Convert 3D to Spherical Coordinates

2024-01-16 20:45 UTC gpt-4 Open in ChatGPT ↗

In Unity, converting a 3D direction vector to its 2D spherical coordinates representation involves a process of deriving the azimuthal angle (θ) and the polar angle (φ). Spherical coordinates are often represented as a radius (r), azimuthal angle (θ), and polar angle (φ). However, in this case, since we’re dealing with direction vectors, the radius (r) is generally considered to be 1 (as direction vectors are typically normalized).

Here is a step-by-step explanation and the corresponding code to achieve this in Unity:

Normalize the Direction Vector: Ensure that the vector is normalized. This means the vector should have a length of 1. This is important because spherical coordinates assume a unit radius.
```
Vector3 direction = new Vector3(x, y, z).normalized;
```
Calculate the Azimuthal Angle (θ): This is the angle between the x-axis and the projection of the vector onto the x-z plane. It’s calculated using the Atan2 function.
```
float theta = Mathf.Atan2(direction.z, direction.x);
```
Calculate the Polar Angle (φ): This is the angle between the y-axis and the direction vector. It can be calculated using the Acos function on the y-component of the vector.
```
float phi = Mathf.Acos(direction.y);
```
Convert Angles to Degrees: (Optional) Unity often works with degrees, so you might want to convert these angles from radians to degrees.
```
theta *= Mathf.Rad2Deg;
phi *= Mathf.Rad2Deg;
```
Handle the Result: Now, theta and phi are your spherical coordinates. You can return them, print them, or use them as needed in your application.

Here’s how the complete code might look:

public static (float theta, float phi) DirectionToSpherical(Vector3 direction)
{
    direction.Normalize(); // Ensure the vector is normalized
    float theta = Mathf.Atan2(direction.z, direction.x);
    float phi = Mathf.Acos(direction.y);

    // Optionally convert to degrees
    theta *= Mathf.Rad2Deg;
    phi *= Mathf.Rad2Deg;

    return (theta, phi);
}

Remember, the output theta and phi are in degrees if you have converted them. These angles represent the direction in spherical coordinates relative to the Unity coordinate system.

To rotate a vector (in this case, the vector $(1, 0, 0)$) so that it aligns with a given direction in Unity, you can use quaternion rotation. In your scenario, the goal is to align the $x$-axis (represented by the vector $(1, 0, 0)$) with a given direction vector.

Here is how you can achieve this:

Normalize the Target Direction: Ensure that the direction vector you want to align with is normalized.
Calculate the Rotation Quaternion: Use Quaternion.FromToRotation to create a rotation that rotates one vector into another. In your case, you’ll rotate from $(1, 0, 0)$ to your target direction.
Apply the Rotation: Rotate the $(1, 0, 0)$ vector using the calculated quaternion.

Here’s the C# code for Unity:

public static Vector3 RotateVectorToMatchDirection(Vector3 targetDirection)
{
    // Normalize the target direction
    targetDirection.Normalize();

    // Create a rotation from (1, 0, 0) to the target direction
    Quaternion rotation = Quaternion.FromToRotation(new Vector3(1, 0, 0), targetDirection);

    // Apply the rotation to the (1, 0, 0) vector
    Vector3 rotatedVector = rotation * new Vector3(1, 0, 0);

    return rotatedVector;
}

In this code, targetDirection is your original direction vector to which you want to align. The function RotateVectorToMatchDirection will return a new vector that has been rotated to match the direction of targetDirection.

Keep in mind that the targetDirection must be a normalized direction vector (i.e., a vector with a length of 1) for this to work correctly. This method uses quaternion rotation, which is a robust and efficient way to handle 3D rotations in Unity.

To decompose a given direction vector into two rotations around two arbitrary, perpendicular axes, you need to create a function that computes how much rotation is needed around each of these axes to align an initial vector (like $(1, 0, 0)$) with the target direction. This task can be a bit complex due to the nature of 3D rotations.

Here’s a conceptual approach to creating such a function in Unity:

Normalize the Inputs: Ensure that the direction vector and the two axes are normalized.
First Rotation: Align the initial vector with the projection of the target direction on the plane defined by the two axes.
Second Rotation: Rotate the now aligned vector to the target direction.
Compute the Rotations: Use quaternion rotations to represent these transformations.

Here is a potential implementation in C# for Unity:

public static (Quaternion rotationAroundAxis1, Quaternion rotationAroundAxis2) DecomposeDirection(Vector3 direction, Vector3 axis1, Vector3 axis2)
{
    // Normalize the inputs
    direction.Normalize();
    axis1.Normalize();
    axis2.Normalize();

    // Project the direction onto the plane defined by the two axes
    Vector3 projectionOnPlane = Vector3.ProjectOnPlane(direction, axis1);

    // First rotation: align the initial vector with the projection on the plane
    Quaternion rotationToPlane = Quaternion.FromToRotation(Vector3.right, projectionOnPlane);

    // Second rotation: align the now aligned vector to the target direction
    Vector3 rotatedProjection = rotationToPlane * Vector3.right;
    Quaternion rotationToDirection = Quaternion.FromToRotation(rotatedProjection, direction);

    // Decompose the second rotation into rotations around axis1 and axis2
    Quaternion rotationAroundAxis1 = Quaternion.AngleAxis(rotationToDirection.eulerAngles.y, axis1);
    Quaternion rotationAroundAxis2 = Quaternion.AngleAxis(rotationToDirection.eulerAngles.x, axis2);

    return (rotationAroundAxis1, rotationAroundAxis2);
}

This function returns two quaternions: one for the rotation around axis1 and one for the rotation around axis2.

A few important notes:

This implementation assumes that axis1 and axis2 are perpendicular to each other and are in the plane orthogonal to the initial vector $(1, 0, 0)$. If they are not, the method will not work as expected.
The decomposition of the final rotation into two parts around the two axes may not be unique, especially in 3D rotations. This function provides one of the possible decompositions.
The function uses Euler angles to decompose the rotation. While Euler angles are intuitive, they can sometimes lead to issues like gimbal lock. Depending on your application, you might need a more robust solution.