Parallelogram

class sknano.core.geometric_regions.Parallelogram(o=None, u=None, v=None)

Bases: sknano.core.geometric_regions.Geometric2DRegion

Geometric2DRegion for a parallelogram.

New in version 0.3.0.

Represents a parallelogram with origin \(o=(o_x, o_y)\) and direction vectors \(\mathbf{u}=(u_x, u_y)\) and \(\mathbf{v}=(v_x, v_y)\).

Parameters:

• o (array_like, optional) – Parallelogram origin. If None, it defaults to o=[0, 0].
• v (u,) – Parallelogram direction vectors stemming from origin o. If None, then the default values are u=[1, 0] and v=[1, 1].

Notes

Parallelogram represents the bounded region \(\left \{o+\lambda_1\mathbf{u}+\lambda_2\mathbf{v}\in R^2 |0\le\lambda_i\le 1\right\}\), where \(\mathbf{u}\) and \(\mathbf{v}\) have to be linearly independent.

Calling Paralleogram with no parameters is equivalent to Parallelogram(o=[0, 0], u=[1, 0], v=[1, 1])

Attributes

area	Paralleogram area, \(A=|\mathbf{u}\times\mathbf{v}|\).
bounding_box	Bounding Cuboid.
center	Alias for centroid.
centroid	Paralleogram centroid, \((c_x, c_y)\).
fmtstr	Format string.
measure	Alias for area, which is the measure of a 2D geometric region.
ndim	Return the dimensions.
o	2D point coordinates \((o_x, o_y)\) of origin.
pmax	Point at maximum extent.
pmin	Point at minimum extent.
u	2D direction vector \(\mathbf{u}=(u_x, u_y)\), with origin o
v	2D direction vector \(\mathbf{v}=(v_x, v_y)\), with origin o

Methods

center_centroid()	Center centroid on origin.
contains(point)	Test region membership of point in Parallelogram.
get_points()	Return list of points from GeometricRegion.points and GeometricRegion.vectors
rotate(**kwargs)	Rotate GeometricRegion points and vectors.
todict()	Returns a dict of the Paralleogram constructor parameters.
translate(t[, fix_anchor_points])	Translate GeometricRegion points and vectors by Vector t.

Attributes Summary

area	Paralleogram area, \(A=|\mathbf{u}\times\mathbf{v}|\).
centroid	Paralleogram centroid, \((c_x, c_y)\).
o	2D point coordinates \((o_x, o_y)\) of origin.
u	2D direction vector \(\mathbf{u}=(u_x, u_y)\), with origin o
v	2D direction vector \(\mathbf{v}=(v_x, v_y)\), with origin o

Methods Summary

contains(point)	Test region membership of point in Parallelogram.
todict()	Returns a dict of the Paralleogram constructor parameters.

Attributes Documentation

area

Paralleogram area, \(A=|\mathbf{u}\times\mathbf{v}|\).

Computed as:

\[A = |\mathbf{u}\times\mathbf{v}|\]

centroid

Paralleogram centroid, \((c_x, c_y)\).

Computed as the 2D point \((c_x, c_y)\) with coordinates:

\[c_x = o_x + \frac{u_x + v_x}{2}\]\[c_y = o_y + \frac{u_y + v_y}{2}\]

where \((o_x, o_y)\), \((u_x, u_y)\), and \((v_x, v_y)\) are the \((x, y)\) coordinates of the origin \(o\) and \((x, y)\) components of the direction vectors \(\mathbf{u}\) and \(\mathbf{v}\), respectively.

Returns:2D Point of centroid. Return type:Point

o

2D point coordinates \((o_x, o_y)\) of origin.

Returns:2D Point coordinates \((o_x, o_y)\) of origin. Return type:Point

u

2D direction vector \(\mathbf{u}=(u_x, u_y)\), with origin o

v

2D direction vector \(\mathbf{v}=(v_x, v_y)\), with origin o

Methods Documentation

contains(point)

Test region membership of point in Parallelogram.

Parameters:point (array_like) – Returns:True if point is within Paralleogram, False otherwise Return type:bool

Notes

A point \((p_x, p_y)\) is within the bounded region of a parallelogram with origin \((o_x, o_y)\) and direction vectors \(\mathbf{u}=(u_x, u_y)\) and \(\mathbf{v}=(v_x, v_y)\) if the following is true:

\[0\le\frac{(p_y - o_y) v_x + (o_x - p_x) v_y}{u_y v_x - u_x v_y} \le 1 \land 0\le\frac{(p_y - o_y) u_x + (o_x - p_x) u_y}{u_x v_y - u_y v_x} \le 1\]

todict()

Returns a dict of the Paralleogram constructor parameters.