c# - Detecting possible projectile vs shield collision -

i'm trying detect, whether projectile going hit shield - , collision happen.

on image, can see situation. though a , b projectiles in approximately same distance shield centre s, 1 of them collide shield , other 1 not.

solution mathematically straightforward use mathematical equation of circle , equation of line of projectile path.

with pen , paper, can calculate that. however, have bad experience implementing analytic geometry directly.

also, return positives projectiles leaving shield. how filter them?

second approach, friends idea, measuring minimal distance between projectile path , shield centre - if l smaller r, collision happens.

it seems easier implement, doesn't let me know collision happen.

i work in 2 dimensions. i'm using c# , unity engine, universal solutions welcome.

the projectile is, of course, considered point 0 size.

use parametric equation of trajectory: x= x0 + t.p + x0, y = t.q + y0 (starting (x0, y0) in direction (p, q)).

plug circle equation (x - xc)^2 + (y - yc)^2 = r^2, , get:

(t.p + dx)^2 + (t.q + dy)^2 = (p^2+q^2).t^2 + 2.(p.dx+q.dy).t + dx^2+dy^2 = r^2. 

this quadratic equation in t. shield hit whenever there positive root.

if projectile has nonzero radius r, imagine deflate projectile 0 , inflate target r + r.