Are They Similar Polygons?

Question

Task

Given the x,y coordinates (coordinates are guaranteed to be integers) of the vertices of two simple polygons in clockwise or anti-clockwise order. Output a Truthy value if both the polygons are similar otherwise a Falsy value

A simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If the sides intersect then the polygon is not simple. Two edges meeting at a corner are required to form a an angle that is not not straight (180°)

Two polygons are similar if either polygon can be rescaled, repositioned, and reflected, so as to coincide precisely with the other polygon.

Testcases

Input

[(0, 0), (1, 0), (1, 1), (0, 1)]
[(-1, 0), (2, 1), (1, 4), (-2, 3)]

Graph

Output

Truthy

---

Input

[(2, 3), (0, 0), (4, 0)]
[(-3, 0), (-2, 2), (0, 1), (1, -1), (-1, -2), (-2, -2)]

Graph

Output

Falsy

---

Input

[(1, 4), (1, 3), (0, 2), (1, 1), (-3, 2)]
[(2, 0), (2, -1), (3, -1), (4, -2), (-2, -2)]

Graph

Output

Falsy

---

Input

[(-1, 0), (2, 1), (1, 4), (-2, 3)]
[(1, 4), (2, 1), (-1, 0), (-2, 3)]

Graph

Output

Truthy

---

Input

[(-2, 0), (-1, 1), (0, 1), (1, 3), (1, 0)]
[(5, 4), (4, 2), (3, 2), (2, 1), (5, 1)]

Graph

Output

Truthy

---

Input

[(2, 13), (4, 8), (2, 3), (0, 8)]
[(0, 0), (5, 0), (8, 4), (3, 4)]

Graph

Output

Falsy

---

Input

[(-1, 0), (-5, 3), (-5, 9), (-1, 6)]
[(0, 0), (5, 0), (8, 4), (3, 4)]

Graph

Output

Falsy

---

Input

[(0, 0), (1, 2), (1, 0)]
[(2, 0), (2, 2), (3, 0)]

Graph

Output

Truthy

---

Answer 1

Mathematica, 97 bytes (SBCS)

Or@@(Equal@@(a@u/#)&/@(a/@NestList[RotateLeft,v,Length@v]))
a@l_:=Norm[#〚1〛-#〚2〛]&/@l~Subsets~{2}

You can try it online!

The function a computes all the pairwise distances between any two vertices of the polygon.

The main function applies this to the first list and to every rotation of the second list, to try and find the corresponding vertices. The rotation of the second list is correct if all the corresponding distances have been scaled accordingly.

Thanks to @J42161217 for saving me some 3 bytes

Answer 2

Python 3, 321 bytes

def s(l,m):
 r,n=range,len(l)
 if n^len(m):return 0
 d=lambda p,q:((p[0]-q[0])**2+(p[1]-q[1])**2)**0.5
 a=lambda p,q,r:(d(p,q)**2+d(q,r)**2-d(p,r)**2)/(2*d(p,q)*d(q,r))
 c=lambda o:[a(o[(i-1)%n],o[i],o[(i+1)%n])for i in r(n)]
 return any([all([c(l)[i]-c(m)[::z][(i+o)%n]<1e-15for i in r(n)])for z in (1,-1)for o in r(n)])

First time golfing!

To determine the similarity between the two shapes, I considered the angle at each vertex (between the line segments formed by each vertex with the previous and next). I initially planned to only look at the distances between each vertex, but the scaling transformations would cause that to be pretty tricky.

Before anything else, I checked to make sure the shapes have the same number of vertices. I calculated the angles for each shape (lambda function c) using the distance formula (lambda function d) and the law of cosines (lambda function a). I omitted the final arccosine from the angle calculation as it was unnecessary for simple comparison. Finally, I looped through the first shape's angles and every possible slice (direction and offset) of the second shape's angles. If all the values match (or have a difference of less than 1e-15, thanks floating point...) for any iteration, the shapes are similar.