I have a polygon consists of lots of points. I want to find the intersection of the polygon and a circle. Providing the circle center of [x0,y0] and radius of r0, I have wrote a rough function to simply solve the quadratic equation of the circle and a line. But what about the efficiency of find the intersection of every line segment of the polygon one by one? Is there more efficient way?
I know sympy already provide the feature to get the intersections of different geometry. But also what about the efficiency of external library like sympy compared to calculate it by my own function, if I want to deal with lots of polygons?
def LineIntersectCircle(p,lsp,lep):
# p is the circle parameter, lsp and lep is the two end of the line
x0,y0,r0 = p
x1,y1 = lsp
x2,y2 = esp
if x1 == x2:
if abs(r0) >= abs(x1 - x0):
p1 = x1, y0 - sqrt(r0**2 - (x1-x0)**2)
p2 = x1, y0 + sqrt(r0**2 - (x1-x0)**2)
inp = [p1,p2]
# select the points lie on the line segment
inp = [p for p in inp if p[1]>=min(y1,y2) and p[1]<=max(y1,y2)]
else:
inp = []
else:
k = (y1 - y2)/(x1 - x2)
b0 = y1 - k*x1
a = k**2 + 1
b = 2*k*(b0 - y0) - 2*x0
c = (b0 - y0)**2 + x0**2 - r0**2
delta = b**2 - 4*a*c
if delta >= 0:
p1x = (-b - sqrt(delta))/(2*a)
p2x = (-b + sqrt(delta))/(2*a)
p1y = k*x1 + b0
p2y = k*x2 + b0
inp = [[p1x,p1y],[p2x,p2y]]
# select the points lie on the line segment
inp = [p for p in inp if p[0]>=min(x1,x2) and p[0]<=max(x1,x2)]
else:
inp = []
return inp
The method for finding the intersection points of a line 𝑦 = 𝑥 𝑚 + 𝑏 and a circle given in general form is as follows: Substitute 𝑚 𝑥 + 𝑏 for 𝑦 in the equation of the circle, here 𝑥 + 𝑦 + 2 𝑥 − 2 𝑦 − 3 = 0 .
In geometry, a line meeting a circle in exactly one point is known as a tangent line, while a line meeting a circle in exactly two points in known as a secant line (Rhoad et al.
Hi A point of contact between a tangent and a circle is the only point touching the circle by this line, The point can be found either by : equating the equations; The line : y = mx +c The circle : (x-a)^2 + (y_b)^2 = r^2 The result will be the value of {x}which can be substituted in the equation of the line to find ...
I guess maybe your question is about how to in theory do this in the fastest manner. But if you want to do this quickly, you should really use something which is written in C/C++.
I am quite used to Shapely, so I will provide an example of how to do this with this library. There are many geometry libraries for python. I will list them at the end of this answer.
from shapely.geometry import LineString
from shapely.geometry import Point
p = Point(5,5)
c = p.buffer(3).boundary
l = LineString([(0,0), (10, 10)])
i = c.intersection(l)
print i.geoms[0].coords[0]
(2.8786796564403576, 2.8786796564403576)
print i.geoms[1].coords[0]
(7.121320343559642, 7.121320343559642)
What is a little bit counter intuitive in Shapely is that circles are the boundries of points with buffer areas. This is why I do p.buffer(3).boundry
Also the intersection i
is a list of geometric shapes, both of them points in this case, this is why I have to get both of them from i.geoms[]
There is another Stackoverflow question which goes into details about these libraries for those interested.
EDIT because comments:
Shapely is based on GEOS (trac.osgeo.org/geos) which is built in C++ and considerably faster than anything you write natively in python. SymPy seems to be based on mpmath (mpmath.org) which also seems to be python, but seems to have lots of quite complex math integrated into it. Implementing that yourself may require a lot of work, and will probably not be as fast as GEOS C++ implementations.
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