423 lines
10 KiB
Python
423 lines
10 KiB
Python
|
"""Parabolic geometrical entity.
|
||
|
|
|
||
|
|
Contains
|
||
|
|
* Parabola
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
from sympy.core import S
|
||
|
|
from sympy.core.sorting import ordered
|
||
|
|
from sympy.core.symbol import _symbol, symbols
|
||
|
|
from sympy.geometry.entity import GeometryEntity, GeometrySet
|
||
|
|
from sympy.geometry.point import Point, Point2D
|
||
|
|
from sympy.geometry.line import Line, Line2D, Ray2D, Segment2D, LinearEntity3D
|
||
|
|
from sympy.geometry.ellipse import Ellipse
|
||
|
|
from sympy.functions import sign
|
||
|
|
from sympy.simplify import simplify
|
||
|
|
from sympy.solvers.solvers import solve
|
||
|
|
|
||
|
|
|
||
|
|
class Parabola(GeometrySet):
|
||
|
|
"""A parabolic GeometryEntity.
|
||
|
|
|
||
|
|
A parabola is declared with a point, that is called 'focus', and
|
||
|
|
a line, that is called 'directrix'.
|
||
|
|
Only vertical or horizontal parabolas are currently supported.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
focus : Point
|
||
|
|
Default value is Point(0, 0)
|
||
|
|
directrix : Line
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
==========
|
||
|
|
|
||
|
|
focus
|
||
|
|
directrix
|
||
|
|
axis of symmetry
|
||
|
|
focal length
|
||
|
|
p parameter
|
||
|
|
vertex
|
||
|
|
eccentricity
|
||
|
|
|
||
|
|
Raises
|
||
|
|
======
|
||
|
|
ValueError
|
||
|
|
When `focus` is not a two dimensional point.
|
||
|
|
When `focus` is a point of directrix.
|
||
|
|
NotImplementedError
|
||
|
|
When `directrix` is neither horizontal nor vertical.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8)))
|
||
|
|
>>> p1.focus
|
||
|
|
Point2D(0, 0)
|
||
|
|
>>> p1.directrix
|
||
|
|
Line2D(Point2D(5, 8), Point2D(7, 8))
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __new__(cls, focus=None, directrix=None, **kwargs):
|
||
|
|
|
||
|
|
if focus:
|
||
|
|
focus = Point(focus, dim=2)
|
||
|
|
else:
|
||
|
|
focus = Point(0, 0)
|
||
|
|
|
||
|
|
directrix = Line(directrix)
|
||
|
|
|
||
|
|
if directrix.contains(focus):
|
||
|
|
raise ValueError('The focus must not be a point of directrix')
|
||
|
|
|
||
|
|
return GeometryEntity.__new__(cls, focus, directrix, **kwargs)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def ambient_dimension(self):
|
||
|
|
"""Returns the ambient dimension of parabola.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
ambient_dimension : integer
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> f1 = Point(0, 0)
|
||
|
|
>>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.ambient_dimension
|
||
|
|
2
|
||
|
|
|
||
|
|
"""
|
||
|
|
return 2
|
||
|
|
|
||
|
|
@property
|
||
|
|
def axis_of_symmetry(self):
|
||
|
|
"""Return the axis of symmetry of the parabola: a line
|
||
|
|
perpendicular to the directrix passing through the focus.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
axis_of_symmetry : Line
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.geometry.line.Line
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.axis_of_symmetry
|
||
|
|
Line2D(Point2D(0, 0), Point2D(0, 1))
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.directrix.perpendicular_line(self.focus)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def directrix(self):
|
||
|
|
"""The directrix of the parabola.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
directrix : Line
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.geometry.line.Line
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> l1 = Line(Point(5, 8), Point(7, 8))
|
||
|
|
>>> p1 = Parabola(Point(0, 0), l1)
|
||
|
|
>>> p1.directrix
|
||
|
|
Line2D(Point2D(5, 8), Point2D(7, 8))
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.args[1]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def eccentricity(self):
|
||
|
|
"""The eccentricity of the parabola.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
eccentricity : number
|
||
|
|
|
||
|
|
A parabola may also be characterized as a conic section with an
|
||
|
|
eccentricity of 1. As a consequence of this, all parabolas are
|
||
|
|
similar, meaning that while they can be different sizes,
|
||
|
|
they are all the same shape.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
https://en.wikipedia.org/wiki/Parabola
|
||
|
|
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.eccentricity
|
||
|
|
1
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The eccentricity for every Parabola is 1 by definition.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return S.One
|
||
|
|
|
||
|
|
def equation(self, x='x', y='y'):
|
||
|
|
"""The equation of the parabola.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
x : str, optional
|
||
|
|
Label for the x-axis. Default value is 'x'.
