# Geometry Primitives

Authors: Mihnea Brebenel, Benjamin Qi

Basic setup for geometry problems.

You should know operations such as the cross product and dot product.

## Standard Problems

### Location of a point

Focus Problem – try your best to solve this problem before continuing!

To check the $P$ location towards the $P_1$ $P_2$ line we use following formula: $(P.y - P_1.y) * (P_2.x - P_1.x) - (P.x - P_1.x) * (P_2.y - P_1.y)$ If it's equal to $0$ it means that $P$, $P_1$ and $P_2$ are collinear. Otherwise the value's sign indicated if $P$ is under the line - negative - or above the line - positive.

Demonstration

#### Implementation

C++

#include <bits/stdc++.h>using namespace std;Code Snippet: Point Class (Click to expand)long long collinear(Point p, Point p1, Point p2) {return 1LL * (p.y - p1.y) * (p2.x - p1.x) -1LL * (p.x - p1.x) * (p2.y - p1.y);}

### Segment intersection

Focus Problem – try your best to solve this problem before continuing!

We can quickly dismiss the segment intersection by treating them as rectangles having the segments as diagonals which can be easily done. If it turns out the rectangles intersect then we just check if the segment's ends are on different sides of the other segment.

#### Implementation

C++

#include <bits/stdc++.h>using namespace std;Code Snippet: Point Class (Click to expand)int sign(long long num) {if (num < 0) {return -1;} else if (num == 0) {return 0;

### Polygon area

Focus Problem – try your best to solve this problem before continuing!

The following algorithm uses the Shoelace formula.

#### Implementation

C++

#include <bits/stdc++.h>using namespace std;Code Snippet: Point Class (Click to expand)int main() {int n;cin >> n;vector<Point> points(n);for (auto &p : points) { cin >> p; }

### Point's location relative to polygon

Focus Problem – try your best to solve this problem before continuing!

We can cast a ray from the point $P$ going in any fixed direction (people usually go to the right). If the point is located on the outside of the polygon the ray will intersect its edges an even number of times. If the point is on the inside of the polygon then it will intersect the edge an odd number of times.

This approach is called ray casting.

#### Implementation

C++

#include <bits/stdc++.h>using namespace std;Code Snippet: Point Class (Click to expand)int sign(long long num) {if (num < 0) {return -1;} else if (num == 0) {return 0;

### Lattice points in polygon

Focus Problem – try your best to solve this problem before continuing!

Let's first focus on the lattice points on the polygon's boundary. We'll process each edge individually. The number of intersections of a line with lattice points is the greatest common divisor of $P_1.x - P_2.x$ and $P_1.y - P_2.y$.

Demonstration

Now that we know the number of lattice points on the boundary we can find the number of lattice points inside the polygon using Pick's theorem. Let's denote $A$ polygon's area, $i$ the number of integer points inside and $b$ the number of integer points on its boundary. Then, according to Pick's theorem, we have the following equation: $A = i + b/2 - 1$. Changing the order a little bit to get $i = A - b/2 + 1$. We've found $b$ and, as you've probably already solved Polygon Area from above, $A$ can be computed using cross-product.

#### Implementation

C++

#include <bits/stdc++.h>using namespace std;Code Snippet: Point Class (Click to expand)int main() {int n;cin >> n;vector<Point> points(n);

Status | Source | Problem Name | Difficulty | Tags | |
---|---|---|---|---|---|

YS | Easy | ||||

Kattis | Easy | ||||

Kattis | Easy | ||||

Kattis | Easy | ||||

Kattis | Easy |

## Resources

Resources | ||||
---|---|---|---|---|

CF | short description of operations | |||

CPH | Complex #s, Points & Lines, Polygons, Distances | |||

CF | code, examples | |||

cp-algo | ||||

CPC | basics, polygon area, point in polygon | |||

CF | some material is quite advanced | |||

CP2 |

## Misc Problems

Some European Olympiads, like the CEOI, Balkan OI, and the Croatian OI, tend to have a lot of geometry problems. These problems tend to be quite difficult as well, so look for problems there when you run out of problems to practice!

Status | Source | Problem Name | Difficulty | Tags | |
---|---|---|---|---|---|

CF | Hard | ||||

Kattis | Hard | ||||

Kattis | Hard |

### Module Progress:

### Join the USACO Forum!

Stuck on a problem, or don't understand a module? Join the USACO Forum and get help from other competitive programmers!