💡 Problem Formulation: Determining if a given list of points resides on a straight line is a common problem in computational geometry. For instance, you may have a list of points like [(1,2), (2,4), (3,6)], and you need a program to tell you if these points aligned linearly. The desired output is a simple boolean indicating True if the points form a straight line, or False otherwise.
Method 1: Slope Comparison
This method involves calculating the slope between consecutive points and ensuring it remains constant throughout the list. If the slope between any two consecutive pairs of points differs, the points do not form a straight line. The mathematical premise is that the linear equation defining a straight line possesses a uniform gradient.
Here’s an example:
def is_straight_line(points):
(x0, y0), (x1, y1) = points[:2]
for (x, y) in points[2:]:
if (x1 - x0) * (y - y0) != (y1 - y0) * (x - x0):
return False
return True
points = [(1, 2), (2, 4), (3, 6)]
print(is_straight_line(points))Output:
True
This code defines a function is_straight_line() that computes the cross multiplication of differences in the x and y coordinates to keep the comparison intact without having to divide (and potentially running into division by zero errors). It works through the list checking if the slopes between all pairs of points are equal.
Method 2: Use of linear algebra (numpy)
By using linear algebra, specifically the numpy library, we can simplify the slope checking process. It involves creating vectors from the points and determining if these vectors are linearly dependent. This is typically checked using the rank of a matrix composed of the vectors.
Here’s an example:
import numpy as np
def is_straight_line(points):
points_array = np.array(points)
x_diff = points_array[1:, 0] - points_array[0, 0]
y_diff = points_array[1:, 1] - points_array[0, 1]
matrix = np.vstack((x_diff, y_diff))
return np.linalg.matrix_rank(matrix) == 1
points = [(1, 2), (2, 4), (3, 6)]
print(is_straight_line(points))Output:
True
This code makes use of numpy’s matrix_rank() function to determine if the points form a straight line. It simply constructs two vectors representing the differences in the x and y coordinates and checks the rank. If the rank is 1, the vectors (and therefore the points) are linearly dependent, indicating a straight line.
Method 3: Linear Regression
A statistical approach, linear regression can be used to find the best fit line for the given points, and then we can check if this line fits perfectly. If there’s any deviation from the perfect fit, the points don’t form a straight line.
Here’s an example:
from sklearn.linear_model import LinearRegression
import numpy as np
def is_straight_line(points):
x = np.array([point[0] for point in points]).reshape(-1, 1)
y = np.array([point[1] for point in points])
reg = LinearRegression().fit(x, y)
return reg.score(x, y) == 1.0
points = [(1, 2), (2, 4), (3, 6)]
print(is_straight_line(points))Output:
True
Here we employ the scikit-learn library’s LinearRegression class to fit our points. The score() method returns the coefficient of determination R² of the prediction. A score of 1.0 indicates all points perfectly lie on the predicted line, meaning they form a straight line.
Method 4: Determinant Method
The determinant method uses the fact that if the area of the triangle formed by any three points is zero, then these points must lie on a straight line. By calculating the determinant of a matrix formed by the points, we can confirm if the set of points form a straight line.
Here’s an example:
def is_straight_line(points):
for i in range(2, len(points)):
if (points[i][0] * (points[1][1] - points[0][1]) +
points[1][0] * (points[0][1] - points[i][1]) +
points[0][0] * (points[i][1] - points[1][1])) != 0:
return False
return True
points = [(1, 2), (2, 4), (3, 6)]
print(is_straight_line(points))Output:
True
The is_straight_line() function checks the determinant of a 3×3 matrix, where the first row is composed of the x-coordinates, the second of the y-coordinates, and the third of ones. A non-zero determinant indicates that the triangle formed by these three points has a non-zero area, hence the points are non-collinear.
Bonus One-Liner Method 5: Python Set
A simple and intuitive one-liner approach is to leverage the uniqueness property of a Python set. By calculating the slopes of lines connecting each point to the first point and placing them in a set, we ensure no duplicates. If the set length is one, all points form a straight line.
Here’s an example:
is_straight_line = lambda pts: len(set((y - pts[0][1]) / (x - pts[0][0]) if x != pts[0][0] else 'inf' for x, y in pts)) == 1 points = [(1, 2), (2, 4), (3, 6)] print(is_straight_line(points))
Output:
True
This lambda function handles division by zero by assigning 'inf' as the slope when x-coordinates are equal. It computes the slope to each point from the first point and checks if all computed slopes are the same within a set.
Summary/Discussion
- Method 1: Slope Comparison. This method is straightforward and does not require any additional libraries. However, it can suffer from floating-point precision issues.
- Method 2: Use of linear algebra (numpy). Using numpy for vector operations is efficient and mathematically clean. The downside is the need for an external library and being overkill for smaller problems.
- Method 3: Linear Regression. Ideal for situations where some error tolerance in the data is needed, but this method is overcomplicated for the problem if the input data is precise.
- Method 4: Determinant Method. This is a robust mathematical solution, but can be computationally more intensive than necessary for a simple problem like this.
- Bonus Method 5: Python Set. Pythonic and succinct, it is great for quick checks but could fail due to division by zero if not handled properly.