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Asked: 2025-06-05T08:14:24+05:30

Determine if two rays intersect based on their endpoints and angles in a programming challenge.

I’ve been diving into some geometry problems recently, and I stumbled upon one that totally had me scratching my head. I thought it might be fun to put it out there and see how you’d tackle it!

Imagine you’re playing around with two rays in a 2D plane, and you need to figure out if they actually intersect or not. Here’s the deal: each ray has a starting point and an angle that indicates its direction. The starting point is defined by its coordinates (x, y), and the angle is given in degrees from the positive x-axis.

Let’s say Ray A starts at point (x1, y1) and extends out at an angle of θ1 degrees, while Ray B starts at point (x2, y2) and extends out at an angle of θ2 degrees. Your job is to determine if these two rays intersect at any point.

Now, here’s the catch: we’re working with infinite rays! So, if one ray goes straight indefinitely, will it ever cross the other ray at some point in space?

I was thinking about how to approach this problem. You could convert the angles into a more usable format, like using trigonometric functions to get the direction vectors. Then, you might need to set up some equations based on the parametric representation of the rays.

If I’m doing my math right, you’d probably have to check for some conditions on the equations you set up: if they yield a solution that results in both rays being intersected at positive lengths (since the rays only move forward from their starting points).

To make things even spicier, what happens if the rays are parallel? Or what if they overlap? Finding a comprehensive solution sounds tricky.

I’d love to hear how you all think about this – do you have any strategies you’d implement, any formulas you think could help? And would you consider throwing this into a programming function? Can’t wait to see your thought processes and solutions!

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    2 Answers

    1. To determine if two infinite rays intersect in a 2D plane, we can utilize their parametric representations. For Ray A starting at point (x1, y1) and extending at an angle θ1, we can express the ray in parametric form as:
      (x1 + t1 * cos(θ1), y1 + t1 * sin(θ1)), where t1 is a non-negative parameter representing the distance along the ray. Similarly, Ray B can be represented as:
      (x2 + t2 * cos(θ2), y2 + t2 * sin(θ2)), with t2 also being a non-negative parameter. To find the intersection, we set the two parametric equations equal to each other and solve for t1 and t2:
      x1 + t1 * cos(θ1) = x2 + t2 * cos(θ2) and y1 + t1 * sin(θ1) = y2 + t2 * sin(θ2). The critical part is that we need both t1 and t2 to be non-negative for the intersection point to lie on the rays.

      If the two rays are parallel (i.e., they have the same direction vector), they will either not intersect (if they start at different points) or overlap entirely (if they start at the same point). To handle edge cases, we need to check the determinant of the system of equations we derived earlier. If it is zero, then the rays are either parallel or collinear. If it is not zero, we can compute t1 and t2. For implementation, you could write a function in Python that calculates the intersection based on these equations, ensuring to return whether a valid intersection exists based on the non-negativity of the parameters.

    2. Oh, this sounds like one of those tricky yet fun problems! So, basically you’ve got two rays shooting out from different points in some direction, and you’re trying to see if they cross paths at all, huh?

      You know what first comes to my mind—maybe it’d be easier to think in terms of vectors! For example, you could start by converting those angles to direction vectors.

      If your ray A starts at (x1, y1) and has an angle θ1, you could write the direction vector as:

      dirA = [cos(θ1°), sin(θ1°)]

      Same goes for Ray B starting at (x2, y2) with angle θ2:

      dirB = [cos(θ2°), sin(θ2°)]

      Then, each ray can be described parametrically like this (t ≥ 0 means the ray only moves forward from the start):

      Ray A points: (x1 + t·cos(θ1°), y1 + t·sin(θ1°))
      
Ray B points: (x2 + s·cos(θ2°), y2 + s·sin(θ2°))

      Your goal is basically: Do these two equations have a common point for some positive values t and s?

      To find that intersection (if there is one), you can just set the x and y equations equal to each other and solve the resulting system:

      x1 + t·cos(θ1°) = x2 + s·cos(θ2°)
      
y1 + t·sin(θ1°) = y2 + s·sin(θ2°)

      From here, you’ll have two equations and two unknowns (t and s), which can be solved either by algebra directly or by using some software (why not code a quick function?). Anyway, if your solutions yield t ≥ 0 and s ≥ 0, yes—they intersect! If any of t or s is negative, nope—that means the intersection occurs behind the ray’s start, not in front.

      Oh, and one more thing: if these equations give you no solution, or if the direction vectors are parallel (meaning the vectors are multiples of each other or exactly opposite), you’ll have to double-check carefully. Either one ray could start exactly where the other ray goes (overlapping), or they never meet at all. Might need a special check for those special cases to make sure your code doesn’t break!

      Speaking about code, it’d be pretty neat to turn this into a function, like ‘doRaysIntersect(x1, y1, θ1, x2, y2, θ2)’, right? Could even share it and see how people start breaking it!

      Anyway, does this help at all or just confuse you more? 🤔 Let me know!