Hag's Khashba
Contest: Codeforces Round 478 (Div. 2)
Difficulty rating: 2600
Tags: geometry

Hag is a very talented person. He has always had an artist inside him but his father forced him to study mechanical engineering.

Yesterday he spent all of his time cutting a giant piece of wood trying to make it look like a goose. Anyway, his dad found out that he was doing arts rather than studying mechanics and other boring subjects. He confronted Hag with the fact that he is a spoiled son that does not care about his future, and if he continues to do arts he will cut his 25 Lira monthly allowance.

Hag is trying to prove to his dad that the wooden piece is a project for mechanics subject. He also told his dad that the wooden piece is a strictly convex polygon with $$$n$$$ vertices.

Hag brought two pins and pinned the polygon with them in the $$$1$$$-st and $$$2$$$-nd vertices to the wall. His dad has $$$q$$$ queries to Hag of two types.

- $$$1$$$ $$$f$$$ $$$t$$$: pull a pin from the vertex $$$f$$$, wait for the wooden polygon to rotate under the gravity force (if it will rotate) and stabilize. And then put the pin in vertex $$$t$$$.
- $$$2$$$ $$$v$$$: answer what are the coordinates of the vertex $$$v$$$.

Please help Hag to answer his father's queries.

You can assume that the wood that forms the polygon has uniform density and the polygon has a positive thickness, same in all points. After every query of the 1-st type Hag's dad tries to move the polygon a bit and watches it stabilize again.

Input format:
The first line contains two integers $$$n$$$ and $$$q$$$ ($$$3\leq n \leq 10\,000$$$, $$$1 \leq q \leq 200000$$$) — the number of vertices in the polygon and the number of queries.

The next $$$n$$$ lines describe the wooden polygon, the $$$i$$$-th line contains two integers $$$x_i$$$ and $$$y_i$$$ ($$$|x_i|, |y_i|\leq 10^8$$$) — the coordinates of the $$$i$$$-th vertex of the polygon. It is guaranteed that polygon is strictly convex and the vertices are given in the counter-clockwise order and all vertices are distinct.

The next $$$q$$$ lines describe the queries, one per line. Each query starts with its type $$$1$$$ or $$$2$$$. Each query of the first type continues with two integers $$$f$$$ and $$$t$$$ ($$$1 \le f, t \le n$$$) — the vertex the pin is taken from, and the vertex the pin is put to and the polygon finishes rotating. It is guaranteed that the vertex $$$f$$$ contains a pin. Each query of the second type continues with a single integer $$$v$$$ ($$$1 \le v \le n$$$) — the vertex the coordinates of which Hag should tell his father.

It is guaranteed that there is at least one query of the second type.

Output format:
The output should contain the answer to each query of second type — two numbers in a separate line. Your answer is considered correct, if its absolute or relative error does not exceed $$$10^{-4}$$$.

Formally, let your answer be $$$a$$$, and the jury's answer be $$$b$$$. Your answer is considered correct if $$$\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-4}$$$

Constraints:
Time limit: 3.0 s
Memory limit: 256.0 MB

Note:
In the first test note the initial and the final state of the wooden polygon.

Red Triangle is the initial state and the green one is the triangle after rotation around $$$(2,0)$$$.

In the second sample note that the polygon rotates $$$180$$$ degrees counter-clockwise or clockwise direction (it does not matter), because Hag's father makes sure that the polygon is stable and his son does not trick him.

Example input:
3 4
0 0
2 0
2 2
1 1 2
2 1
2 2
2 3

Example output:
3.4142135624 -1.4142135624
2.0000000000 0.0000000000
0.5857864376 -1.4142135624

Example input:
3 2
-1 1
0 0
1 1
1 1 2
2 1

Example output:
1.0000000000 -1.0000000000

Editorial:
In this problem, we calculate the center of mass of the polygon (ie its centroid) and then we make it our reference point. we maintain its coordinates after each and every rotation.
Now when the pin is taken out from a vertex it leaves the other pin as a pivot for rotation we calculate the angle of current rotation. calculate the new coordinates of the center of mass after rotation, we should also store the initial distances from the center of mass, and the angle that the polygon had rotated around itself.
from the angle, coordinates of the center of mass and the initial distances from the center of mass it is possible to calculate the coordinates of any point in the polygon.
Note: when calculating the center of mass we should shift the polygon to the (0,0) because in some algorithms it uses point (0,0) in triangulation the polygon. if the polygon is very far from (0,0) accuracy will be lost. so better to either shift the polygon to (0,0) or use the first point of the polygon to form the sub-triangles of the polygon when calculating the center of mass.