Using Parabolic Trajectories For Game Projectiles
Modeling Projectile Motion
Projectile motion in games often follows a parabolic trajectory. This allows game developers to simulate physics similar to the real world, with objects affected by gravity over time. Parabolic motion produces natural looking arcs for thrown or launched game objects.
Explaining parabolic trajectories in games
Parabolic trajectories result from an object being given an initial velocity vector while also being influenced by gravity over the course of its flight. This produces a distinctive arc shape to the object’s path. In games, modeling projectiles in this way creates immersive worlds that move and behave realistically.
The key factors defining a parabolic trajectory are the initial launch velocity (speed and angle), gravity, and any other forces such as air resistance or drag. By programming these physics effects, game developers can simulate the smooth arcing paths seen in reality as balls fly through the air or arrows soar towards targets.
Setting up the equation of motion
The foundation for modeling projectile motion is the kinematic equations that relate an object’s position, velocity, acceleration and time. Specifically, the equation defines the vertical and horizontal components of motion separately. This allows the trajectory to be broken down into x and y coordinates over time.
The horizontal motion equation ignores gravity and drag:
x = x0 + vxt
Where x0 is the starting horizontal position, vx is horizontal launch velocity, and t is elapsed time.
The vertical equation includes acceleration downward due to gravity:
y = y0 + vyt + (1/2)at2
Here, y0 is initial height, vy is vertical velocity, a is gravitational acceleration (-9.81 m/s), and t is time.
By calculating x and y over small time steps, the parabolic trajectory can be plotted one coordinate at a time. More advanced physics with drag forces can also be incorporated as needed.
Example code for calculating projectile motion
Here is some basic code for modeling parabolic motion in a game:
float gravity = -9.81f;
void UpdateProjectile(Projectile p) {
p.timeElapsed += Time.deltaTime;
float x = p.initialPosition.x + p.initialVelocity.x * p.timeElapsed;
float y = p.initialPosition.y +
p.initialVelocity.y * p.timeElapsed +
0.5 * gravity * p.timeElapsed * p.timeElapsed;
p.currentPosition = new Vector2(x, y);
}
This steps through the kinematic equations each frame to plot the updated x,y coordinates. Given an initial position, velocity, and constant gravity, the parabolic arc can be produced over time.
Implementing Gravity and Drag
Adding gravity to affect trajectory over time
The force of gravity causes a downwards vertical acceleration on projectiles during flight. This acceleration is what causes the object to eventually fall back to the ground. In games, modeling gravity is essential for realistic parabolic trajectories.
A constant gravitational force can be applied each frame as vertical acceleration on the object. Typically this acceleration value is set to -9.81 m/s^2, approximating Earth’s gravity:
vector.y += gravity * Time.deltaTime;
When integrated over time, this produces increasing downward velocity and displacement on the y-axis, resulting in a parabolic arc.
For non-Earth environments, the gravitational constant can be adjusted. For example, smaller physics objects may benefit from reduced gravity for a more pronounced trajectory.
Modeling air resistance with drag equations
Besides gravity, projectiles are also affected by drag forces when moving through air. Air resistance has the effect of reducing velocity over time, causing the trajectory arc to bend downwards more rapidly.
Drag can be modeled by applying a force proportional to the object’s current velocity. The drag equation is:
Fd = -1/2 ρ v2 Cd A
Where ρ is air density, v is velocity, Cd is drag coefficient, and A is reference area of the object. By setting these parameters, drag will drain energy from the system over time.
Code for applying gravity and drag forces
void ApplyPhysicsForces(Projectile projectile) {
// Apply constant gravitational acceleration
projectile.velocityY += gravity * Time.deltaTime;
// Calculate drag force
float drag = 0.5 * airDensity * projectile.velocitySq * projectile.dragCoeff * projectile.area;
// Apply drag to reduce velocity over time
projectile.velocityX -= drag * projectile.velocityX;
projectile.velocityY -= drag * projectile.velocityY;
}
Tuning drag coefficients and air density allows for realistic physics – light objects like arrows experience more air resistance, while heavy cannonballs cut through air more easily.
Applying Parabolic Motion to Game Objects
Making game objects move along parabolic paths
Once projectile physics are established through kinematic equations and forces like gravity and drag, game objects can have parabolic motion applied to their transformations each frame.
This is done by setting the object’s initial position and velocities based on a spawn point and launch parameters. Then each frame, the physics update steps are performed to calculate the new position from the equations of motion.
Over time, tracing the object’s updated positions will plot out a smooth parabolic curve through space, just like a thrown ball in the real world.
Setting initial launch velocity and angle
An initial launch velocity vector defines the starting speeds and direction that determine the size, shape and orientation of the trajectory parabola. Key parameters are:
- Speed – Directly scales range and arc height
- Angle – Controls how much the arc curves up then down
- x/y Components – Horizontal and vertical launch velocities
Launch speed can be constant, or vary based on input timing and charging. Launch angles typically range from 30-60 degrees for efficient ballistic trajectories. The x and y components can be calculated from direction angles and speeds.
Sample code for moving game objects in parabolic motion
void FireProjectile(Vector3 launchPosition, Vector2 launchVelocity) {
GameObject projectile = Instantiate(projectilePrefab, launchPosition, Quaternion.identity);
projectile.GetComponent<Rigidbody>().velocity = launchVelocity;
projectile.StartCoroutine(UpdateTrajectory(projectile));
}
IEnumerator UpdateTrajectory(GameObject obj) {
while(true) {
ApplyPhysicsForces(obj);
obj.transform.position += obj.velocity * Time.deltaTime;
yield return null;
}
}
This handles spawning, launching, and updating projectiles over time. By repeatedly applying velocity changes and moving the object based on physics calculations, realistic parabolic trajectories emerge.
Advanced Usage and Optimization
Efficiently calculating trajectories
For efficiency with many active projectiles, avoid expensive per-frame recalculations. Parabolic trajectories can be pre-computed based on predefined launch parameters. Store the resulting position arrays to reference instead of recalculating.
Interpolation between pre-calculated trajectory points allows fluid motion without heavy math each frame. Optimization strategies like object pooling can manage memory and performance across many simultaneous projectile objects.
Optimizing for large numbers of projectiles
Drawing trajectories through basic physics calculations can become costly when games involve dozens or hundreds of parabolic projectiles at once. Optimization strategies include:
- Object Pooling – Reuse inactive projectile instances instead of destroying
- Position Interpolation – Precache trajectory arrays instead of per-frame math
- GPU Physics – Calculate trajectories in parallel on the GPU for efficiency
These approaches alleviate pressure on the CPU by reducing per-frame calculations and leveraging GPU processing where possible for massive paralleization.
Improving performance with caching and pooling
Avoiding discarding and respawning gameobjects can save significant CPU work when dealing with many projectiles over a scene’s lifetime. Object pooling keeps unused projectiles in memory ready to be activated as needed.
Caching pre-computed trajectory data also reduces calculations, transforming positions by interpolating between precached values instead of repeated math. For large barrages of thousands of projectiles, these optimizations keep gameplay smooth and reactive.
