MATLAB Implementation for Calculating Intersection Points of Two Circles

When calculating the intersection coordinates of two circles in MATLAB without using symbolic computation functions like `solve`, a purely numerical approach can be implemented. The core methodology involves establishing equations based on geometric relationships and solving intersection points using vector operations.

Problem Formulation Given two circles with centers at ( (x_1, y_1) ) and ( (x_2, y_2) ), and radii ( r_1 ) and ( r_2 ) respectively, the intersection coordinates must satisfy both circle equations: [ (x - x_1)^2 + (y - y_1)^2 = r_1^2 \ (x - x_2)^2 + (y - y_2)^2 = r_2^2 ]

Geometric Solution Steps Distance Validation: Calculate the distance ( d ) between circle centers using `d = norm([x2-x1, y2-y1])`. If ( d > r_1 + r_2 ) (separated) or ( d < |r_1 - r_2| ) (contained), no intersection exists. Coordinate Transformation: Translate the coordinate system to the first circle center ( (x_1, y_1) ), then rotate using a rotation matrix so the second center lies on the new X-axis, simplifying calculations. Intersection Calculation: In the new coordinate system, apply the Pythagorean theorem to derive Y-values of intersections, compute corresponding X-values, then transform back to the original coordinate system using inverse rotation and translation.

MATLAB Implementation Key Points Use vector subtraction `[dx, dy] = [x2-x1, y2-y1]` to compute center distance and direction angle with `theta = atan2(dy, dx)`. Implement coordinate transformation through rotation matrix operations, avoiding direct solution of nonlinear equations. Handle special cases: tangent circles (single solution) with `abs(d - (r1+r2)) < tol` and intersecting circles (dual solutions) with separate coordinate calculations for both intersection points.

This approach eliminates symbolic computation overhead, making it suitable for efficient numerical calculations in geometric graphics processing or real-time systems. The implementation typically involves functions like `norm()` for distance calculation, `atan2()` for angle computation, and matrix operations for coordinate transformations.