Draw perpendicular line of fixed length at a point of another line
Question:
I have two points A (10,20) and B (15,30). The points generate a line AB. I need to draw a perpendicular line, CD, on point B with a length of 6 (each direction 3 units) in Python.
I already have some properties of line AB using the following code:
from scipy import stats
x = [10,15]
y = [20,30]
slope, intercept, r_value, p_value, std_err = stats.linregress(x,y)
How can I calculate the location of C and D. I need their X and Y value.
The value of C and D will be used for accomplishing another objective using the Shapely library.
Answers:
If slope is the slope of AB, then the slope of CD is -1/slope. This is equal to vertical change over horizontal change: dy/dx = -1/slope. This gives that dx = -slope*dx. And by Pythagorean Theorem, you have 3**2 = dy**2+dx**2. Substitute for dx, and you get
3**2 = (-slope*dy)**2+dy**2
3**2 = (slope**2 + 1)*dy**2
dy**2 = 3**2/(slope**2+1)
dy = math.sqrt(3**2/(slope**2+1))
Then you can get dx = -slope*dy. Finally, you can use dx and dy to get C and D. So the code would be:
import math
dy = math.sqrt(3**2/(slope**2+1))
dx = -slope*dy
C[0] = B[0] + dx
C[1] = B[1] + dy
D[0] = B[0] - dx
D[1] = B[1] - dy
(Note that although math.sqrt returns only one number, in general there is a positive and negative square root. C corresponds to the positive square root, and D to the negative).
You probably should use vectors to calculate the position of the points.
- create the
vector AB
- calculate its
normalized perpendicular
- add or subtract 3 times this to
B
with the help of a simple, reusable Vector class, the calculation is trivial, and reads like English:
Find the points perpendicular to AB at distance 3 from point B:
P1 = B + (B-A).perp().normalized() * 3
P2 = B + (B-A).perp().normalized() * 3
class Vector:
def __init__(self, x, y):
self.x = x
self.y = y
def __sub__(self, other):
return Vector(self.x - other.x, self.y - other.y)
def __add__(self, other):
return Vector(self.x + other.x, self.y + other.y)
def dot(self, other):
return self.x * other.x + self.y * other.y
def norm(self):
return self.dot(self)**0.5
def normalized(self):
norm = self.norm()
return Vector(self.x / norm, self.y / norm)
def perp(self):
return Vector(1, -self.x / self.y)
def __mul__(self, scalar):
return Vector(self.x * scalar, self.y * scalar)
def __str__(self):
return f'({self.x}, {self.y})'
A = Vector(10, 20)
B = Vector(15, 30)
AB = B - A
AB_perp_normed = AB.perp().normalized()
P1 = B + AB_perp_normed * 3
P2 = B - AB_perp_normed * 3
print(f'Point{P1}, and Point{P2}')
output:
Point(17.683281572999746, 28.658359213500127), and Point(12.316718427000252, 31.341640786499873)
Since you are interested in using Shapely, the easiest way to get the perpendicular line that I can think of, is to use parallel_offset method to get two parallel lines to AB, and connect their endpoints:
from shapely.geometry import LineString
a = (10, 20)
b = (15, 30)
cd_length = 6
ab = LineString([a, b])
left = ab.parallel_offset(cd_length / 2, 'left')
right = ab.parallel_offset(cd_length / 2, 'right')
c = left.boundary[1]
d = right.boundary[0] # note the different orientation for right offset
cd = LineString([c, d])
And the coordinates of CD:
>>> c.x, c.y
(12.316718427000252, 31.341640786499873)
>>> d.x, d.y
(17.683281572999746, 28.658359213500127)
If we have a line parallel to x-axis or y-axis, then we can add the following two methods to the class Vector.
def perptoy(self):
return Vector(1, 0)
def perptox(self):
return Vector(1, -10000) // any big number will do instead of -inf
AB = B - A
if (A.y == B.y):
AB_perp_normed = AB.perptox().normalized();
elif (A.x == B.x):
AB_perp_normed = AB.perptoy().normalized();
else:
AB_perp_normed = AB.perp().normalized();
And the rest follows the same as in the earlier post.
