Parallelepiped

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

Bases: sknano.core.geometric_regions.Geometric3DRegion

Geometric3DRegion for a parallelepiped.

New in version 0.3.0.

Represents a parallelepiped with origin $(o_x, o_y, o_z)$ and direction vectors $\mathbf{u}=(u_x, u_y, u_z)$, $\mathbf{v}=(v_x, v_y, v_z)$, and $\mathbf{w}=(w_x, w_y, w_z)$.

Parameters:

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

Notes

Parallelepiped represents the bounded region $\left\{o+\lambda_1\mathbf{u}+\lambda_2\mathbf{v}+ \lambda_3\mathbf{w}\in R^3|0\le\lambda_i\le 1\right\}$, where $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ have to be linearly independent.

Calling Parallelepiped with no parameters is equivalent to Parallelepiped(o=[0, 0, 0], u=[1, 0, 0], v=[1, 1, 0], w=[0, 1, 1]).

Attributes

a	Alias for Parallelepiped.u.length.
abc	Alias for Parallelepiped.lengths.
b	Alias for Parallelepiped.v.length.
bounding_box	Bounding Cuboid.
c	Alias for Parallelepiped.w.length.
center	Alias for centroid.
centroid	Parallelepiped centroid, $(c_x, c_y, c_z)$.
fmtstr	Format string.
lengths	tuple of side lengths
measure	Alias for volume, which is the measure of a 3D geometric region.
ndim	Return the dimensions.
o	3D point coordinates $(o_x, o_y, o_z)$ of origin.
pmax	Point at maximum extent.
pmin	Point at minimum extent.
u	3D direction vector $\mathbf{u}=(u_x, u_y, u_z)$, with origin o
v	3D direction vector $\mathbf{v}=(v_x, v_y, v_z)$, with origin o
volume	Parallelepiped volume, $V$.
w	3D direction vector $\mathbf{w}=(w_x, w_y, w_z)$, with origin o

Methods

center_centroid()	Center centroid on origin.
contains(point)	Test region membership of point in Parallelepiped.
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 Parallelepiped constructor parameters.
translate(t[, fix_anchor_points])	Translate GeometricRegion points and vectors by Vector t.

Attributes Summary

a	Alias for Parallelepiped.u.length.
abc	Alias for Parallelepiped.lengths.
b	Alias for Parallelepiped.v.length.
c	Alias for Parallelepiped.w.length.
centroid	Parallelepiped centroid, $(c_x, c_y, c_z)$.
lengths	tuple of side lengths
o	3D point coordinates $(o_x, o_y, o_z)$ of origin.
u	3D direction vector $\mathbf{u}=(u_x, u_y, u_z)$, with origin o
v	3D direction vector $\mathbf{v}=(v_x, v_y, v_z)$, with origin o
volume	Parallelepiped volume, $V$.
w	3D direction vector $\mathbf{w}=(w_x, w_y, w_z)$, with origin o

Methods Summary

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

Attributes Documentation

a

Alias for Parallelepiped.u.length.

abc

Alias for Parallelepiped.lengths.

b

Alias for Parallelepiped.v.length.

c

Alias for Parallelepiped.w.length.

centroid

Parallelepiped centroid, $(c_x, c_y, c_z)$.

Computed as the 3D point $(c_x, c_y, c_z)$ with coordinates:

$$c_x = o_x + \frac{u_x + v_x + w_x}{2}$$ $$c_y = o_y + \frac{u_y + v_y + w_y}{2}$$ $$c_z = o_z + \frac{u_z + v_z + w_z}{2}$$

where $(o_x, o_y, o_z)$, $(u_x, u_y, u_z)$, $(v_x, v_y, v_z)$, and $(w_x, w_y, w_z)$ are the $(x, y, z)$ coordinates of the origin $o$ and $(x, y, z)$ components of the direction vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$, respectively.

Returns:3D Point of centroid. Return type:Point

lengths

tuple of side lengths

o

3D point coordinates $(o_x, o_y, o_z)$ of origin.

Returns:3D Point coordinates $(o_x, o_y, o_z)$ of origin. Return type:Point

u

3D direction vector $\mathbf{u}=(u_x, u_y, u_z)$, with origin o

v

3D direction vector $\mathbf{v}=(v_x, v_y, v_z)$, with origin o

volume

Parallelepiped volume, $V$.

Computed as:

$$V = |\mathbf{u}\cdot\mathbf{v}\times\mathbf{w}|$$

w

3D direction vector $\mathbf{w}=(w_x, w_y, w_z)$, with origin o

Methods Documentation

contains(point)

Test region membership of point in Parallelepiped.

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

Notes

A point $(p_x, p_y, p_z)$ is within the bounded region of a parallelepiped with origin $(o_x, o_y, o_z)$ and direction vectors $\mathbf{u}=(u_x, u_y, u_z)$, $\mathbf{v}=(v_x, v_y, v_z)$, and $\mathbf{w}=(w_x, w_y, w_z)$ if the following is true:

$$0\le\frac{ v_z (w_x (p_y - o_y) + w_y (o_x - p_x)) + w_z (v_x (o_y - p_y) + v_y (p_x - o_x)) + o_z (v_y w_x - v_x w_y) + p_z (v_x w_y - v_y w_x)}{ u_z (v_x w_y - v_y w_x) + u_y (v_z w_x - v_x w_z) + u_x (v_y w_z - v_z w_y)}\le 1 \land$$ $$0\le\frac{ u_z (w_x (p_y - o_y) + w_y (o_x - p_x)) + w_z (u_x (o_y - p_y) + u_y (p_x - o_x)) + o_z (u_y w_x - u_x w_y) + p_z (u_x w_y - u_y w_x)}{ u_z (v_y w_x - v_x w_y) + u_y (v_x w_z - v_z w_x) + u_x (v_z w_y - v_y w_z)}\le 1 \land$$ $$0\le\frac{ u_z (v_x (p_y - o_y) + v_y (o_x - p_x)) + v_z (u_x (o_y - p_y) + u_y (p_x - o_x)) + o_z (u_y v_x - u_x v_y) + p_z (u_x v_y - u_y v_x)}{ u_z (v_x w_y - v_y w_x) + u_y (v_z w_x - v_x w_z) + u_x (v_y w_z - v_z w_y)}\le 1$$

todict()

Returns a dict of the Parallelepiped constructor parameters.