`Rectangle`#

To define a rectangle, as for parallelograms, give 3 vertices (defining two orthogonal segments)

Figure made with TikZ

There is a parameter for each of them: _v1 , _v2 , and _v4 , as for Parallelogram . These parameters take 2D or 3D points.

For rectangles in plane z=0, where sides are parallel to x-axis and y-axis, the rectangle may be defined by its center (\(c\) in the figure) and its lengths or \(p_1\) (recalled origin in this case) and its lengths. To do it, use _center, _xlength and _ylength or _origin, _xlength and _ylength keys. _origin and _center keys take 2D or 3D points and _xlength and _ylength keys take one single positive value.

There is another possibility : defining the rectangle by its bounds : _xmin , _xmax , _ymin and _ymax keys taking one single value.

_nnodes can take one single value or an explicit list of 2 or 4 values (or a Numbers object) and _hsteps can take one real value or an explicit list of 4 real values (or a Reals object). If required, give the names of main domain and side domains as explained in Geometry definition:

Rectangle r1(_v1=Point(0.,0.), _v2=Point(2.,0.), _v4=Point(0.,1.) , _nnodes={20, 10},
             _domain_name="Omega", _side_names={"Gamma1", "Gamma2", "Gamma1", "Gamma2"});
Rectangle r2(_center=Point(1.,0.5), _xlength=2., _ylength=1. , _nnodes={20, 10},
             _domain_name="Omega", _side_names={"Gamma1", "Gamma2", "Gamma1", "Gamma2"});
Rectangle r3(_origin=Point(0.,0.), _xlength=2., _ylength=1. , _nnodes={20, 10},
             _domain_name="Omega" , _side_names={"Gamma1", "Gamma2", "Gamma1", "Gamma2"});
Rectangle r4(_xmin=0., _xmax=2., _ymin=0., _ymax=1., _nnodes={20, 10},
             _domain_name="Omega", _side_names={"Gamma1", "Gamma2", "Gamma1", "Gamma2"});

This is 4 definitions of the same Rectangle object.

Let’s summarize information about geometrical keys on rectangles:

key(s)	authorized types	examples
`_center` , `_origin`	`Point`	`_origin = Point(0.,0.), _center = Point(1.,0.5)`
`_v1`, `_v2`, `_v4`	`Point`	`_v1 = Point (0.,0.), _v2 = Point(2.,0.), _v4 = Point (0.,1.)`
`_xlength` , `_ylength`	single unsigned integer or real positive value	`_xlength =2, _ylength =1`
`_xmin`, `_xmax`, `_ymin`, `_ymax`	single integer or real value	`_xmin =0, _xmax =2, _ymin =0, _ymax=1`

Hint

Parametrization of the rectangle \((v_1,v_2,v_4)\) is :

\[(u,v)\in[0,1]\times [0,1]\longmapsto v_1+u(v_2-v_1)+v(v_4-v_1) \quad\mathrm{linear}.\]

Rectangle#

`Rectangle`#