Lissajous Curves - 3D Voxel Animation Learning Example

This guide walks you through how to generate a looping 3D voxel animation of lissajous curves using SpatialStudio. The script creates beautiful mathematical curves that trace elegant paths through 3D space, then saves the animation to a .splv file.

What this script does

Creates a 3D scene of size 128×128×128
Generates multiple lissajous curves, each with:
- Smooth parametric motion following the equations: x=sin(at), y=sin(bt), z=sin(ct)
- Trailing particle effects that fade over time
- Vibrant colors that shift as the curves evolve
Animates the mathematical curves for 8 seconds at 30 FPS
Outputs the file lissajous.splv that you can play in your viewer

How it works (simplified)

Voxel volume Each frame is a 3D grid filled with RGBA values (SIZE × SIZE × SIZE × 4).
Lissajous equations The curves follow parametric equations where different frequency ratios create unique 3D patterns.
Particle trails Each curve leaves behind glowing particles that gradually fade, creating beautiful trail effects.
Color cycling Colors shift smoothly through the HSV spectrum as the curves trace their paths.
Animation loop A normalized time variable t cycles from 0 → 2π, ensuring the motion loops smoothly.
Encoding Frames are passed into splv.Encoder, which writes them into the .splv video file.

Try it yourself

Install requirements first:

pip install spatialstudio numpy tqdm

Then copy this script into lissajous.py and run:

python lissajous.py

Full Script

import numpy as np
from spatialstudio import splv
from tqdm import tqdm
import colorsys

# Scene setup
SIZE, FPS, SECONDS = 128, 30, 8
FRAMES = FPS * SECONDS
CENTER_X = CENTER_Y = CENTER_Z = SIZE // 2
OUT_PATH = "../outputs/lissajous.splv"

# Lissajous settings
CURVE_COUNT = 5
TRAIL_LENGTH = 60
CURVE_SCALE = 40

def add_voxel(volume, x, y, z, color, alpha=255):
    if 0 <= x < SIZE and 0 <= y < SIZE and 0 <= z < SIZE:
        volume[x, y, z, :3] = color
        volume[x, y, z, 3] = alpha

def hsv_to_rgb(h, s, v):
    r, g, b = colorsys.hsv_to_rgb(h, s, v)
    return (int(r * 255), int(g * 255), int(b * 255))

def generate_lissajous_point(t, curve_id):
    # Different frequency ratios for each curve
    freq_ratios = [
        (1, 2, 3), (2, 3, 1), (3, 1, 2), 
        (1, 3, 2), (2, 1, 3)
    ]
    a, b, c = freq_ratios[curve_id % len(freq_ratios)]
    
    # Phase offsets for variety
    phase_x = curve_id * 0.5
    phase_y = curve_id * 0.7
    phase_z = curve_id * 0.3
    
    x = np.sin(a * t + phase_x) * CURVE_SCALE
    y = np.sin(b * t + phase_y) * CURVE_SCALE
    z = np.sin(c * t + phase_z) * CURVE_SCALE
    
    return (
        int(CENTER_X + x),
        int(CENTER_Y + y),
        int(CENTER_Z + z)
    )

def generate_curve_trail(volume, curve_id, current_t, frame):
    trail_points = []
    
    # Generate trail points going back in time
    for i in range(TRAIL_LENGTH):
        trail_t = current_t - (i * 0.1)
        x, y, z = generate_lissajous_point(trail_t, curve_id)
        
        # Calculate fade based on distance from head
        fade = 1.0 - (i / TRAIL_LENGTH)
        fade = fade ** 2  # Exponential fade for better visual effect
        
        # Color shifts over time and varies by curve
        hue = (current_t * 0.1 + curve_id * 0.2) % 1.0
        saturation = 0.8 + 0.2 * np.sin(current_t * 2 + curve_id)
        value = fade * (0.5 + 0.5 * np.sin(current_t * 1.5 + i * 0.1))
        
        color = hsv_to_rgb(hue, saturation, value)
        alpha = int(fade * 255)
        
        # Add glow effect around main point
        for dx in range(-2, 3):
            for dy in range(-2, 3):
                for dz in range(-2, 3):
                    distance = np.sqrt(dx*dx + dy*dy + dz*dz)
                    if distance <= 2:
                        glow_fade = (2 - distance) / 2
                        glow_alpha = int(alpha * glow_fade)
                        if glow_alpha > 0:
                            add_voxel(volume, x+dx, y+dy, z+dz, color, glow_alpha)

def generate_curve_head(volume, curve_id, t):
    x, y, z = generate_lissajous_point(t, curve_id)
    