|
||
|
|
y : str, optional
|
||
|
|
Label for the y-axis. Default value is 'y'.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
equation : SymPy expression
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.equation()
|
||
|
|
-x**2 - 16*y + 64
|
||
|
|
>>> p1.equation('f')
|
||
|
|
-f**2 - 16*y + 64
|
||
|
|
>>> p1.equation(y='z')
|
||
|
|
-x**2 - 16*z + 64
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = _symbol(x, real=True)
|
||
|
|
y = _symbol(y, real=True)
|
||
|
|
|
||
|
|
m = self.directrix.slope
|
||
|
|
if m is S.Infinity:
|
||
|
|
t1 = 4 * (self.p_parameter) * (x - self.vertex.x)
|
||
|
|
t2 = (y - self.vertex.y)**2
|
||
|
|
elif m == 0:
|
||
|
|
t1 = 4 * (self.p_parameter) * (y - self.vertex.y)
|
||
|
|
t2 = (x - self.vertex.x)**2
|
||
|
|
else:
|
||
|
|
a, b = self.focus
|
||
|
|
c, d = self.directrix.coefficients[:2]
|
||
|
|
t1 = (x - a)**2 + (y - b)**2
|
||
|
|
t2 = self.directrix.equation(x, y)**2/(c**2 + d**2)
|
||
|
|
return t1 - t2
|
||
|
|
|
||
|
|
@property
|
||
|
|
def focal_length(self):
|
||
|
|
"""The focal length of the parabola.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
focal_lenght : number or symbolic expression
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
The distance between the vertex and the focus
|
||
|
|
(or the vertex and directrix), measured along the axis
|
||
|
|
of symmetry, is the "focal length".
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
https://en.wikipedia.org/wiki/Parabola
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.focal_length
|
||
|
|
4
|
||
|
|
|
||
|
|
"""
|
||
|
|
distance = self.directrix.distance(self.focus)
|
||
|
|
focal_length = distance/2
|
||
|
|
|
||
|
|
return focal_length
|
||
|
|
|
||
|
|
@property
|
||
|
|
def focus(self):
|
||
|
|
"""The focus of the parabola.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
focus : Point
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.geometry.point.Point
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> f1 = Point(0, 0)
|
||
|
|
>>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.focus
|
||
|
|
Point2D(0, 0)
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.args[0]
|
||
|
|
|
||
|
|
def intersection(self, o):
|
||
|
|
"""The intersection of the parabola and another geometrical entity `o`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
o : GeometryEntity, LinearEntity
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
intersection : list of GeometryEntity objects
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Ellipse, Line, Segment
|
||
|
|
>>> p1 = Point(0,0)
|
||
|
|
>>> l1 = Line(Point(1, -2), Point(-1,-2))
|
||
|
|
>>> parabola1 = Parabola(p1, l1)
|
||
|
|
>>> parabola1.intersection(Ellipse(Point(0, 0), 2, 5))
|
||
|
|
[Point2D(-2, 0), Point2D(2, 0)]
|
||
|
|
>>> parabola1.intersection(Line(Point(-7, 3), Point(12, 3)))
|
||
|
|
[Point2D(-4, 3), Point2D(4, 3)]
|
||
|
|
>>> parabola1.intersection(Segment((-12, -65), (14, -68)))
|
||
|
|
[]
|
||
|
|
|
||
|
|
"""
|
||
|
|
x, y = symbols('x y', real=True)