I have two points A (10,20) and B (15,30). The points generate a line AB. I need to draw a perpendicular line, CD, on point B with a length of 6 (each direction 3 units) in Python.
I already have some properties of line AB using the following code:
from scipy import stats
x = [10,15]
y = [20,30]
slope, intercept, r_value, p_value, std_err = stats.linregress(x,y)
How can I calculate the location of C and D. I need their X and Y value.
The value of C and D will be used for accomplishing another objective using the Shapely library.
If slope is the slope of AB, then the slope of CD is -1/slope. This is equal to vertical change over horizontal change: dy/dx = -1/slope. This gives that dx = -slope*dx. And by Pythagorean Theorem, you have 3**2 = dy**2+dx**2. Substitute for dx, and you get
3**2 = (-slope*dy)**2+dy**2
3**2 = (slope**2 + 1)*dy**2
dy**2 = 3**2/(slope**2+1)
dy = math.sqrt(3**2/(slope**2+1))
Then you can get dx = -slope*dy. Finally, you can use dx and dy to get C and D. So the code would be:
import math
dy = math.sqrt(3**2/(slope**2+1))
dx = -slope*dy
C[0] = B[0] + dx
C[1] = B[1] + dy
D[0] = B[0] - dx
D[1] = B[1] - dy
(Note that although math.sqrt returns only one number, in general there is a positive and negative square root. C corresponds to the positive square root, and D to the negative).
You probably should use vectors to calculate the position of the points.
- create the
vector AB - calculate its
normalized perpendicular - add or subtract 3 times this to
B
with the help of a simple, reusable Vector class, the calculation is trivial, and reads like English:
Find the points perpendicular to AB at distance 3 from point B:
P1 = B + (B-A).perp().normalized() * 3
P2 = B + (B-A).perp().normalized() * 3
class Vector:
def __init__(self, x, y):
self.x = x
self.y = y
def __sub__(self, other):
return Vector(self.x - other.x, self.y - other.y)
def __add__(self, other):
return Vector(self.x + other.x, self.y + other.y)
def dot(self, other):
return self.x * other.x + self.y * other.y
def norm(self):
return self.dot(self)**0.5
def normalized(self):
norm = self.norm()
return Vector(self.x / norm, self.y / norm)
def perp(self):
return Vector(1, -self.x / self.y)
def __mul__(self, scalar):
return Vector(self.x * scalar, self.y * scalar)
def __str__(self):
return f'({self.x}, {self.y})'
A = Vector(10, 20)
B = Vector(15, 30)
AB = B - A
AB_perp_normed = AB.perp().normalized()
P1 = B + AB_perp_normed * 3
P2 = B - AB_perp_normed * 3
print(f'Point{P1}, and Point{P2}')
output:
Point(17.683281572999746, 28.658359213500127), and Point(12.316718427000252, 31.341640786499873)
Since you are interested in using Shapely, the easiest way to get the perpendicular line that I can think of, is to use parallel_offset method to get two parallel lines to AB, and connect their endpoints:
from shapely.geometry import LineString
a = (10, 20)
b = (15, 30)
cd_length = 6
ab = LineString([a, b])
left = ab.parallel_offset(cd_length / 2, 'left')
right = ab.parallel_offset(cd_length / 2, 'right')
c = left.boundary[1]
d = right.boundary[0] # note the different orientation for right offset
cd = LineString([c, d])
And the coordinates of CD:
>>> c.x, c.y
(12.316718427000252, 31.341640786499873)
>>> d.x, d.y
(17.683281572999746, 28.658359213500127)
If we have a line parallel to x-axis or y-axis, then we can add the following two methods to the class Vector.
def perptoy(self):
return Vector(1, 0)
def perptox(self):
return Vector(1, -10000) // any big number will do instead of -inf
AB = B - A
if (A.y == B.y):
AB_perp_normed = AB.perptox().normalized();
elif (A.x == B.x):
AB_perp_normed = AB.perptoy().normalized();
else:
AB_perp_normed = AB.perp().normalized();
And the rest follows the same as in the earlier post.