A simple Math class to calculate a point in 2D or 3D space lying on a parabola. And a more complex parabola controller that you can put on an object.

MathParabola.cs

using UnityEngine;
using System;

public class MathParabola
{

    public static Vector3 Parabola(Vector3 start, Vector3 end, float height, float t)
    {
        Func<float, float> f = x => -4 * height * x * x + 4 * height * x;

        var mid = Vector3.Lerp(start, end, t);

        return new Vector3(mid.x, f(t) + Mathf.Lerp(start.y, end.y, t), mid.z);
    }

    public static Vector2 Parabola(Vector2 start, Vector2 end, float height, float t)
    {
        Func<float, float> f = x => -4 * height * x * x + 4 * height * x;

        var mid = Vector2.Lerp(start, end, t);

        return new Vector2(mid.x, f(t) + Mathf.Lerp(start.y, end.y, t));
    }

}

ParabolaController.cs

using UnityEngine;
using System.Collections.Generic;

public class ParabolaController : MonoBehaviour
{
    /// <summary>
    /// Animation Speed
    /// </summary>
    public float Speed = 1;

    /// <summary>
    /// Start of Parabola
    /// </summary>
    public GameObject ParabolaRoot;

    /// <summary>
    /// Autostart Animation
    /// </summary>
    public bool Autostart = true;

    /// <summary>
    /// Animate
    /// </summary>
    public bool Animation = true;

    //next parabola event
    internal bool nextParbola = false;

    //animation time
    protected float animationTime = float.MaxValue;

    //gizmo
    protected ParabolaFly gizmo;

    //draw
    protected ParabolaFly parabolaFly;

    void OnDrawGizmos()
    {
        if (gizmo == null)
        {
            gizmo = new ParabolaFly(ParabolaRoot.transform);
        }

        gizmo.RefreshTransforms(1f);
        if ((gizmo.Points.Length - 1) % 2 != 0)
            return;

        int accur = 50;
        Vector3 prevPos = gizmo.Points[0].position;
        for (int c = 1; c <= accur; c++)
        {
            float currTime = c * gizmo.GetDuration() / accur;
            Vector3 currPos = gizmo.GetPositionAtTime(currTime);
            float mag = (currPos - prevPos).magnitude * 2;
            Gizmos.color = new Color(mag, 0, 0, 1);
            Gizmos.DrawLine(prevPos, currPos);
            Gizmos.DrawSphere(currPos, 0.01f);
            prevPos = currPos;
        }
    }


    // Use this for initialization
    void Start()
    {

        parabolaFly = new ParabolaFly(ParabolaRoot.transform);

        if (Autostart)
        {
            RefreshTransforms(Speed);
            FollowParabola();
        }
    }

    // Update is called once per frame
    void Update()
    {
        nextParbola = false;

        if (Animation && parabolaFly != null && animationTime < parabolaFly.GetDuration())
        {
            int parabolaIndexBefore;
            int parabolaIndexAfter;
            parabolaFly.GetParabolaIndexAtTime(animationTime, out parabolaIndexBefore);
            animationTime += Time.deltaTime;
            parabolaFly.GetParabolaIndexAtTime(animationTime, out parabolaIndexAfter);

            transform.position = parabolaFly.GetPositionAtTime(animationTime);

            if (parabolaIndexBefore != parabolaIndexAfter)
                nextParbola = true;

            //if (transform.position.y > HighestPoint.y)
            //HighestPoint = transform.position;
        }
        else if (Animation && parabolaFly != null && animationTime > parabolaFly.GetDuration())
        {
            animationTime = float.MaxValue;
            Animation = false;
        }

    }

    public void FollowParabola()
    {
        RefreshTransforms(Speed);
        animationTime = 0f;
        transform.position = parabolaFly.Points[0].position;
        Animation = true;
        //HighestPoint = points[0].position;
    }

    public Vector3 getHighestPoint(int parabolaIndex)
    {
        return parabolaFly.getHighestPoint(parabolaIndex);
    }

    public Transform[] getPoints()
    {
        return parabolaFly.Points;
    }

    public Vector3 GetPositionAtTime(float time)
    {
        return parabolaFly.GetPositionAtTime(time);
    }

    public float GetDuration()
    {
        return parabolaFly.GetDuration();
    }

    public void StopFollow()
    {
        animationTime = float.MaxValue;
    }
    /// <summary>
    /// Returns children transforms, sorted by name.
    /// </summary>
    public void RefreshTransforms(float speed)
    {
        parabolaFly.RefreshTransforms(speed);
    }


    public static float DistanceToLine(Ray ray, Vector3 point)
    {
        //see:http://answers.unity3d.com/questions/62644/distance-between-a-ray-and-a-point.html
        return Vector3.Cross(ray.direction, point - ray.origin).magnitude;
    }

    public static Vector3 ClosestPointInLine(Ray ray, Vector3 point)
    {
        return ray.origin + ray.direction * Vector3.Dot(ray.direction, point - ray.origin);
    }

    public class ParabolaFly
    {

        public Transform[] Points;
        protected Parabola3D[] parabolas;
        protected float[] partDuration;
        protected float completeDuration;

        public ParabolaFly(Transform ParabolaRoot)
        {

            List<Component> components = new List<Component>(ParabolaRoot.GetComponentsInChildren(typeof(Transform)));
            List<Transform> transforms = components.ConvertAll(c => (Transform)c);

            transforms.Remove(ParabolaRoot.transform);
            transforms.Sort(delegate (Transform a, Transform b)
            {
                return a.name.CompareTo(b.name);
            });

