Calculations

Calculations

• Values of a range

  The values a, a + s, a + 2s … up to b.

  x_k = a + k × s

  CraftScript array arithmetic cs-tensor.js

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• Sum, product, mean, maximum and minimum

  Reduces an array along one dimension.

  mean = (Σ xₖ) ÷ n

  CraftScript array arithmetic cs-tensor.js

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• Triangle normal

  The unit normal of one triangle and twice its area.

  n = (b − a) × (c − a); unit normal = n ÷ |n|

  Planar faces of a mesh faces.js

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• Coplanar triangle groups

  Joins triangles that share an edge and lie in one plane.

  n₁ · n₂ > 1 − 1e-5 and |d₁ − d₂| < 1e-5 × max(1, max |coordinate|)

  Planar faces of a mesh faces.js

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• Groups from the kernel's faces

  Uses the kernel's own face of each triangle as the group.

  Planar faces of a mesh faces.js

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• Face outline and area in its plane

  The outline loops of a planar face in two dimensions and the face's area.

  area = ½ Σ (x_j y_i − x_i y_j)

  Planar faces of a mesh faces.js

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• Dropping collinear outline vertices

  Removes outline vertices that lie on a straight run.

  |u × v| ÷ (|u| |v|) < 1e-6 and u · v > 0

  Planar faces of a mesh faces.js

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• One subdivision step

  New face, edge and vertex points of the cage refined once.

  smooth vertex = Q ÷ n + 2R ÷ n + (n − 3) v ÷ n; crease vertex = (a + 6v + b) ÷ 8; edge = mean of its ends and the two face points

  Freeform bodies: subdivision surface and cage edits form.js

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• Limit position of a vertex

  Where a vertex ends up on the smooth surface.

  smooth: (n² v + 4 Σ e + Σ f) ÷ (n (n + 5)); crease: (a + 4v + b) ÷ 6; corner: v

  Freeform bodies: subdivision surface and cage edits form.js

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• Limit surface normal at a vertex

  The unit normal of the smooth surface at an interior vertex.

  A = 1 + cos(2π ÷ n) + cos(π ÷ n) √(2 (9 + cos(2π ÷ n))); normal = t₁ × t₂ ÷ |t₁ × t₂|

  Freeform bodies: subdivision surface and cage edits form.js

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• Bézier patch of one quad

  The 16 control points of the bicubic patch for one face of the refined cage.

  b = M G Mᵀ with rows of M: (1, 4, 1, 0) ÷ 6, (0, 4, 2, 0) ÷ 6, (0, 2, 4, 0) ÷ 6, (0, 1, 4, 1) ÷ 6; across a sharp edge: 2 × edge − inner

  Freeform bodies: subdivision surface and cage edits form.js

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• Point on a patch

  Evaluates a bicubic Bézier patch at (u, v).

  S(u, v) = Σᵢ Σⱼ Bᵢ(u) Bⱼ(v) b_ij, B = ((1 − u)³, 3u(1 − u)², 3u²(1 − u), u³)

  Freeform bodies: subdivision surface and cage edits form.js

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• Normal of a patch

  The unit normal of a patch at (u, v).

  n = ∂S/∂u × ∂S/∂v, normalized

  Freeform bodies: subdivision surface and cage edits form.js

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• Refinement level for conversion

  How many times the cage is refined before it is turned into patches.

  level = max(1 if any face is not a quad, ⌈largest semi-sharp weight − 1e-12⌉), at most 4

  Freeform bodies: subdivision surface and cage edits form.js

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• How well the patches meet

  The largest gap and the largest angle between normals across shared patch edges.

  gap = max |p₀ − p₁|; angle = acos(clamp(n₀ · n₁, −1, 1)) × 180 ÷ π

  Freeform bodies: subdivision surface and cage edits form.js

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• Fitting a cage to a target surface

  Moves cage vertices until their limit points lie on a sphere or torus.

  p ← p + (proj(L) − L), L the vertex's limit point

  Freeform bodies: subdivision surface and cage edits form.js

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• Torus cage

  The vertices of a torus-shaped starting cage.

