/math-essentials

name: math-essentials
description: Use when implementing game math — vectors, transforms, interpolation, curves, random number generation, and common geometric recipes

Game Math in Godot 4.3+

All examples target Godot 4.3+ with no deprecated APIs. GDScript is shown first, then C#.

> **Related skills:** **player-controller** for movement physics, **ai-navigation** for pathfinding math, **camera-system** for camera interpolation, **tween-animation** for easing curves, **physics-system** for collision math.

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1. Vector Operations

Essential Vector Methods

| Method | Returns | Description | |-------------------------|-----------|-----------------------------------------------| | `length()` | `float` | Magnitude of the vector | | `length_squared()` | `float` | Squared magnitude (faster, skip sqrt) | | `normalized()` | `Vector` | Unit vector (length 1) in the same direction | | `distance_to(b)` | `float` | Distance between two points | | `distance_squared_to(b)` | `float` | Squared distance (faster for comparisons) | | `direction_to(b)` | `Vector` | Normalized direction from this to b | | `angle_to(b)` | `float` | Angle in radians between two vectors | | `angle_to_point(b)` | `float` | Angle from this point to b (2D) | | `dot(b)` | `float` | Dot product | | `cross(b)` | `float/Vector3` | Cross product (2D returns float, 3D returns vector) | | `rotated(angle)` | `Vector2` | Rotated by radians (2D) | | `move_toward(to, delta)` | `Vector` | Move toward target by at most delta | | `clamp(min, max)` | `Vector` | Clamp each component | | `snapped(step)` | `Vector` | Snap to grid | | `reflect(normal)` | `Vector` | Reflect off a surface | | `bounce(normal)` | `Vector` | Bounce off a surface (inverted reflect) | | `slide(normal)` | `Vector` | Slide along a surface |

Direction and Distance

# Get direction from A to B (normalized)
var dir: Vector2 = global_position.direction_to(target.global_position)

# Get distance
var dist: float = global_position.distance_to(target.global_position)

# Use squared distance for comparisons (faster — avoids sqrt)
if global_position.distance_squared_to(target.global_position) < detection_range * detection_range:
    chase_target()

Vector2 dir = GlobalPosition.DirectionTo(target.GlobalPosition);
float dist = GlobalPosition.DistanceTo(target.GlobalPosition);

if (GlobalPosition.DistanceSquaredTo(target.GlobalPosition) < detectionRange * detectionRange)
    ChaseTarget();

Dot Product

The dot product tells you how aligned two vectors are.

# Is the target in front of us? (dot > 0 = in front, < 0 = behind)
var forward: Vector2 = Vector2.RIGHT.rotated(rotation)
var to_target: Vector2 = global_position.direction_to(target.global_position)
var dot: float = forward.dot(to_target)

if dot > 0.7:  # roughly within ~45° cone
    print("Target is ahead")
elif dot < -0.7:
    print("Target is behind")

Vector2 forward = Vector2.Right.Rotated(Rotation);
Vector2 toTarget = GlobalPosition.DirectionTo(target.GlobalPosition);
float dot = forward.Dot(toTarget);

if (dot > 0.7f) GD.Print("Target is ahead");

Cross Product (3D)

The cross product gives a vector perpendicular to two input vectors.

# Get the surface normal from two edge vectors
var edge1: Vector3 = vertex_b - vertex_a
var edge2: Vector3 = vertex_c - vertex_a
var normal: Vector3 = edge1.cross(edge2).normalized()

Vector3 edge1 = vertexB - vertexA;
Vector3 edge2 = vertexC - vertexA;
Vector3 normal = edge1.Cross(edge2).Normalized();

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2. Transforms

Transform2D

A 2D transform holds position, rotation, and scale.

# Get the global transform
var xform: Transform2D = global_transform

# Convert between local and global space
var local_point: Vector2 = to_local(global_point)
var world_point: Vector2 = to_global(local_point)

# Apply transform to a point
var transformed: Vector2 = xform * Vector2(10, 0)  # point in local space → global

# Inverse transform
var local: Vector2 = xform.affine_inverse() * global_point

Transform2D xform = GlobalTransform;
Vector2 localPoint = ToLocal(globalPoint);
Vector2 worldPoint = ToGlobal(localPoint);
Vector2 transformed = xform * new Vector2(10, 0);
Vector2 local = xform.AffineInverse() * globalPoint;

Transform3D & Basis

# Basis holds rotation and scale as 3 column vectors
var basis: Basis = global_transform.basis

# Forward direction (looking along -Z in Godot)
var forward: Vector3 = -basis.z
var right: Vector3 = basis.x
var up: Vector3 = basis.y

# Look at a target
look_at(target.global_position, Vector3.UP)

# Rotate around an axis
rotate_y(deg_to_rad(90.0))
rotate_object_local(Vector3.UP, deg_to_rad(45.0))

# Interpolate between two transforms (smooth transition)
var a: Transform3D = $Start.global_transform
var b: Transform3D = $End.global_transform
global_transform = a.interpolate_with(b, 0.5)  # halfway

Basis basis = GlobalTransform.Basis;
Vector3 forward = -basis.Z;
Vector3 right = basis.X;
Vector3 up = basis.Y;

LookAt(target.GlobalPosition, Vector3.Up);
RotateY(Mathf.DegToRad(90.0f));

Transform3D a = GetNode<Node3D>("Start").GlobalTransform;
Transform3D b = GetNode<Node3D>("End").GlobalTransform;
GlobalTransform = a.InterpolateWith(b, 0.5f);

is_orthonormal() (Godot 4.7+)

`Basis.is_orthonormal()` (const) returns `true` if the basis is *orthogonal* (axes perpendicular to each other) **and** *normalized* (every axis has length `