If the circles have the same center, they intersect at no point if they have different radius, or at all points if they have the same radius.
Leaving that case aside, we can assume without loss of generality that one circle is centered at (0, 0)
and has radius 1
, while the other one is centered at (x, 0)
with x > 0
and has radius R
with R ≥ 1
. We're left with three cases.
-
If
R
is outside the closed intervalx-1..x+1
, the circles don't intersect. -
If
R
is on either endpoint of that interval,x-1 | x+1
, the circles intersect at exactly one point, either(+1, 0)
or(-1, 0)
, respectively. -
Finally, if
R
is strictly inside the intervalx-1..x+1
, the two intersection points are of the form(cos ±α, sin ±α)
, whereα
satisfies (Pythagoras)R² == (x - cos α)² + sin²α == x² - 2 x cos α + cos²α + sin²α == x² - 2 x cos α + 1
. Thusα = acos (x² - R² + 1)/(2 x)
.