Line from Two Points

easyFREERANSACGeometryCV: Model Fitting and Optimization

Implement a function to compute the parameters of a line given two points. The concept of fitting a line to a set of points is fundamental in Model Fitting and Optimization, particularly in the context of RANSAC, where it's used to robustly estimate the parameters of a model from noisy data.

To find the line parameters, we can use the slope-intercept form of a line, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. Given two points $(x_1, y_1)$ and $(x_2, y_2)$ , we can calculate the slope $m$ as the ratio of the difference in $y$ -coordinates to the difference in $x$ -coordinates.

Calculate the difference in $y$ -coordinates and $x$ -coordinates between the two points.
Compute the slope $m$ using the differences calculated in step 1.
Use one of the points to solve for the y-intercept $b$ .

m = \frac{y_2 - y_1}{x_2 - x_1}, \quad b = y_1 - m \cdot x_1

This technique is widely used in computer vision applications, such as line detection in images.

Example

Input

line_from_points((0,0), (2,4))

Output

[2.0, 0.0]

Reasoning

The inputs p1 = (0,0) and p2 = (2,4) are two points on the line, each written as $(x, y)$ : so $x_1 = 0, y_1 = 0$ and $x_2 = 2, y_2 = 4$ .
Slope: moving from the first point to the second, $x$ rises by $\Delta x = 2 - 0 = 2$ while $y$ rises by $\Delta y = 4 - 0 = 4$ . The slope is rise over run, $m = \frac{4}{2} = 2.0$ , so each unit step in $x$ adds 2 to $y$ .
Intercept: the line must pass through $(x_1, y_1)$ , so $0 = 2.0 \cdot 0 + b$ , giving $b = 0.0$ . Checking with the other point: $2.0 \cdot 2 + 0.0 = 4$ , as required.
Output: the pair $[m, b]$ = [2.0, 0.0], i.e. the line $y = 2x$ , with both values as decimals rounded to 4 decimal places.
Notice that swapping the two points flips the signs of both $\Delta y$ and $\Delta x$ , so the slope (and the line) is unchanged.

Constraints

Assume x1 ≠ x2 (non-vertical line)
Return [m, b] rounded to 4 decimal places

More from CV: Model Fitting and Optimization

RANSAC Homography Estimationmedium RANSAC Line Fittingmedium RANSAC Line Fithard Count Inlierseasy Weighted Least Squareshard

Concept primer

You are fitting the equation of a line in 2D, using exactly two points, which is the minimal data needed to uniquely define a non-vertical line. In the broader RANSAC context, this is the “minimal sample” used to propose a line model before checking it against many noisy points.

1. Background Knowledge

A line in 2D can be written in slope-intercept form: $y = mx + b$ where:
$m$ is the slope (how steep the line is),
$b$ is the y-intercept (where the line crosses the y-axis, i.e., at $x = 0$ ).
Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ , there is exactly one line that passes through both (unless it’s a vertical line, which is a special case where the slope is undefined). In RANSAC for line fitting, you repeatedly:
randomly pick two points,
compute the line through them,
then measure how many other points agree with this line (inliers).

2. Algorithm / Approach

The general pattern to compute a line from two points:

Use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Plug one point into $y = mx + b$ to solve for $b$ : $b = y_1 - m \cdot x_1$
Return $(m, b)$ as the line parameters.

In more robust systems (like RANSAC), this is wrapped inside a loop that tests many point pairs, but the core computation of $m$ and $b$ is what you’re implementing here.

3. Step-by-Step Strategy

Read inputs: $x_1, y_1, x_2, y_2$ .
Check denominator:

Compute $\Delta x = x_2 - x_1$ .
If $\Delta x = 0$ , the line is vertical (slope infinite); depending on the problem, you may:
avoid such pairs, or
handle this as a special case (e.g., return some flag or separate representation).

Compute slope:

$m = (y_2 - y_1) / (x_2 - x_1)$ .

Compute intercept:

$b = y_1 - m \cdot x_1$ .

Output:

Return or print $m$ and $b$ .

Example (in a C++/Python-like pseudocode):

def line_from_two_points(x1, y1, x2, y2):
 dx = x2 - x1
 if dx == 0:
 # vertical line: x = x1
 # handle according to problem spec
 raise ValueError("Vertical line: slope undefined")

 m = (y2 - y1) / dx
 b = y1 - m * x1
 return m, b

4. Common Pitfalls

Division by zero: when $x_1 = x_2$ , $\Delta x = 0$ , and the slope formula is undefined. You must check this.
Integer division: in languages like C++/Java/Python 2, if you use integers, (y2 - y1) / (x2 - x1) can truncate to an integer. Use floating point (double, float, or float64) to get a real-valued slope.
Precision issues: large coordinates or very close $x_1$ and $x_2$ can lead to numerical instability; for this easy problem it usually doesn’t matter, but be aware conceptually.
Point order doesn’t matter: using $(x_2, y_2)$ instead of $(x_1, y_1)$ to compute $b$ should give the same result; if it doesn’t, there’s likely a bug.

5. Time & Space Complexity

Time complexity: $O(1)$
Just a constant number of arithmetic operations.
Space complexity: $O(1)$
Only a few scalar variables are used, regardless of input size.

solution.py

Python 3.13

Test results