  ((R + r cos b) cos a, r sin b, (R + r cos b) sin a), a = 2π i ÷ n, b = 2π j ÷ m

  Freeform bodies: subdivision surface and cage edits form.js

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• Mirror partners

  Finds each vertex's partner across the symmetry plane.

  q = p mirrored; partner when |v − q| ≤ 1e-6 × size

  Freeform bodies: subdivision surface and cage edits form.js

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• Rebuilding the mirrored half

  Keeps one side of the symmetry plane and mirrors it.

  mirror: x' = 2 × at − x

  Freeform bodies: subdivision surface and cage edits form.js

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• Moving, rotating and scaling vertices

  Applies a 4 × 4 transform to the selected vertices, with an optional soft falloff and mirrored motion.

  p' = p + w (M p − p); soft weight w = (1 − (d ÷ R)²)²

  Freeform bodies: subdivision surface and cage edits form.js

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• Extruding faces

  Moves a region of faces along its averaged normals and adds side walls.

  new vertex = v + dist × (Σ face normals at v) ÷ |Σ face normals at v|

  Freeform bodies: subdivision surface and cage edits form.js

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• Face normal

  The area-weighted normal of a cage face with any number of corners.

  n_x = Σ (p_y − q_y)(p_z + q_z), and cyclic

  Freeform bodies: subdivision surface and cage edits form.js

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• Flattening vertices onto a plane

  Projects the selected vertices onto a given plane or their best-fit plane.

  p' = p − n ((p − o) · n)

  Freeform bodies: subdivision surface and cage edits form.js

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• Best-fit plane normal

  The direction in which a set of centered points varies least.

  A = Σ p pᵀ; normal = eigenvector of the smallest eigenvalue

  Freeform bodies: subdivision surface and cage edits form.js

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• Adding a face through points

  Adds a face, reusing existing vertices that are close to the given points.

  reuse a vertex when |q − p| ≤ 1e-6

  Freeform bodies: subdivision surface and cage edits form.js

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• Bridging two loops

  Joins two loops of equal size with rows of quads.

  offset = argmin Σ |aᵢ − b₍offset − i₎|²; row vertex = a + (b − a) × r ÷ segments

  Freeform bodies: subdivision surface and cage edits form.js

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• Tolerance and roughness text

  The text of a tolerance value or an Ra value on the drawing.

  text = value rounded to 4 decimals

  Drawing annotations: tolerance frames, datums and surface finish gdt.js

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• Circle and arc points

  The points of a circle or arc in a symbol.

  (c_x + r cos t, c_y + r sin t), t = a₀ + sweep × i ÷ n

  Drawing annotations: tolerance frames, datums and surface finish gdt.js

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• Annotation layout

  The frame, compartments, leader, datum triangle and surface texture legs of one annotation.

  frame height = 2h; datum triangle height = side × √3 ÷ 2; texture legs at 60°: offset = leg ÷ tan(π ÷ 3)

  Drawing annotations: tolerance frames, datums and surface finish gdt.js

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• Point on the handle axis nearest the pointer ray

  How far along the handle's axis the pointer is, read from the pointer's ray.

  t = (b·e − c·d) ÷ (a·c − b²)

  Depth and angle set by dragging a handle handle-math.js

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• Dragged extrude depth

  The depth stored on the Extrude after a drag.

  depth = max(step, round((v₀ + scale × (t − t₀)) ÷ step) × step), step = max(0.01, 10^⌊log₁₀|v₀|⌋ ÷ 10)

  Depth and angle set by dragging a handle handle-math.js

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• Dragged revolve angle

  The angle stored on the Revolve after a drag.

  angle = atan2(q·w, q·u) in degrees, unwrapped toward the previous value, rounded to 1°, kept in 1…360

  Depth and angle set by dragging a handle handle-math.js

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• Sketch plane from a picked flat face

  The plane reference stored when a sketch is put on a picked flat face that has no stable name.

  o = each coordinate rounded to 9 decimals; n = each component rounded to 12 decimals

  Values stored from picks in the workspace partcraft.js

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• Mate plane from a picked face of an instance

  The plane stored in an assembly mate when a flat face of a placed part is picked.

