def normal_lines(self, p, prec=None):
"""Normal lines between `p` and the ellipse.
Parameters
==========
p : Point
Returns
=======
normal_lines : list with 1, 2 or 4 Lines
Examples
========
>>> from sympy import Line, Point, Ellipse
>>> e = Ellipse((0, 0), 2, 3)
>>> c = e.center
>>> e.normal_lines(c + Point(1, 0))
[Line(Point(0, 0), Point(1, 0))]
>>> e.normal_lines(c)
[Line(Point(0, 0), Point(0, 1)), Line(Point(0, 0), Point(1, 0))]
Off-axis points require the solution of a quartic equation. This
often leads to very large expressions that may be of little practical
use. An approximate solution of `prec` digits can be obtained by
passing in the desired value:
>>> e.normal_lines((3, 3), prec=2)
[Line(Point(-38/47, -85/31), Point(9/47, -21/17)),
Line(Point(19/13, -43/21), Point(32/13, -8/3))]
Whereas the above solution has an operation count of 12, the exact
solution has an operation count of 2020.
"""
p = Point(p)
# XXX change True to something like self.angle == 0 if the arbitrarily
# rotated ellipse is introduced.
# https://github.com/sympy/sympy/issues/2815)
if True:
rv = []
if p.x == self.center.x:
rv.append(Line(self.center, slope=oo))
if p.y == self.center.y:
rv.append(Line(self.center, slope=0))
if rv:
# at these special orientations of p either 1 or 2 normals
# exist and we are done
return rv
# find the 4 normal points and construct lines through them with
# the corresponding slope
x, y = Dummy('x', real=True), Dummy('y', real=True)
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
norm = -1 / dydx
slope = Line(p, (x, y)).slope
seq = slope - norm
points = []
if prec is not None:
yis = solve(seq, y)[0]
xeq = eq.subs(y, yis).as_numer_denom()[0].expand()
try:
iv = list(zip(*Poly(xeq).intervals()))[0]
# bisection is safest here since other methods may miss root
xsol = [
S(nroot(lambdify(x, xeq), i, solver="anderson"))
for i in iv
]
points = [
Point(i,
solve(eq.subs(x, i), y)[0]).n(prec) for i in xsol
]
except PolynomialError:
pass
if not points:
points = solve((seq, eq), (x, y))
# complicated expressions may not be decidably real so evaluate to
# check whether they are real or not
points = [
Point(i).n(prec) if prec is not None else Point(i)
for i in points if all(j.n(2).is_real for j in i)
]
slopes = [norm.subs(zip((x, y), pt.args)) for pt in points]
if prec is not None:
slopes = [
i.n(prec) if i not in (-oo, oo, zoo) else i for i in slopes
]
return [Line(pt, slope=s) for pt, s in zip(points, slopes)]
def normal_lines(self, p, prec=None):
"""Normal lines between `p` and the ellipse.
Parameters
==========
p : Point
Returns
=======
normal_lines : list with 1, 2 or 4 Lines
Examples
========
>>> from sympy import Line, Point, Ellipse
>>> e = Ellipse((0, 0), 2, 3)
>>> c = e.center
>>> e.normal_lines(c + Point(1, 0))
[Line(Point(0, 0), Point(1, 0))]
>>> e.normal_lines(c)
[Line(Point(0, 0), Point(0, 1)), Line(Point(0, 0), Point(1, 0))]
Off-axis points require the solution of a quartic equation. This
often leads to very large expressions that may be of little practical
use. An approximate solution of `prec` digits can be obtained by
passing in the desired value:
>>> e.normal_lines((3, 3), prec=2)
[Line(Point(-38/47, -85/31), Point(9/47, -21/17)),
Line(Point(19/13, -43/21), Point(32/13, -8/3))]
Whereas the above solution has an operation count of 12, the exact
solution has an operation count of 2020.
"""
p = Point(p)
# XXX change True to something like self.angle == 0 if the arbitrarily
# rotated ellipse is introduced.
# https://github.com/sympy/sympy/issues/2815)
if True:
rv = []
if p.x == self.center.x:
rv.append(Line(self.center, slope=oo))
if p.y == self.center.y:
rv.append(Line(self.center, slope=0))
if rv:
# at these special orientations of p either 1 or 2 normals
# exist and we are done
return rv
# find the 4 normal points and construct lines through them with
# the corresponding slope
x, y = Dummy('x', real=True), Dummy('y', real=True)
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
norm = -1/dydx
slope = Line(p, (x, y)).slope
seq = slope - norm
points = []
if prec is not None:
yis = solve(seq, y)[0]
xeq = eq.subs(y, yis).as_numer_denom()[0].expand()
try:
iv = list(zip(*Poly(xeq).intervals()))[0]
# bisection is safest here since other methods may miss root
xsol = [S(nroot(lambdify(x, xeq), i, solver="anderson"))
for i in iv]
points = [Point(i, solve(eq.subs(x, i), y)[0]).n(prec)
for i in xsol]
except PolynomialError:
pass
if not points:
points = [Point(*s) for s in solve((seq, eq), (x, y))]
# complicated expressions may not be decidably real so evaluate to
# check whether they are real or not
points = [i.n(prec) if prec is not None else i
for i in points if all(j.n(2).is_real for j in i.args)]
slopes = [norm.subs(zip((x, y), pt.args)) for pt in points]
if prec is not None:
slopes = [i.n(prec) if i not in (-oo, oo, zoo) else i
for i in slopes]
return [Line(pt, slope=s) for pt,s in zip(points, slopes)]