# Trait cgmath::EuclideanSpace [−] [src]

```pub trait EuclideanSpace: Copy + Clone where    Self: Array<Element = Self::Scalar>,    Self: Add<Self::Diff, Output = Self>,    Self: Sub<Self, Output = Self::Diff>,    Self: Mul<Self::Scalar, Output = Self>,    Self: Div<Self::Scalar, Output = Self>,    Self: Rem<Self::Scalar, Output = Self>, {
type Scalar: BaseNum;
type Diff: VectorSpace<Scalar = Self::Scalar>;
fn origin() -> Self;
fn from_vec(v: Self::Diff) -> Self;
fn to_vec(self) -> Self::Diff;
fn dot(self, v: Self::Diff) -> Self::Scalar;

fn midpoint(self, other: Self) -> Self { ... }
fn centroid(points: &[Self]) -> Self { ... }
}```

Points in a Euclidean space with an associated space of displacement vectors.

# Point-Vector distinction

`cgmath` distinguishes between points and vectors in the following way:

• Points are locations relative to an origin
• Vectors are displacements between those points

For example, to find the midpoint between two points, you can write the following:

```use cgmath::Point3;

let p0 = Point3::new(1.0, 2.0, 3.0);
let p1 = Point3::new(-3.0, 1.0, 2.0);
let midpoint: Point3<f32> = p0 + (p1 - p0) * 0.5;```

Breaking the expression up, and adding explicit types makes it clearer to see what is going on:

```let dv: Vector3<f32> = p1 - p0;
let half_dv: Vector3<f32> = dv * 0.5;
let midpoint: Point3<f32> = p0 + half_dv;```

## Converting between points and vectors

Points can be converted to and from displacement vectors using the `EuclideanSpace::{from_vec, to_vec}` methods. Note that under the hood these are implemented as inlined a type conversion, so should not have any performance implications.

## Associated Types

### `type Scalar: BaseNum`

The associated scalar over which the space is defined.

Due to the equality constraints demanded by `Self::Diff`, this is effectively just an alias to `Self::Diff::Scalar`.

### `type Diff: VectorSpace<Scalar = Self::Scalar>`

The associated space of displacement vectors.

## Required Methods

### `fn origin() -> Self`

The point at the origin of the Euclidean space.

### `fn from_vec(v: Self::Diff) -> Self`

Convert a displacement vector to a point.

This can be considered equivalent to the addition of the displacement vector `v` to to `Self::origin()`.

### `fn to_vec(self) -> Self::Diff`

Convert a point to a displacement vector.

This can be seen as equivalent to the displacement vector from `Self::origin()` to `self`.

### `fn dot(self, v: Self::Diff) -> Self::Scalar`

This is a weird one, but its useful for plane calculations.

## Provided Methods

### `fn midpoint(self, other: Self) -> Self`

Returns the middle point between two other points.

```use cgmath::prelude::*;
use cgmath::Point3;

let p = Point3::midpoint(
Point3::new(1.0, 2.0, 3.0),
Point3::new(3.0, 1.0, 2.0),
);```

### `fn centroid(points: &[Self]) -> Self`

Returns the average position of all points in the slice.

```use cgmath::prelude::*;
use cgmath::Point2;

let triangle = [
Point2::new(1.0, 1.0),
Point2::new(2.0, 3.0),
Point2::new(3.0, 1.0),
];

let centroid = Point2::centroid(&triangle);```

## Implementors

• `impl<S: BaseNum> EuclideanSpace for Point1<S>`
• `impl<S: BaseNum> EuclideanSpace for Point2<S>`
• `impl<S: BaseNum> EuclideanSpace for Point3<S>`