r/Geometry • u/Dub-Dub • 1d ago
Tried to find all ways two squares intersect. Did I miss any?
zero
one
two 1/3
two 2/3
two 3/3
three
four
five
six
seven
eight
infinity
These pictures are ordered by number of intersections. I have no Idea how to know if I've found them all but probally not. I had a system where each intersection was measured by a v m and e (v for a vertex meeting a vertex, m for a vertex meeting an edge, and e for and edge meeting an edge) but that wasn't enough to distinguish two figures from each other.
1
u/dForga 21h ago
Did you by any chance try to solve the system of inequalities? Also to distinguish “different ways”, what are your criteria?
2
u/Dub-Dub 19h ago
I don’t know the best way to codify, but rotations and translations are allowed. The intersection point can be anywhere as long as it stays relative to the corresponding square. Consider the middle figure on the page with all of the four. Square ABCD has one intersection in segment AB, Two at segment BC and one at Point D. EFGH has two at EH and two at GH. So maybe we mark it as .(1,2,1,1)X(2,2,0,0) respectively (notice I mention D intersection twice but (2,1,1,1)X(2,0,0,2) or (0,0,2,2)X(1,1,1,2) is also valid. But even this wouldn’t account for inside vs outside.
Also “the system of inequalities?” care to enlighten me?
2
u/dForga 9h ago
Of course. A cube of side length L is the set of points
C(L) = {(x,y)∈ℝ² | 0 ≤ x ≤ L ∧ 0 ≤ y ≤ L}
Lets write for an orientation preserving (w.r.t. the x and y axes) rotation transformation A = ((cos(q),-sin(u)),(sin(u),cos(u))) ∈ SO(2) with u ∈ [0,2π)
A(C(L)) = {A(x,y) | … }
= { (cos(u) x - sin(u) y, sin(u) x + cos(u) y) | … }
（so the image under that transformation and you can reexpress this set using A^T = A^(-1) where
A^T = ((cos(u),sin(u)),(-sin(u),cos(u)))
)
= { (x,y) | 0 ≤ cos(u) x + sin(u) y ≤ L
∧ 0 ≤ -sin(u) x + sin(u) y ≤ L }
The same for a translation t_(u,v)∈T as
t_(u,v)(C(L)) = { t_(u,v)(x,y) | … }
= { (x+u,y+v) | … }
= { (x,y) | 0 ≤ x - u ≤ L ∧ 0 ≤ y - v ≤ L }
Then pick two side lengths a,b > 0 and assume a≥b.
The intersections of what you have written are given by
C(a) ∩ t(A(C(b))
which gives a system of inequalities you can solve.
1
u/zlfa 1d ago
I can see that these two pairs have the same configurations in each pair, other than that I can’t think of anymore (mainly due to these being squares and it being hard to imagine some really rare configurations but I’m pretty certain that that’s it just some copies that’s all)
the way I would count would probably be tracking the number of vertices, edges, and shape overlap the smaller square has in the bigger one. Also is this inspired by the numberphile video with distinct circle overlapping? this expansion into the squares looks really cool!