|
||
|
|
parabola_eq = self.equation()
|
||
|
|
if isinstance(o, Parabola):
|
||
|
|
if o in self:
|
||
|
|
return [o]
|
||
|
|
else:
|
||
|
|
return list(ordered([Point(i) for i in solve(
|
||
|
|
[parabola_eq, o.equation()], [x, y], set=True)[1]]))
|
||
|
|
elif isinstance(o, Point2D):
|
||
|
|
if simplify(parabola_eq.subs([(x, o._args[0]), (y, o._args[1])])) == 0:
|
||
|
|
return [o]
|
||
|
|
else:
|
||
|
|
return []
|
||
|
|
elif isinstance(o, (Segment2D, Ray2D)):
|
||
|
|
result = solve([parabola_eq,
|
||
|
|
Line2D(o.points[0], o.points[1]).equation()],
|
||
|
|
[x, y], set=True)[1]
|
||
|
|
return list(ordered([Point2D(i) for i in result if i in o]))
|
||
|
|
elif isinstance(o, (Line2D, Ellipse)):
|
||
|
|
return list(ordered([Point2D(i) for i in solve(
|
||
|
|
[parabola_eq, o.equation()], [x, y], set=True)[1]]))
|
||
|
|
elif isinstance(o, LinearEntity3D):
|
||
|
|
raise TypeError('Entity must be two dimensional, not three dimensional')
|
||
|
|
else:
|
||
|
|
raise TypeError('Wrong type of argument were put')
|
||
|
|
|
||
|
|
@property
|
||
|
|
def p_parameter(self):
|
||
|
|
"""P is a parameter of parabola.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
p : number or symbolic expression
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
The absolute value of p is the focal length. The sign on p tells
|
||
|
|
which way the parabola faces. Vertical parabolas that open up
|
||
|
|
and horizontal that open right, give a positive value for p.
|
||
|
|
Vertical parabolas that open down and horizontal that open left,
|
||
|
|
give a negative value for p.
|
||
|
|
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
https://www.sparknotes.com/math/precalc/conicsections/section2/
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.p_parameter
|
||
|
|
-4
|
||
|
|
|
||
|
|
"""
|
||
|
|
m = self.directrix.slope
|
||
|
|
if m is S.Infinity:
|
||
|
|
x = self.directrix.coefficients[2]
|
||
|
|
p = sign(self.focus.args[0] + x)
|
||
|
|
elif m == 0:
|
||
|
|
y = self.directrix.coefficients[2]
|
||
|
|
p = sign(self.focus.args[1] + y)
|
||
|
|
else:
|
||
|
|
d = self.directrix.projection(self.focus)
|
||
|
|
p = sign(self.focus.x - d.x)
|
||
|
|
return p * self.focal_length
|
||
|
|
|
||
|
|
@property
|
||
|
|
def vertex(self):
|
||
|
|
"""The vertex of the parabola.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
vertex : Point
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.geometry.point.Point
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Parabola, Point, Line
|
||
|
|
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
|
||
|
|
>>> p1.vertex
|
||
|
|
Point2D(0, 4)
|
||
|
|
|
||
|
|
"""
|
||
|
|
focus = self.focus
|
||
|
|
m = self.directrix.slope
|
||
|
|
if m is S.Infinity:
|
||
|
|
vertex = Point(focus.args[0] - self.p_parameter, focus.args[1])
|
||
|
|
elif m == 0:
|
||
|
|
vertex = Point(focus.args[0], focus.args[1] - self.p_parameter)
|
||
|
|
else:
|
||
|
|
vertex = self.axis_of_symmetry.intersection(self)[0]
|
||
|
|
return vertex
|