            Points = transforms.ToArray();

            //check if odd
            if ((Points.Length - 1) % 2 != 0)
                throw new UnityException("ParabolaRoot needs odd number of points");

            //check if larger is needed
            if (parabolas == null || parabolas.Length < (Points.Length - 1) / 2)
            {
                parabolas = new Parabola3D[(Points.Length - 1) / 2];
                partDuration = new float[parabolas.Length];
            }

        }

        public Vector3 GetPositionAtTime(float time)
        {
            int parabolaIndex;
            float timeInParabola;
            GetParabolaIndexAtTime(time, out parabolaIndex, out timeInParabola);

            var percent = timeInParabola / partDuration[parabolaIndex];
            return parabolas[parabolaIndex].GetPositionAtLength(percent * parabolas[parabolaIndex].Length);
        }

        public void GetParabolaIndexAtTime(float time, out int parabolaIndex)
        {
            float timeInParabola;
            GetParabolaIndexAtTime(time, out parabolaIndex, out timeInParabola);
        }

        public void GetParabolaIndexAtTime(float time, out int parabolaIndex, out float timeInParabola)
        {
            //f(x) = ax² + bx + c
            timeInParabola = time;
            parabolaIndex = 0;

            //determine parabola
            while (parabolaIndex < parabolas.Length - 1 && partDuration[parabolaIndex] < timeInParabola)
            {
                timeInParabola -= partDuration[parabolaIndex];
                parabolaIndex++;
            }
        }

        public float GetDuration()
        {
            return completeDuration;
        }

        public Vector3 getHighestPoint(int parabolaIndex)
        {
            return parabolas[parabolaIndex].getHighestPoint();
        }

        /// <summary>
        /// Returns children transforms, sorted by name.
        /// </summary>
        public void RefreshTransforms(float speed)
        {
            if (speed <= 0f)
                speed = 1f;

            if (Points != null)
            {

                completeDuration = 0;

                //create parabolas
                for (int i = 0; i < parabolas.Length; i++)
                {
                    if (parabolas[i] == null)
                        parabolas[i] = new Parabola3D();

                    parabolas[i].Set(Points[i * 2].position, Points[i * 2 + 1].position, Points[i * 2 + 2].position);
                    partDuration[i] = parabolas[i].Length / speed;
                    completeDuration += partDuration[i];
                }


            }

        }
    }

    public class Parabola3D
    {
        public float Length { get; private set; }

        public Vector3 A;
        public Vector3 B;
        public Vector3 C;

        protected Parabola2D parabola2D;
        protected Vector3 h;
        protected bool tooClose;

        public Parabola3D()
        {
        }

        public Parabola3D(Vector3 A, Vector3 B, Vector3 C)
        {
            Set(A, B, C);
        }

        public void Set(Vector3 A, Vector3 B, Vector3 C)
        {
            this.A = A;
            this.B = B;
            this.C = C;
            refreshCurve();
        }

        public Vector3 getHighestPoint()
        {
            var d = (C.y - A.y) / parabola2D.Length;
            var e = A.y - C.y;

            var parabolaCompl = new Parabola2D(parabola2D.a, parabola2D.b + d, parabola2D.c + e, parabola2D.Length);

            Vector3 E = new Vector3();
            E.y = parabolaCompl.E.y;
            E.x = A.x + (C.x - A.x) * (parabolaCompl.E.x / parabolaCompl.Length);
            E.z = A.z + (C.z - A.z) * (parabolaCompl.E.x / parabolaCompl.Length);

            return E;
        }

        public Vector3 GetPositionAtLength(float length)
        {
            //f(x) = ax² + bx + c
            var percent = length / Length;

            var x = percent * (C - A).magnitude;
            if (tooClose)
                x = percent * 2f;

            Vector3 pos;

            pos = A * (1f - percent) + C * percent + h.normalized * parabola2D.f(x);
            if (tooClose)
                pos.Set(A.x, pos.y, A.z);

            return pos;
        }

        private void refreshCurve()
        {

            if (Vector2.Distance(new Vector2(A.x, A.z), new Vector2(B.x, B.z)) < 0.1f &&
                Vector2.Distance(new Vector2(B.x, B.z), new Vector2(C.x, C.z)) < 0.1f)
                tooClose = true;
            else
                tooClose = false;

            Length = Vector3.Distance(A, B) + Vector3.Distance(B, C);

            if (!tooClose)
            {
                refreshCurveNormal();
            }
            else
            {
                refreshCurveClose();
            }
        }


        private void refreshCurveNormal()
        {
            //                        .  E   .
            //                   .       |       point[1]
            //             .             |h         |       .
            //         .                 |       ___v1------point[2]
            //      .            ______--vl------    
            // point[0]---------
            //

