesh point to world coordinates and Y up for the file.

  p′ = placement matrix × p; (x, y, z) → (x, z, −y)

  OBJ import and export io/obj.js

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• Texture coordinates with a map's scale and offset

  Applies a material's map scale and offset to a texture coordinate on import.

  u′ = u × scale_u + offset_u; v′ = v × scale_v + offset_v

  OBJ import and export io/obj.js

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• Rounding of every parameter

  Rounds each number of a converted material.

  v → the number nearest v with 7 significant digits

  Imported material parameters io/pbr.js

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• Film thickness and dispersion

  A thin film's thickness in micrometers and a glass's Abbe number.

  thickness (µm) = maximum thickness (nm) ÷ 1000; Abbe number = 20 ÷ dispersion

  Imported material parameters io/pbr.js

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• Emission strength

  Splits an emissive color into a color with a peak of 1 and a luminance.

  peak = largest channel; color = channels ÷ peak; luminance = peak × strength

  Imported material parameters io/pbr.js

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• Shininess to roughness

  A roughness from a Blinn–Phong shininess exponent.

  roughness = (2 ÷ (s + 2))^¼, clamped to [0, 1]

  Imported material parameters io/pbr.js

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• Knot span

  Finds the knot interval that holds a parameter.

  3DM curves and surfaces: points on them io/3dm/geom.js

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• de Boor's algorithm

  A point of a B-spline from its control points and knots.

  a = (t − k₀) ÷ (k₁ − k₀); d_j = (1 − a) d_(j−1) + a d_j, repeated degree times

  3DM curves and surfaces: points on them io/3dm/geom.js

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• Rational point

  Divides a homogeneous point by its weight.

  (x, y, z) = (X, Y, Z) ÷ w

  3DM curves and surfaces: points on them io/3dm/geom.js

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• Point on a NURBS surface

  A surface point from its control net.

  de Boor along v for each active row, then along u, then the division by the weight

  3DM curves and surfaces: points on them io/3dm/geom.js

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• Point on a curve

  A point on a line, arc, polyline, NURBS curve or chain of curves at a parameter.

  arc: a = angle mapped from the parameter; p = o + r cos a × x + r sin a × y

  3DM curves and surfaces: points on them io/3dm/geom.js

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• A curve as a polyline

  Cuts a curve into chords that stay within a tolerance of it, keeping every knot and corner.

  a piece is halved while its middle or quarter points lie farther than tol from the chord, at most 10 times

  3DM curves and surfaces: points on them io/3dm/geom.js

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• Rotation about an axis

  Turns a point about a line by an angle, for surfaces of revolution.

  p′ = o + v cos a + (z × v) sin a + z (z · v)(1 − cos a)

  3DM curves and surfaces: points on them io/3dm/geom.js

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• Point on a surface

  A point on a plane, NURBS surface, surface of revolution or sum surface.

  plane: o + a x + b y; revolution: the profile point turned about the axis; sum: p(u) + q(v) + base

  3DM curves and surfaces: points on them io/3dm/geom.js

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• Surface normal

  The unit normal of a surface at a point.

  ∂u ≈ S(u + h, v) − S(u − h, v); ∂v likewise; n = ∂u × ∂v, made unit; h = 1e-5 × parameter range

  3DM curves and surfaces: points on them io/3dm/geom.js

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• Selection center

  The mean position of the selected vertices: the pivot for rotate, scale and spin.

  c = (Σ pᵢ) × (1 ÷ n)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Point on an axis nearest the pointer

  The point of a line through w along dir that is nearest the pointer's ray; used for axis-locked moves and extrude distances.

  s = (b·e − c·d) ÷ (a·c − b²); point = w + d₁·s

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Grab, rotate and scale

  Builds the move applied to every affected vertex in world space.

  grab: p′ = p + d; rotate: p′ = R(k, θ)(p − P) + P; scale: p′ = P + f·(p − P), or along a locked axis a: p′ = p + a·((p − P)·a)·(f − 1)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Rotation about an axis

  Turns a point about the axis k through c by the angle a.

  q = p − c; p′ = c + q·cos a + (n × q)·sin a + n·(n·q)·(1 − cos a)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Extrude distance

  Moves the new vertices of an extrusion by the dragged or typed distance along the region's normal, or along each face's own normal.

  d = (line point now − line point at start)·dir; p′ = p + dirL·d, dirL = M⁻¹·dir

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Knife point on an edge

  Where along an edge a knife click lands, as a fraction of the edge from its start.

  s = clamp((q − a)·(b − a) ÷ L², 0, 1)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Bisect plane

  The cutting plane through two clicked points as seen from the camera, in the mesh's own space.

  n = unit((p₂ − p₁) × ray direction); plane = {M⁻¹·p₁, unit(M⁻¹ applied to n as a direction)}

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Spin center, axis and angle

  The center, axis and total angle handed to the spin operator, in the mesh's own space.

  center = M⁻¹·c; axis = unit(M⁻¹ applied to the world axis); angle = typed × π ÷ 180, or −(a₁ − a₀)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

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• Vertex normals

  A unit normal per vertex from the faces round it, weighted by face area.

  N(v) = unit(Σ Newell normal of each face at v)

  Inset, bevel and moves along normals mesh/ops.js

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• Inset corner

  Where a face corner goes when both of its edges move inward by d.

  bis = unit(in₁ + in₂); c = bis·in₁; k = d ÷ c; p′ = p + bis·k

  Inset, bevel and moves along normals mesh/ops.js

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• Inset of a region's border

  The new border vertices of a region inset, each from the two border edges that meet at it, raised by the depth.

  n = unit(n(face in) + n(face out)); p′ = insetCorner(p, a, b, n, thickness) + n·depth

  Inset, bevel and moves along normals mesh/ops.js

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• Bevel: slide along plain edges

  How far a new point slides along an edge that is not bevelled, so the bevel is the stated width inside the face.

  sin = |u_in × u_out|; slide = width ÷ sin; point = v + u·min(slide, cap)

  Inset, bevel and moves along normals mesh/ops.js

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• Bevel: where two bevelled edges meet

  The point inside a face where the offset lines of two bevelled edges cross.

  k = width ÷ |u_in × u_out|; point = v + (u_in + u_out)·k

  Inset, bevel and moves along normals mesh/ops.js

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• Bevel: profile points

  The points between the two sides of a bevelled edge when it has more than one segment.

  M = (p₀ + p₁) ÷ 2; Q = M + (p_c − M)·2·profile; B(t) = (1 − t)²·p₀ + 2t(1 − t)·Q + t²·p₁, t = k ÷ segments

  Inset, bevel and moves along normals mesh/ops.js

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• Shrink or fatten

  Moves each listed vertex along its normal by d.

  p′ = p + N(v)·d

  Inset, bevel and moves along normals mesh/ops.js

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