def tangent_lines(self, p):
"""Tangent lines between `p` and the ellipse.
If `p` is on the ellipse, returns the tangent line through point `p`.
Otherwise, returns the tangent line(s) from `p` to the ellipse, or
None if no tangent line is possible (e.g., `p` inside ellipse).
Parameters
==========
p : Point
Returns
=======
tangent_lines : list with 1 or 2 Lines
Raises
======
NotImplementedError
Can only find tangent lines for a point, `p`, on the ellipse.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Line
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line(Point(3, 0), Point(3, -12))]
>>> # This will plot an ellipse together with a tangent line.
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy import Point, Ellipse
>>> e = Ellipse(Point(0,0), 3, 2)
>>> t = e.tangent_lines(e.random_point())
>>> p = Plot()
>>> p[0] = e # doctest: +SKIP
>>> p[1] = t # doctest: +SKIP
"""
if self.encloses_point(p):
return []
if p in self:
delta = self.center - p
rise = (self.vradius ** 2)*delta.x
run = -(self.hradius ** 2)*delta.y
p2 = Point(simplify(p.x + run),
simplify(p.y + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
maj = self.hradius
test = (2*maj -
Point.distance(f1, p) -
Point.distance(f2, p))
else:
test = self.radius - Point.distance(self.center, p)
if test.is_number and test.is_positive:
return []
# else p is outside the ellipse or we can't tell. In case of the
# latter, the solutions returned will only be valid if
# the point is not inside the ellipse; if it is, nan will result.
x, y = Dummy('x'), Dummy('y')
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
slope = Line(p, Point(x, y)).slope
tangent_points = solve([slope - dydx, eq], [x, y])
# handle horizontal and vertical tangent lines
if len(tangent_points) == 1:
assert tangent_points[0][
0] == p.x or tangent_points[0][1] == p.y
return [Line(p, p + Point(1, 0)), Line(p, p + Point(0, 1))]
# others
return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]
def tangent_lines(self, p):
"""Tangent lines between `p` and the ellipse.
If `p` is on the ellipse, returns the tangent line through point `p`.
Otherwise, returns the tangent line(s) from `p` to the ellipse, or
None if no tangent line is possible (e.g., `p` inside ellipse).
Parameters
----------
p : Point
Returns
-------
tangent_lines : list with 1 or 2 Lines
Raises
------
NotImplementedError
Can only find tangent lines for a point, `p`, on the ellipse.
See Also
--------
Point
Line
Examples
--------
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line(Point(3, 0), Point(3, -12))]
>>> # This will plot an ellipse together with a tangent line.
>>> from sympy import Point, Ellipse, Plot
>>> e = Ellipse(Point(0,0), 3, 2)
>>> t = e.tangent_lines(e.random_point()) # doctest: +SKIP
>>> p = Plot() # doctest: +SKIP
>>> p[0] = e # doctest: +SKIP
>>> p[1] = t # doctest: +SKIP
"""
from sympy import solve
if self.encloses_point(p):
return []
if p in self:
rise = (self.vradius ** 2) * (self.center[0] - p[0])
run = (self.hradius ** 2) * (p[1] - self.center[1])
p2 = Point(simplify(p[0] + run), simplify(p[1] + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
maj = self.hradius
test = 2 * maj - Point.distance(f1, p) - Point.distance(f2, p)
else:
test = self.radius - Point.distance(self.center, p)
if test.is_number and test.is_positive:
return []
# else p is outside the ellipse or we can't tell. In case of the
# latter, the solutions returned will only be valid if
# the point is not inside the ellipse; if it is, nan will result.
x, y = Dummy("x"), Dummy("y")
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
slope = Line(p, Point(x, y)).slope
tangent_points = solve([w.as_numer_denom()[0] for w in [slope - dydx, eq]], [x, y])
# handle horizontal and vertical tangent lines
if len(tangent_points) == 1:
assert tangent_points[0][0] == p[0] or tangent_points[0][1] == p[1]
return [Line(p, Point(p[0] + 1, p[1])), Line(p, Point(p[0], p[1] + 1))]
# others
return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]
def tangent_lines(self, p):
"""Tangent lines between `p` and the ellipse.
If `p` is on the ellipse, returns the tangent line through point `p`.
Otherwise, returns the tangent line(s) from `p` to the ellipse, or
None if no tangent line is possible (e.g., `p` inside ellipse).
Parameters
----------
p : Point
Returns
-------
tangent_lines : list with 1 or 2 Lines
Raises
------
NotImplementedError
Can only find tangent lines for a point, `p`, on the ellipse.
See Also
--------
Point
Line
Examples
--------
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line(Point(3, 0), Point(3, -12))]
>>> # This will plot an ellipse together with a tangent line.
>>> from sympy import Point, Ellipse, Plot
>>> e = Ellipse(Point(0,0), 3, 2)
>>> t = e.tangent_lines(e.random_point()) # doctest: +SKIP
>>> p = Plot() # doctest: +SKIP
>>> p[0] = e # doctest: +SKIP
>>> p[1] = t # doctest: +SKIP
"""
if self.encloses_point(p):
return []
if p in self:
rise = (self.vradius ** 2)*(self.center[0] - p[0])
run = (self.hradius ** 2)*(p[1] - self.center[1])
p2 = Point(simplify(p[0] + run),
simplify(p[1] + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
maj = self.hradius
test = (2*maj -
Point.distance(f1, p) -
Point.distance(f2, p))
else:
test = self.radius - Point.distance(self.center, p)
if test.is_number and test.is_positive:
return []
# else p is outside the ellipse or we can't tell. In case of the
# latter, the solutions returned will only be valid if
# the point is not inside the ellipse; if it is, nan will result.
x, y = Dummy('x'), Dummy('y')
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
slope = Line(p, Point(x, y)).slope
tangent_points = solve([slope - dydx, eq], [x, y])
# handle horizontal and vertical tangent lines
if len(tangent_points) == 1:
assert tangent_points[0][0] == p[0] or tangent_points[0][1] == p[1]
return [Line(p, Point(p[0]+1, p[1])), Line(p, Point(p[0], p[1]+1))]
# others
return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]