How To Draw A Parabola Between Two Points Godot

How to Draw a Parabola Between Two Points in Godot

This article will guide you through creating a parabola between two arbitrary points in Godot Engine. We'll leverage Godot's built-in functionalities and a bit of mathematical magic to achieve a smooth, visually appealing curve. This technique is useful for various game development scenarios, such as projectile trajectories, pathfinding, or creating aesthetically pleasing shapes.

Understanding the Parabola Equation

A parabola's shape is defined by a quadratic equation. The general form is y = ax² + bx + c. To define a parabola between two points, we need to find the values of a, b, and c. We will use two points and the apex (highest or lowest point) to solve for these coefficients. For simplicity, we'll assume the parabola's apex lies midway between the two given points. This gives us a symmetrical parabola.

Implementing in Godot using Line2D

We'll use Godot's Line2D node to draw the parabola. We can't directly draw a parabola with a single Line2D node; instead, we'll generate numerous points along the curve and connect them.

Steps:

1. Create the Scene: Add a Line2D node to your scene. This will be our parabola. We'll call it Parabola.

2. Define the Points: You'll need two Vector2 variables to store the start and end points of your parabola. You can set these in the Godot editor or dynamically in your script. Let's call them pointA and pointB.

3. Calculate the Apex: The apex (highest or lowest point) will be the midpoint between pointA and pointB. Calculate it as follows:

   var apex = (pointA + pointB) / 2
   

4. Calculate Coefficients (a, b, c): We'll use the three points (pointA, apex, pointB) to solve the system of three equations with three unknowns (a, b, c). Since we assume a symmetrical parabola, the equation simplifies considerably. We can use the pointA and apex coordinates to derive a:

   var a = (apex.y - pointA.y) / pow(apex.x - pointA.x, 2)
   

   Then b and c can be derived using the apex coordinates. Since this is a symmetric parabola with the apex on the midpoint between our two points, b will be equal to 0. c can be calculated as:

   var b = 0
   var c = apex.y - a * pow(apex.x, 2)
   

5. Generate Points for the Line2D: We'll generate multiple points along the parabola and add them to the Line2D's points array. This determines the resolution of the curve. A higher number of points will result in a smoother curve.

   func generate_parabola_points(num_points: int):
       var points = []
       for i in range(num_points + 1):
           var x = lerp(pointA.x, pointB.x, float(i) / num_points)
           var y = a * pow(x, 2) + b * x + c
           points.append(Vector2(x, y))
       return points
   

6. Update the Line2D: Finally, update the Line2D's points property with the generated points.

   $Parabola.points = generate_parabola_points(50) # 50 points for a smooth curve
   

Complete GDScript Example:

extends Node2D

export var pointA = Vector2(100, 100)
export var pointB = Vector2(300, 200)

onready var Parabola = $Parabola

func _ready():
    var apex = (pointA + pointB) / 2
    var a = (apex.y - pointA.y) / pow(apex.x - pointA.x, 2)
    var b = 0
    var c = apex.y - a * pow(apex.x, 2)

    func generate_parabola_points(num_points: int):
        var points = []
        for i in range(num_points + 1):
            var x = lerp(pointA.x, pointB.x, float(i) / num_points)
            var y = a * pow(x, 2) + b * x + c
            points.append(Vector2(x, y))
        return points

    Parabola.points = generate_parabola_points(50)

Remember to adjust the number of points in generate_parabola_points to control the smoothness of the parabola. This approach provides a fundamental understanding and implementation for drawing parabolas in Godot. Further refinements can be made to handle more complex scenarios, such as non-symmetric parabolas or parabolas with varying apex positions. Experiment and adapt this code to fit your specific game needs.