    # Bright head color
    hue = (t * 0.1 + curve_id * 0.2) % 1.0
    color = hsv_to_rgb(hue, 1.0, 1.0)
    
    # Larger, brighter head
    for dx in range(-3, 4):
        for dy in range(-3, 4):
            for dz in range(-3, 4):
                distance = np.sqrt(dx*dx + dy*dy + dz*dz)
                if distance <= 3:
                    brightness = max(0, 1 - distance/3)
                    bright_color = tuple(int(c * brightness) for c in color)
                    add_voxel(volume, x+dx, y+dy, z+dz, bright_color)

def add_sparkle_effects(volume, t):
    # Add some random sparkles for visual interest
    sparkle_count = 20
    for i in range(sparkle_count):
        # Pseudo-random positions based on time and index
        seed = t * 10 + i * 137  # 137 for better distribution
        sx = int((np.sin(seed * 0.7) * 0.4 + 0.5) * SIZE)
        sy = int((np.sin(seed * 0.9) * 0.4 + 0.5) * SIZE)
        sz = int((np.sin(seed * 1.1) * 0.4 + 0.5) * SIZE)
        
        # Sparkle brightness varies over time
        sparkle_brightness = (np.sin(t * 5 + i) + 1) * 0.5
        if sparkle_brightness > 0.7:  # Only show bright sparkles
            alpha = int((sparkle_brightness - 0.7) * 255 / 0.3)
            add_voxel(volume, sx, sy, sz, (255, 255, 255), alpha)

def generate_scene(volume, t, frame):
    # Generate all curves
    for curve_id in range(CURVE_COUNT):
        generate_curve_trail(volume, curve_id, t, frame)
        generate_curve_head(volume, curve_id, t)
    
    # Add sparkle effects
    add_sparkle_effects(volume, t)

enc = splv.Encoder(SIZE, SIZE, SIZE, framerate=FPS, outputPath=OUT_PATH, motionVectors="off")

for frame in tqdm(range(FRAMES), desc="Generating lissajous curves"):
    volume = np.zeros((SIZE, SIZE, SIZE, 4), dtype=np.uint8)
    t = (frame / FRAMES) * 4*np.pi  # Two full cycles for more complex patterns
    generate_scene(volume, t, frame)
    enc.encode(splv.Frame(volume, lrAxis="x", udAxis="y", fbAxis="z"))

enc.finish()
print(f"Created {OUT_PATH}")

Next steps

Change CURVE_COUNT to generate more or fewer curves.
Modify the freq_ratios to create different mathematical patterns.
Adjust TRAIL_LENGTH to make longer or shorter particle trails.
Experiment with CURVE_SCALE to make the curves larger or smaller.
Try different color schemes by modifying the HSV values.

Mathematical background

Lissajous curves are the result of combining two or more sinusoidal motions. In 3D, we use three equations:

X(t) = sin(a×t + φₓ)
Y(t) = sin(b×t + φᵧ)
Z(t) = sin(c×t + φᵤ)

The relationship between the frequencies (a, b, c) determines the shape complexity. When these ratios are simple integers, you get closed loops. More complex ratios create intricate, never-quite-repeating patterns that are mesmerizing to watch!

examples

Snake 3D

Creating Yoyo Animation

Creating Wooden Planks Animation

Creating Windstorm Animation

Creating Windmill Animation

Creating Whale Animation

Creating Waterfall Animation

Creating Volcano Animation

Creating Vaporwave Animation

Creating Tree Animation

Creating Tornado Animation

Creating Sun Animation

Creating Stone Wall Animation

Creating Spring Animation

Creating Spider Animation

Creating Spaceship Animation

Creating Solar System Narration.Txt Animation

Creating Solar System Educational Animation

Creating Solar System Animation

Creating Snowglobe Animation

Creating Snake Animation

Creating Script Generation Warning.Md Animation

Creating Sailboat Animation

Creating Robot Animation

Creating River Rapids Animation

Creating Rainbow Animation

Creating Prism Animation

Creating Pinwheel Animation

Creating Pine Tree Animation

Creating Phoenix Animation

Creating Pendulum Animation

Creating Octopus Animation

Creating Mushroom Animation

Creating Metronome Animation

Creating Meteor Animation

Creating Metal Plates Animation

Creating Lissajous Animation

Creating Lightning Animation

Creating Lighthouse Animation

Creating Lava Lamp Animation

Creating Lantern Animation

Creating Kite Animation

Creating Kaleidoscope Animation

Creating Jellyfish Animation

Creating Inner Planets Orbit Animation

Creating Hourglass Animation

Creating Hive Animation

Creating Grass Animation

Creating Geyser Animation

Creating Gear Animation

Creating Fountain Animation

Creating Flower Animation

Creating Fish Animation

Creating Fireworks Animation

Creating Fireflies Lissajous Fast.Splv Animation

Creating Fern Animation

Creating Earth Orbit Animation

Creating Dragon Animation

Creating Disco Ball Animation

Creating Demon Animation

Creating Crystallization Animation

Creating Crystal Animation

Creating Cosmic Formation Animation

Creating Coral Reef Animation

Creating Compass Animation

Creating Cloud Animation

Creating Clock Animation

Creating Castle Animation

Creating Carousel Animation

Creating Candle Animation

Creating Campfire Animation

Creating Cactus Animation

Creating Butterfly Animation

Creating Bush Animation

Creating Brick Wall Animation

Creating Blackhole Animation

Creating Beehive Animation

Creating Aurora Animation

🎈 Floating Balloons