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improved the definitions

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Bernhard 2023-03-26 12:20:30 +02:00
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docs/algebra_definition.rst
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Algebraic definition
========================
Objects and object arithmetic
----------------------------------
Objects and arithmetic
------------------------
Set definitions:
**Set definitions:**
:math:`P` is the set of all ``Part`` objects ``p`` with ``p._dim = 3``
:math:`C^3` is the set of all ``Part`` objects ``p`` with ``p._dim = 3``
:math:`S` is the set of all ``Sketch`` objects ``s`` with ``s._dim = 2``
:math:`C^2` is the set of all ``Sketch`` objects ``s`` with ``s._dim = 2``
:math:`C` is the set of all ``Curve`` objects ``c`` with ``c._dim = 3``
:math:`C^1` is the set of all ``Curve`` objects ``c`` with ``c._dim = 3``
Neutral elements:
**Neutral elements:**
:math:`p_0` := ``p0`` is the empty ``Part`` object ``p0`` with ``p0._dim = 3`` and ``p0.wrapped = None``
:math:`c^3_0` is the empty ``Part`` object ``p0 = Part()`` with ``p0._dim = 3`` and ``p0.wrapped = None``
:math:`s_0` := ``s0`` is the empty ``SKetch`` object ``s0`` with ``s0._dim = 2`` and ``s0.wrapped = None``
:math:`c^2_0` is the empty ``Sketch`` object ``s0 = Sketch()`` with ``s0._dim = 2`` and ``s0.wrapped = None``
:math:`c^1_0` is the empty ``Curve`` object ``c0 = Curve()`` with ``c0._dim = 1`` and ``c0.wrapped = None``
:math:`c_0` := ``c0`` is the empty ``Curve`` object ``c0`` with ``c0._dim = 1`` and ``c0.wrapped = None``
**Sets of predefined basic shapes:**
:math:`P_D := \lbrace` ``Part``, ``Box``, ``Cylinder``, ``Cone``, ``Sphere``, ``Torus``, ``Wedge``, ``Hole``, ``CounterBoreHole``, ``CounterSinkHole`` :math:`\rbrace`
:math:`B^3 := \lbrace` ``Part``, ``Box``, ``Cylinder``, ``Cone``, ``Sphere``, ``Torus``, ``Wedge``, ``Hole``, ``CounterBoreHole``, ``CounterSinkHole`` :math:`\rbrace`
:math:`S_D := \lbrace` ``Sketch``, ``Rectangle``, ``Circle``, ``Ellipse``, ``Rectangle``, ``Polygon``, ``RegularPolygon``, ``Text``, ``Trapezoid``, ``SlotArc``, ``SlotCenterPoint``, ``SlotCenterToCenter``, ``SlotOverall`` :math:`\rbrace`
:math:`B^2 := \lbrace` ``Sketch``, ``Rectangle``, ``Circle``, ``Ellipse``, ``Rectangle``, ``Polygon``, ``RegularPolygon``, ``Text``, ``Trapezoid``, ``SlotArc``, ``SlotCenterPoint``, ``SlotCenterToCenter``, ``SlotOverall`` :math:`\rbrace`
:math:`C_D := \lbrace` ``Curve``, ``Bezier``, ``PolarLine``, ``Polyline``, ``Spline``, ``Helix``, ``CenterArc``, ``EllipticalCenterArc``, ``RadiusArc``, ``SagittaArc``, ``TangentArc``, ``ThreePointArc``, ``JernArc`` :math:`\rbrace`
:math:`B^1 := \lbrace` ``Curve``, ``Bezier``, ``PolarLine``, ``Polyline``, ``Spline``, ``Helix``, ``CenterArc``, ``EllipticalCenterArc``, ``RadiusArc``, ``SagittaArc``, ``TangentArc``, ``ThreePointArc``, ``JernArc`` :math:`\rbrace`
with :math:`B^3 \subset C^3, B^2 \subset C^2` and :math:`B^1 \subset C^1`
with :math:`P_D \subset P, S_D \subset S` and :math:`C_D \subset C`
**Operations:**
:math:`+: C^n \times C^n \rightarrow C^n` with :math:`(a,b) \mapsto a + b`, :math:`\;` for :math:`n=1,2,3`
:math:`+: P \times P \rightarrow P` with :math:`(a,b) \mapsto a + b`
:math:`\; a + b :=` ``a.fuse(b)`` for each operation
:math:`+: S \times S \rightarrow S` with :math:`(a,b) \mapsto a + b`
:math:`-: C^n \rightarrow C^n` with :math:`a \mapsto -a`, :math:`\;` for :math:`n=1,2,3`
:math:`+: C \times C \rightarrow C` with :math:`(a,b) \mapsto a + b`
and :math:`a + b :=` ``a.fuse(b)`` for each operation
:math:`\; b + (-a) :=` ``b.cut(a)`` for each operation (implicit definition)
:math:`-: P \rightarrow P` with :math:`a \mapsto -a`
:math:`\&: C^n \times C^n \rightarrow C^n` with :math:`(a,b) \mapsto a \; \& \; b`, :math:`\;` for :math:`n=2,3`
:math:`-: S \rightarrow S` with :math:`a \mapsto -a`
:math:`\; a \; \& \; b :=` ``a.intersect(b)`` for each operation
:math:`-: C \rightarrow C` with :math:`a \mapsto -a`
* :math:`\&` is not defined for :math:`n=1` in build123d
* The following relationship holds: :math:`a \; \& \; b = (a + b) + -(a + (-b)) + -(b + (-a))`
and :math:`b + (-a) :=` ``b.cut(a)`` for each operation (implicit definition)
:math:`\&: P \times P \rightarrow P` with :math:`(a,b) \mapsto a \& b`
:math:`\&: S \times S \rightarrow S` with :math:`(a,b) \mapsto a \& b`
:math:`\&: C \times C \rightarrow C` with :math:`(a,b) \mapsto a \& b`
and :math:`a \; \& \; b :=` ``a.intersect(b)`` for each operation
Note: :math:`a \; \& \; b = (a + b) + -(a + (-b)) + -(b + (-a))`
**Abelian groups**
:math:`( P, p_0, +, -)`, :math:`( S, s_0, +, -)`, :math:`( C, c_0, +, -)` are abelian groups
Notes:
:math:`( C^n, \; c^n_0, \; +, \; -)` :math:`\;` are abelian groups for :math:`n=1,2,3`.
