Using Vector Cross Products For Fast 2D Turn Calculations
The problem: Slow turning calculations
In 2D video games, quickly rotating sprites to face their movement direction is a common requirement. However, the calculations involved in determining the angle and applying the rotation can be computationally expensive. When done inefficiently, these rotations can consume a large amount of processor time and cause sprites to stutter or turn sluggishly.
The root of this issue lies in the use of basic trigonometric functions. Calculating angles with sin, cos or atan2 can involve lengthy mathematical operations. Additionally, applying the rotations requires converting angles to quaternion representations before rotation matrices can be used. This process must be done every frame for every sprite that needs to rotate.
With dozens or even hundreds of rotating sprites, these calculations can quickly add up and strain the game’s performance. What’s needed is a faster way to find angles and apply smooth 2D rotations so sprites can turn quickly and responsively.
Introducing vector cross products
Definition and visualization
This is where vector cross products come in. Cross products provide an efficient way to calculate the angle between two vectors. Geometrically, when two vectors are crossed, the resulting vector is perpendicular to the plane containing the input vectors.
The length of this perpendicular vector corresponds to the sine of the angle between the original two vectors. Additionally, the sign of the output vector indicates whether the rotation from vector A to vector B is clockwise or counter-clockwise.
By taking the vectors pointing in a sprite’s current facing direction and desired movement direction, their cross product vector will point directly out of the screen. The length of this vector reveals the angle of rotation needed to line up the vectors.
Efficient angle and rotation calculations
Code-wise, cross products allow very fast angle calculations. We can take the Vector3 CrossProduct method to quickly find a signed angle we can use to rotate a sprite towards its movement direction. Since vectors encapsulate xyz components, we get this information almost for free from the positions of our game objects.
Additionally, these angles can be immediately used with Rotate or TransformRotation functions to adjust sprite orientations. No conversion to quaternions is necessary, keeping our code simple and clean.
By leveraging vector cross products this way, we skip right to the final angle value in one fast operation. This accelerates the critical turning calculations our games need to keep sprites responsive during gameplay.
Implementation in Unity
Creating vector variables
To see cross products in action, we will walk through an implementation example in Unity. First, we need to create Vector3 variables to represent our sprite’s facing direction and movement direction. We initialize these based on transform forward vectors and input move directions.
// Facing direction
Vector3 facingDir = transform.forward;
// Movement direction from inputs
Vector2 moveInput = new Vector2(Input.GetAxis("Horizontal"),
Input.GetAxis("Vertical"));
Vector3 moveDir = new Vector3(moveInput.x, 0, moveInput.y);
Calculating angles with Cross()
With our direction vectors set up, we can now take their cross product to find the angle between them. The signed angle value will indicate how far and in which direction to rotate to line up the vectors.
// Take cross product of facing/movement directions
Vector3 cross = Vector3.Cross(facingDir, moveDir);
// Extract the angle between them
float turnAngle = Vector3.SignedAngle(facingDir, moveDir, Vector3.forward);
That’s it! In two quick lines we’ve found the exact rotation necessary to orient our sprite. We can now use this angle immediately to perform the rotation.
Rotating gameobjects smoothly
Here we take the turning angle and apply it to smoothly swing the sprite towards the target movement direction. Using Rotate allows us to incrementally transition over time for natural turning animations.
// Rotate towards movement direction
transform.Rotate(0, turnAngle * Time.deltaTime);
And there we have it – fast responding 2D turning logic powered by vector cross product angles. Our sprite now rotates smoothly to face the player’s intended direction as soon as input is detected.
Optimization and considerations
Now that we have the basic implementation down, let’s go over some optimization techniques and special cases to be aware of.
Normalizing vectors
If your direction vectors have inconsistent magnitudes, you may want to normalize them before taking the cross product. This will constrain them to length 1 and ensure the angle calculations are consistent.
facingDir.Normalize();
moveDir.Normalize();
Vector3 cross = Vector3.Cross(facingDir, moveDir);
Handling edge cases
If your move direction is ever exactly 0, the cross product will result in a 0 angle as well. Make sure to check for and skip angle calculations in this edge case to avoid rotating unpredictably.
if (moveDir == Vector3.zero) {
continue;
}
// Take cross product
Performance gains
Since cross products require only a few fast vector operations, they are much faster than traditional angle calculations. You can expect 3-4x faster turns compared to methods using trig functions and quaternions. This frees up precious cycles to handle more sprites or improve other systems.
You will see the biggest gains when turning large batches of sprites simultaneously. Cross products really shine for fast parallel rotations across large groups of gameobjects.
Example code
Here is a full code sample implementing smooth sprite turning using optimized cross product rotations.
Vector3 class usage
Key Vector3 methods used:
Vector3.Cross // Take cross product
Vector3.SignedAngle // Extract signed angle
Vector3.Normalize() // Normalize vector length
transform.forward // Facing direction vector
transform.Rotate() // Apply rotation
Applying rotations
Main turning logic:
// Input direction
Vector2 inputDir = new Vector2(Input.GetAxis("Horizontal"),
Input.GetAxis("Vertical"));
// Facing direction
Vector3 facingDir = transform.forward;
// Movement direction
Vector3 moveDir = new Vector3(inputDir.x, 0, inputDir.y);
moveDir.Normalize();
// Zero input check
if (moveDir == Vector3.zero) {
continue;
}
// Angle between vectors
float turnAngle = Vector3.SignedAngle(facingDir, moveDir, Vector3.forward);
// Rotate sprite smoothly
transform.Rotate(0, turnAngle * Time.deltaTime);
Putting it all together
The complete code in action on a simple player sprite:
void Update() {
// Input direction
Vector2 inputDir = new Vector2(Input.GetAxis("Horizontal"),
Input.GetAxis("Vertical"));
// Facing direction
Vector3 facingDir = transform.forward;
// Movement direction
Vector3 moveDir = new Vector3(inputDir.x, 0, inputDir.y);
moveDir.Normalize();
// Zero input check
if (moveDir == Vector3.zero) {
return;
}
// Angle between vectors
float turnAngle = Vector3.SignedAngle(facingDir, moveDir,
Vector3.forward);
// Rotate sprite towards movement
transform.Rotate(0, turnAngle * Time.deltaTime);
}
This allows the sprite to smoothly rotate to face the target direction resulting from player input. The rotations are powered by efficient cross product angles for fast turning calculations.
Conclusion: Leveraging vectors for fast 2D turns
Rotating 2D sprites responsively is crucial for enjoyable gameplay feel. However, the math behind turning calculations can be costly. Converting angles to quaternions and applying trig functions every frame adds up quickly.
Vector cross products provide a fast and elegant solution to this problem. By finding the angle between direction vectors directly, we skip right to the final rotation amount in one vector operation.
Combined with SmoothDamp angle rotations, cross products enable buttery smooth turns perfect for controlling on-screen characters. Your players will appreciate the instant visual feedback as sprites orient themselves fluidly towards user input.
So next time you need to add snappy rotations to your 2D gameobjects, reach for vector cross products. Your code and framerates will thank you!
