Drawing random paths with quadratic bezier curves

Question

I want to draw smooth* and random paths that my objects would follow and I decided to go with quadratic bezier curves (but I'm open to other ideas).

My code is moving my objects in a random, but not smooth* way.

Preview: https://youtu.be/Eg9PEKuH4zA

My question is: how can I make the direction changes smoother? Should I completely ditch my Bezier solution or is there a way I could polish my code to achieve what I want?

*smooth == no abrupt direction changes

My code:

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

public class Lights : MonoBehaviour {
    public float paintHeight = -90.0f;
    public static int NUMBER_OF_LIGHTS = 3;
    private static int BEZIER_PATH_POINTS = 100;
    private float GOLDEN_COLOR_RATIO = 0.618033988749895f;
    private Light[] lights = new Light[NUMBER_OF_LIGHTS];

    Vector3 RandomPoint() {
        float obj_width = gameObject.GetComponent<RectTransform>().rect.width;
        float screenX = Random.Range(-obj_width / 2, obj_width / 2);

        float obj_height = gameObject.GetComponent<RectTransform>().rect.height;
        float screenY = Random.Range(-obj_height / 2, obj_height / 2);

        return new Vector3(screenX, screenY, -paintHeight);
    }


    Vector3 QuadraticBezierPoint(Vector3 startPoint, Vector3 endPoint, Vector3 vertexPoint, float t) {
        /*
         *         vertex
         *        /╲
         *       /   ╲
         *      /  p   ╲
         *     /  . .    ╲
         *    / .      ·   ╲
         *   /·           ·  ╲
         *  start           · end
         *
         *  0 < t < 1
         *
         * B(t) = (1 - t)^2 * P0 + 2 * (1-t) * t * P1 + t^2 * P2
         * 
         */
        return Mathf.Pow((1 - t), 2) * startPoint + 2 * (1 - t) * t * vertexPoint + Mathf.Pow(t, 2) * endPoint;
    }

    Color RandomColor() {
        float h = Random.Range(0.0f, 1.0f) + GOLDEN_COLOR_RATIO;
        h %= 1;
        return Color.HSVToRGB(h, 0.99f, 0.99f);
    }

    void Start() {
        for (int i = 0; i < NUMBER_OF_LIGHTS; i++) {
            GameObject light_obj = new GameObject();
            Light light = light_obj.AddComponent<Light>();
            light.type = LightType.Point;
            light.range = 10.0f;
            light.intensity = 3.5f;
            light.renderMode = LightRenderMode.ForcePixel;
            light.name = "Light" + i;
            light.transform.parent = gameObject.transform;
            lights[i] = light;
        }

        StartCoroutine("Move");
    }

    IEnumerator Move () {
        Dictionary<string, Vector3>[] light_points = new Dictionary<string, Vector3>[NUMBER_OF_LIGHTS];
        Dictionary<string, Color>[] light_colors = new Dictionary<string, Color>[NUMBER_OF_LIGHTS];
        for (int i = 0; i < NUMBER_OF_LIGHTS; i++) {
            light_points[i] = new Dictionary<string, Vector3>();
            light_colors[i] = new Dictionary<string, Color>();
            //light_points[i]["startPoint"] = RandomPoint();
            //light_points[i]["vertexPoint"] = RandomPoint();
            light_points[i]["endPoint"] = RandomPoint();
            light_colors[i]["nextColor"] = RandomColor();
        }

        while(true) {
            for (int i = 0; i < NUMBER_OF_LIGHTS; i++) {
                light_points[i]["startPoint"] = light_points[i]["endPoint"];
                light_points[i]["vertexPoint"] = RandomPoint();
                light_points[i]["endPoint"] = RandomPoint();

                light_colors[i]["currentColor"] = light_colors[i]["nextColor"];
                light_colors[i]["nextColor"] = RandomColor();
            }

            for (int i = 0; i < BEZIER_PATH_POINTS; i++) {
                float percent = (float)i / BEZIER_PATH_POINTS;
                for (int j = 0; j < NUMBER_OF_LIGHTS; j++) {
                    lights[j].transform.localPosition = QuadraticBezierPoint(
                        light_points[j]["startPoint"],
                        light_points[j]["endPoint"],
                        light_points[j]["vertexPoint"],
                        percent
                    );
                    lights[j].color = Color.Lerp(light_colors[j]["currentColor"], light_colors[j]["nextColor"], percent);
                }
                yield return new WaitForSeconds(0.02f);
            }
        }
    }
}

Answer 1

If you have a curve going from A to B with some middle point, you'll have to select a C accordingly so that the angle (A,B,C) at B is not too small or too big.

Additionally you'll have to select the middle points in an appropriate way, see e.g. https://www.particleincell.com/2012/bezier-splines/ , so that the change in direction is not too big.