Unity: Rotating a RigidBody2D to face the cursor, smoothly

FaceCursor.cs

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class FaceCursor : MonoBehaviour {
    // 100 seems to work fine on an object with mass around 1kg.
    [Tooltip("Amount of torque to apply when turning")]
    public float turnSpeed;

    private Rigidbody2D body;
    void Start() {
        body = GetComponent<Rigidbody2D>();
    }

    void FixedUpdate() {
        var gameObjectToMouse = Input.mousePosition - Camera.main.WorldToScreenPoint(transform.position);

        var targetAngle = wrapAngleAroundZero(Mathf.Atan2(gameObjectToMouse.y, gameObjectToMouse.x) * Mathf.Rad2Deg);
        var currentAngle = wrapAngleAroundZero(transform.eulerAngles.z);

        var torque = turnSpeed * Time.fixedDeltaTime;
        // I have no idea what this actually is or what to call it, but it works...
        var angularDelta = torque / body.inertia;

        // How long would it take us to stop? We need this in case we should actually be slamming on the brakes.
        var timeRequiredToStop = Mathf.Abs(body.angularVelocity / angularDelta * Time.fixedDeltaTime);

        // Which direction should we go? Depends on which way is faster.
        // This doesn't factor in current speed or direction, but eh, close enough.
        var timeUntilDestinationReachedCCW = (targetAngle - currentAngle) / Mathf.Abs(body.angularVelocity);
        var timeUntilDestinationReachedCW = -timeUntilDestinationReachedCCW;

        // This is best explained with an example.
        // If you're at -135°, travelling at -45°/s, and trying to reach -90°, how long will that take?
        // Since you're going counterclockwise, that's (-90 - -135) / -45 = -1 second, which doesn't make any
        // sense, except indicates that you overshot it. So you have to go all the way around, which takes
        // 360 / 45 = 8 seconds. So if you add 8 to -1, you get 7 seconds, which makes sense.
        var circleRotationTime = 360 / Mathf.Abs(body.angularVelocity);
        if (timeUntilDestinationReachedCCW < 0) timeUntilDestinationReachedCCW += circleRotationTime;
        if (timeUntilDestinationReachedCW < 0) timeUntilDestinationReachedCW += circleRotationTime;
        var timeUntilDestination = Mathf.Min(timeUntilDestinationReachedCCW, timeUntilDestinationReachedCW);

        if (timeRequiredToStop > timeUntilDestination) {
            // If we're close, slam on the brakes!
            // If it takes us 2.5 seconds to stop, but 2.3 seconds until we arrive, well we better try to stop.
            body.AddTorque(-1 * Mathf.Sign(body.angularVelocity) * torque);
        } else if (timeUntilDestinationReachedCW < timeUntilDestinationReachedCCW) {
            body.AddTorque(-torque);
        } else {
            body.AddTorque(torque);
        }
    }

    // Clamps to [-180, 180]
    private static float wrapAngleAroundZero(float a) {
        if (a >= 0) {
            float rotation = a % 360;
            if (rotation > 180) rotation -= 360;
            return rotation;
        } else {
            float rotation = -a % 360;
            if (rotation > 180) rotation -= 360;
            return -rotation;
        }
    }
}