Algebraic definition

Objects and arithmetic

Set definitions:

\(C^3\) is the set of all Part objects p with p._dim = 3

\(C^2\) is the set of all Sketch objects s with s._dim = 2

\(C^1\) is the set of all Curve objects c with c._dim = 1

Neutral elements:

\(c^3_0\) is the empty Part object p0 = Part() with p0._dim = 3 and p0.wrapped = None

\(c^2_0\) is the empty Sketch object s0 = Sketch() with s0._dim = 2 and s0.wrapped = None

\(c^1_0\) is the empty Curve object c0 = Curve() with c0._dim = 1 and c0.wrapped = None

Sets of predefined basic shapes:

\(B^3 := \lbrace\) Part, Box, Cylinder, Cone, Sphere, Torus, Wedge, Hole, CounterBoreHole, CounterSinkHole \(\rbrace\)

\(B^2 := \lbrace\) Sketch, Rectangle, Circle, Ellipse, Rectangle, Polygon, RegularPolygon, Text, Trapezoid, SlotArc, SlotCenterPoint, SlotCenterToCenter, SlotOverall \(\rbrace\)

\(B^1 := \lbrace\) Curve, Bezier, FilletPolyline, PolarLine, Polyline, Spline, Helix, CenterArc, EllipticalCenterArc, ParabolicCenterArc, HyperbolicCenterArc, RadiusArc, SagittaArc, TangentArc, ThreePointArc, JernArc \(\rbrace\)

with \(B^3 \subset C^3, B^2 \subset C^2\) and \(B^1 \subset C^1\)

Operations:

\(+: C^n \times C^n \rightarrow C^n\) with \((a,b) \mapsto a + b\), \(\;\) for \(n=1,2,3\)

\(\; a + b :=\) a.fuse(b) for each operation

\(-: C^n \rightarrow C^n\) with \(a \mapsto -a\), \(\;\) for \(n=1,2,3\)

\(\; b + (-a) :=\) b.cut(a) for each operation (implicit definition)

\(\&: C^n \times C^n \rightarrow C^n\) with \((a,b) \mapsto a \; \& \; b\), \(\;\) for \(n=2,3\)

\(\; a \; \& \; b :=\) a.intersect(b) for each operation

• \(\&\) is not defined for \(n=1\) in build123d

• The following relationship holds: \(a \; \& \; b = (a + b) + -(a + (-b)) + -(b + (-a))\)

Abelian groups

\(( C^n, \; c^n_0, \; +, \; -)\) \(\;\) are abelian groups for \(n=1,2,3\).

• The implementation a - b = a.cut(b) needs to be read as \(a + (-b)\) since the group does not have a binary - operation. As such, \(a - (b - c) = a + -(b + -c)) \ne a - b + c\)

• This definition also includes that neither - nor & are commutative.

Locations, planes and location arithmetic

Set definitions:

\(L := \lbrace\) Location((x, y, z), (a, b, c)) \(: x,y,z \in R \land a,b,c \in R \rbrace\;\)

with \(a,b,c\) being angles in degrees.

\(P := \lbrace\) Plane(o, x, z) \(: o,x,z ∈ R^3 \land \|x\| = \|z\| = 1\rbrace\)

with o being the origin and x, z the x- and z-direction of the plane.

Neutral element: \(\; l_0 \in L\): Location()

Operations:

\(*: L \times L \rightarrow L\) with \((l_1,l_2) \mapsto l_1 * l_2\)

\(\; l_1 * l_2 :=\) l1 * l2 (multiply two locations)

\(*: P \times L \rightarrow P\) with \((p,l) \mapsto p * l\)

\(\; p * l :=\) Plane(p.location * l) (move plane \(p \in P\) to location \(l \in L\))

Inverse element: \(\; l^{-1} \in L\): l.inverse()

Placing objects onto planes

\(*: P \times C^n \rightarrow C^n \;\) with \((p,c) \mapsto p * c\), \(\;\) for \(n=1,2,3\)

Locate an object \(c \in C^n\) onto plane \(p \in P\), i.e. c.moved(p.location)

Placing objects at locations

\(*: L \times C^n \rightarrow C^n \;\) with \((l,c) \mapsto l * c\), \(\;\) for \(n=1,2,3\)

Locate an object \(c \in C^n\) at location \(l \in L\), i.e. c.moved(l)