  o rounded to 9 decimals, n to 12 decimals, when the face has no stable name

  Values stored from picks in the workspace partcraft.js

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• Anchor point on a drawing view

  The point in a view's own coordinates where a note, dimension or detail view is attached.

  u = u₀ + (x − x_view) ÷ scale, snapped to the nearest drawn line within 2.5 ÷ scale, then rounded to 4 decimals

  Values stored from picks in the workspace partcraft.js

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• Default section plane offset

  The offset of a new section view's cutting plane: the middle of the part's bounding box along the view direction.

  offset = dir · (min + max) ÷ 2, rounded to 4 decimals

  Values stored from picks in the workspace partcraft.js

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• Scale factors

  The three factors a scale feature multiplies by, from the feature or from its shrinkage allowance.

  Scale feature for printing print-features.js

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• Scale matrix about a fixed point

  The 4 × 4 matrix that scales a body about the center, minimum corner, base center or origin of its box.

  a = (min + max) ÷ 2 (center); translation = a × (1 − f)

  Scale feature for printing print-features.js

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• Scaled triangles

  Applies the scale to every corner of a body's triangles, with the box found from the triangles themselves.

  x′ = f_x·x + t_x; y′ = f_y·y + t_y; z′ = f_z·z + t_z

  Scale feature for printing print-features.js

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• Rotation about an axis

  Turns a vector about a unit axis by an angle; used for a plane at an angle to another.

  v′ = v·cos a + (u × v)·sin a + u·(u · v)·(1 − cos a)

  Reference planes, axes and points refgeom.js

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• Point where three planes meet

  The single point that lies on three planes.

  solve [nₐ; n_b; n_c] · p = (nₐ·oₐ, n_b·o_b, n_c·o_c)

  Reference planes, axes and points refgeom.js

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• Axis where two planes meet

  The line of intersection of two planes: its direction and a point on it.

  d = nₐ × n_b; solve nₐ·x = cₐ, n_b·x = c_b, d·x = d·(oₐ + o_b) ÷ 2

  Reference planes, axes and points refgeom.js

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• Offset plane and plane at an angle

  A plane parallel to another at a distance, and a plane turned about an axis by an angle in degrees.

  o′ = o + n·d; n′ = rotate(n ⊥ axis, axis, angle × π ÷ 180)

  Reference planes, axes and points refgeom.js

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• Plane through three points

  The plane through three points, with its center at their mean.

  n = (b − a) × (c − a); center = (a + b + c) ÷ 3

  Reference planes, axes and points refgeom.js

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• Mid plane between two planes

  The plane half way between two planes, or the one that bisects them when they are not parallel.

  n = (nₐ − n_b) ÷ |nₐ − n_b|; c = (oₐ·nₐ − o_b·n_b) ÷ |nₐ − n_b|; parallel and same facing: o = oₐ + n·((o_b − oₐ)·n) ÷ 2

  Reference planes, axes and points refgeom.js

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• Plane tangent to a cylinder

  The plane that touches a cylindrical face on the side of a chosen point.

  radial = w − d·(w · d); n = radial ÷ |radial|; o = foot on the axis + n·r

  Reference planes, axes and points refgeom.js

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• Plane through a line and a point

  The plane that contains a straight edge or axis and one more point.

  n = d × (p − o)

  Reference planes, axes and points refgeom.js

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• Where a line meets a plane or a cylinder

  The point where a straight edge or axis crosses a plane, a flat face or a cylindrical face.

  plane: t = ((o_s − o) · n) ÷ (d · n); cylinder: A·t² + B·t + C = 0, t = (−B ± √(B² − 4AC)) ÷ 2A

  Reference planes, axes and points refgeom.js

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• Point on a curve at a parameter

  The point of a line, circle, arc or spline at a given parameter.

  line: a + (b − a)·t; circle: c + r·(cos t, sin t); arc: c + r·(cos(a₀ + t), sin(a₀ + t))

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Parameter of the nearest point

  The parameter on a curve of the point nearest a given point.

  line: t = ((p − a) · d) ÷ (d · d); circle and arc: angle = atan2(p_y − c_y, p_x − c_x)

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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