A Spirograph is a geometric drawing device that produces mathematical curves known as hypotrochoids (internal rolling) and epitrochoids (external rolling). It works by rolling a smaller circle (rolling circle, radius r) inside or outside a larger fixed circle (radius R), while a pen at distance d from the center of the rolling circle traces intricate patterns. The ratio R/r determines the number of "petals" or lobes in the resulting design.

R (Fixed Circle Radius): Sets the overall size of the pattern. Larger R creates bigger designs.
r (Rolling Circle Radius): Controls the complexity. The ratio R/r determines how many lobes the pattern has before it closes. For example, R=100, r=60 gives a 5-lobed pattern (ratio 5:3).
d (Pen Offset): The distance from the rolling circle's center to the pen. When d < r, the pattern is smoother with rounded lobes. When d = r, you get sharp cusps (a true hypocycloid). When d > r, the pattern creates loops and self-intersections.

Hypocycloid (Inside mode): The rolling circle rolls along the inside of the fixed circle. This is the classic Spirograph experience, producing patterns that stay within the fixed circle's boundary.
Epicycloid (Outside mode): The rolling circle rolls along the outside of the fixed circle. This creates larger, more expansive patterns that extend beyond the fixed circle, often resembling flower petals radiating outward.

The pattern closes when the rolling circle completes an exact number of rotations around the fixed circle. If the ratio R/r is a rational number (like 5/3 or 7/2), the pattern will eventually return to its starting point after a finite number of orbits. Specifically, if R/r = p/q in lowest terms, the pattern closes after q full orbits. If R/r is irrational, the pattern theoretically never closes and would fill the entire available space over infinite time.

Yes! Click the PNG button to download your current design as a high-quality PNG image. The download captures exactly what you see on the canvas. For best results, let the drawing animation complete fully before downloading to get the complete closed pattern. You can also use the Clear button to start fresh and experiment with different parameter combinations.

• Experiment with ratios: Try R/r ratios like 3/1 (triangle), 5/3 (5 petals), 7/3 (7 petals), or 8/5 for fascinating results.
• Adjust pen offset: Setting d close to r creates dramatic sharp cusps. Smaller d values produce softer, more rounded shapes.
• Use Rainbow mode: The color-shifting effect makes patterns visually stunning and highlights the mathematical structure.
• Try both modes: Switch between Inside and Outside rolling to see completely different pattern families.
• Thin lines work best: A line width of 1-2px usually produces the most elegant results for complex patterns.