            //lower v1
            Ray rl = new Ray(A, C - A);
            var v1 = ClosestPointInLine(rl, B);

            //get A=(x1,y1) B=(x2,y2) C=(x3,y3)
            Vector2 A2d, B2d, C2d;

            A2d.x = 0f;
            A2d.y = 0f;
            B2d.x = Vector3.Distance(A, v1);
            B2d.y = Vector3.Distance(B, v1);
            C2d.x = Vector3.Distance(A, C);
            C2d.y = 0f;

            parabola2D = new Parabola2D(A2d, B2d, C2d);

            //lower v
            //var p = parabola.E.x / parabola.Length;
            //Vector3 vl = points[0].position * (1f - p) + points[2].position * p;

            //h
            h = (B - v1) / Vector3.Distance(v1, B) * parabola2D.E.y;
        }

        private void refreshCurveClose()
        {
            //distance to x0 - x2 line = |(x1-x0)x(x1-x2)|/|x2-x0|
            var fac01 = (A.y <= B.y) ? 1f : -1f;
            var fac02 = (A.y <= C.y) ? 1f : -1f;

            Vector2 A2d, B2d, C2d;

            //get A=(x1,y1) B=(x2,y2) C=(x3,y3)
            A2d.x = 0f;
            A2d.y = 0f;

            //b = sqrt(c²-a²)
            B2d.x = 1f;
            B2d.y = Vector3.Distance((A + C) / 2f, B) * fac01;

            C2d.x = 2f;
            C2d.y = Vector3.Distance(A, C) * fac02;

            parabola2D = new Parabola2D(A2d, B2d, C2d);
            h = Vector3.up;
        }
    }

    public class Parabola2D
    {
        public float a { get; private set; }
        public float b { get; private set; }
        public float c { get; private set; }

        public Vector2 E { get; private set; }
        public float Length { get; private set; }

        public Parabola2D(float a, float b, float c, float length)
        {
            this.a = a;
            this.b = b;
            this.c = c;

            setMetadata();
            this.Length = length;
        }

        public Parabola2D(Vector2 A, Vector2 B, Vector2 C)
        {
            //f(x) = ax² + bx + c
            //a = (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / ((x1 - x2)(x1 - x3)(x3 - x2))
            //b = (x1²(y2 - y3) + x2²(y3 - y1) + x3²(y1 - y2))/ ((x1 - x2)(x1 - x3)(x2 - x3))
            //c = (x1²(x2y3 - x3y2) + x1(x3²y2 - x2²y3) + x2x3y1(x2 - x3))/ ((x1 - x2)(x1 - x3)(x2 - x3))
            var divisor = ((A.x - B.x) * (A.x - C.x) * (C.x - B.x));
            if (divisor == 0f)
            {
                A.x += 0.00001f;
                B.x += 0.00002f;
                C.x += 0.00003f;
                divisor = ((A.x - B.x) * (A.x - C.x) * (C.x - B.x));
            }
            a = (A.x * (B.y - C.y) + B.x * (C.y - A.y) + C.x * (A.y - B.y)) / divisor;
            b = (A.x * A.x * (B.y - C.y) + B.x * B.x * (C.y - A.y) + C.x * C.x * (A.y - B.y)) / divisor;
            c = (A.x * A.x * (B.x * C.y - C.x * B.y) + A.x * (C.x * C.x * B.y - B.x * B.x * C.y) + B.x * C.x * A.y * (B.x - C.x)) / divisor;

            b = b * -1f;//hack

            setMetadata();
            Length = Vector2.Distance(A, C);
        }

        public float f(float x)
        {
            return a * x * x + b * x + c;
        }

        private void setMetadata()
        {
            //derive
            //a*x²+b*x+c = 0
            //2ax+b=0
            //x = -b/2a
            var x = -b / (2 * a);
            E = new Vector2(x, f(x));
        }



    }

}

Hi @ditzel
Thanks for providing this useful code!
I am trying to use it to drop items in my game, so I know exactly where the items land. But I can't seem to get the object to follow a downward parabola.
It seem that there is something in the code that looks for the lowest point in the parabola root and sets that as a starting point. Is this true, and if so can it be reversed so the highest point is the starting point?

Cheers and thanks again :)

Thanks for code!
I am trying to use it in WebGL but seems its not working, could you please let me know if anything need to be changed to make it work?

Thanks!

Thanks, @ditzel, Is there a way for the cube following the parabola to align to the parabola whilst following?

Many Thanks.

Hello!

Thank you so much for sharing this useful code.

can you please let me know from where have you derived this equation - ( -4 * height * x * x + 4 * height * x;)

Thanks in advance :)

Thanks for the code!

How would I be able to rotate the object to align with the direction of movement on the parabola?

Thanks in advance :D

Thank you so much, it was very usefull <3

Thanks a lot of work

thanks a lot for the code!

thanks for this code!