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Spaces & Cross Product

Spaces​

  • World Space
  • Local Space

Every transform has their own local space.

Local space is just another coordinate system.

Basis vectors usually have a unit of 1.

Space Transform​

Below shows how to transform a point between different spaces.

public Vector2 WorldSpacePoint;
public Vector2 LocalSpacePoint;
private void OnDrawGizmos()
{
    Vector2 position = transform.position;
    Vector2 up = transform.up;
    Vector2 right = transform.right;


    Vector2 LocalToWorld(Vector2 localSpacePoint, Vector2 up, Vector2 right)
    {
        return position + (right * localSpacePoint.x + up * localSpacePoint.y);
    }
    
    Vector2 WorldToLocal(Vector2 worldSpacePoint, Vector2 up, Vector2 right)
    {
        Vector2 originToPoint = worldSpacePoint - position;
        float x = Vector2.Dot(originToPoint, right);
        float y = Vector2.Dot(originToPoint, up);
        return new Vector2(x,y);
    }

    Gizmos.DrawSphere(LocalToWorld(LocalSpacePoint, up, right), 0.1f);

    Vector2 localPoint = WorldToLocal(WorldSpacePoint, up, right);
    objTf.localPosition = localPoint;

    DrawBasisVector(position, up, right);
    DrawBasisVector(Vector2.zero, Vector2.up, Vector2.right);
}

Matrix​

Game engine uses a matrix to store rotation, scale and position of a GameObject's transform. see 1:52:00 of the video.

We can use matrix to transform vector from local space to world space or vice versa.

Cross Product​

The cross product of two vectors is a vector that is perpendicular to the two input vectors.

Cross product visualization

In Unity, the direction of the cross product is determined by the left hand rule.

  • 👍 Thumb finger - x
  • 👆 Index finger - y
  • 🖕 Middle finger - z

The magnitude of the resultant vector is the area of the parallelogram determiend by the two input vectors.

Cross product magnitude

info

Use cases:

  • Place items on terrain.

References​