* The implementation ``a - b = a.cut(b)`` needs to be read as :math:`a + (-b)` since the group does not have a binary ``-`` operation. As such, :math:`a - (b - c) = a + -(b + -c)) \ne a - b + c`
* This definition also includes that neither ``-`` nor ``&`` are commutative.
Locations, planes and location arithmentic
---------------------------------------------
:math:`L := \lbrace` ``Location((x, y, z), (a, b, c))`` :math:`: x,y,z \in R \land a,b,c \in R\rbrace` with :math:`a,b,c` being angles in degrees
**Set definitions:**
:math:`Q := \lbrace` ``Plane(o, x, z)`` :math:`: 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.
:math:`L := \lbrace` ``Location((x, y, z), (a, b, c))`` :math:`: x,y,z \in R \land a,b,c \in R \rbrace\;`
with :math:`a,b,c` being angles in degrees.
:math:`*: L \times L \rightarrow L` (multiply two locations :math:`l_1, l_2 \in L`, i.e. ``l1 * l2``)
:math:`P := \lbrace` ``Plane(o, x, z)`` :math:`: 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.
:math:`*: Q \times L \rightarrow Q` (move plane :math:`p \in Q` to location :math:`l \in L`, i.e. ``Plane(p.to_location() * l)``)
Neutral element: :math:`\; l_0 \in L`: ``Location()``
Neutral element: :math:`l_0 \in L`: ``Location()``
**Operations:**
Inverse element: :math:`l^{-1} \in L`: ``l.inverse()``
:math:`*: L \times L \rightarrow L` with :math:`(l_1,l_2) \mapsto l_1 * l_2`
:math:`\; l_1 * l_2 :=` ``l1 * l2`` (multiply two locations)
:math:`*: P \times L \rightarrow P` with :math:`(p,l) \mapsto p * l`
:math:`\; p * l :=` ``Plane(p.to_location() * l)`` (move plane :math:`p \in P` to location :math:`l \in L`)
Inverse element: :math:`\; l^{-1} \in L`: ``l.inverse()``
**Placing objects onto planes**
:math:`*: Q \times P \rightarrow P` (locate an object :math:`p \in P` onto plane :math:`q \in Q`, i.e. ``p.moved(q.to_location())``)
:math:`*: Q \times S \rightarrow S` (locate an object :math:`s \in S` onto plane :math:`q \in Q`, i.e. ``s.moved(q.to_location())``)
:math:`*: Q \times C \rightarrow C` (locate an object :math:`c \in C` onto plane :math:`q \in Q`, i.e. ``c.moved(q.to_location())``)
:math:`*: P \times C^n \rightarrow C^n \;` with :math:`(p,c) \mapsto p * c`, :math:`\;` for :math:`n=1,2,3`
Locate an object :math:`c \in C^n` onto plane :math:`p \in P`, i.e. ``c.moved(p.to_location())``
**Placing objects at locations**
:math:`*: L \times P \rightarrow P` (locate an object :math:`p \in P` at location :math:`l \in L`, i.e. ``p.moved(l)``)
:math:`*: L \times C^n \rightarrow C^n \;` with :math:`(l,c) \mapsto l * c`, :math:`\;` for :math:`n=1,2,3`
:math:`*: L \times S \rightarrow S` (locate an object :math:`s \in S` at location :math:`l \in L`, i.e. ``s.moved(l)``)
:math:`*: L \times C \rightarrow C` (locate an object :math:`c \in C` at location :math:`l \in L`, i.e. ``c.moved(l)``)
Locate an object :math:`c \in C^n` at location :math:`l \in L`, i.e. ``c.moved(l)``