When does the pattern close and repeat to form a finite shape?

The curve closes and repeats, forming a finite rosette, when the ratio R/r is a rational number (a fraction of two integers). The number of times the rolling circle must complete a full revolution before the pen returns to its starting point determines the number of lobes or points in the final design. The simulator often provides a hint about this period.

What is the difference between a hypotrochoid and an epitrochoid?

A hypotrochoid is generated when the rolling circle moves along the *inside* of the fixed circle, like a coin rolling inside a hula hoop. An epitrochoid is generated when the rolling circle moves along the *outside* of the fixed circle, like a planet's epicycle. The parametric equations and the resulting families of shapes are distinct for each case.

What happens if the pen distance 'd' is equal to the rolling radius 'r'?

When d = r, the pen is located on the circumference of the rolling circle. In this special case, a hypotrochoid becomes a hypocycloid and an epitrochoid becomes an epicycloid. These curves have sharp cusps instead of loops or smoothed corners, as the pen touches the fixed circle during its motion.

Are these curves just for toys, or do they appear in real-world applications?

Absolutely. Trochoidal shapes are fundamental in engineering and nature. They describe the motion of gears (forming the shape of gear teeth), the path of a piston in a Wankel rotary engine, and even the orbits of celestial bodies in certain historical astronomical models. The Spirograph is an accessible introduction to this important geometry.

Spirograph (Trochoids)

Hypo- or epitrochoid: fixed R, rolling r, pen d; hue trail and period hints.

About this model

A Spirograph toy creates intricate, looping patterns by tracing a point attached to a smaller circle rolling around the inside or outside of a larger, fixed circle. This simulator mathematically models that process as a family of curves known as trochoids. Specifically, it generates hypotrochoids (where the small circle rolls inside the large circle) and epitrochoids (where it rolls outside). The curve's shape is determined by three parameters: the radius R of the fixed circle, the radius r of the rolling circle, and the distance d from the center of the rolling circle to the tracing pen point. The parametric equations governing the pen's position (x, y) over time t are: For an epitrochoid: x = (R + r) cos(t) - d cos(((R + r)/r) t); y = (R + r) sin(t) - d sin(((R + r)/r) t). For a hypotrochoid: x = (R - r) cos(t) + d cos(((R - r)/r) t); y = (R - r) sin(t) - d sin(((R - r)/r) t). The model simplifies reality by assuming perfect rolling without slip, ignoring friction and physical toy constraints. By interacting with R, r, and d, students explore how these ratios control the curve's symmetry, number of lobes, and periodicity. The visual feedback, including hue trails and period hints, helps learners connect abstract trigonometric functions to tangible geometric patterns, reinforcing concepts of parametric equations, periodic motion, and the relationship between rational numbers (like R/r) and closed curves.

Who it's for: High school and early undergraduate students studying pre-calculus, trigonometry, or parametric equations, as well as educators seeking a dynamic visualization of cyclic curves.

Key terms

Trochoid
Hypotrochoid
Epitrochoid
Parametric Equations
Rolling Circle
Locus
Periodicity
Spirograph

How it works

Trochoids are the clean cousin of Lissajous: one parameter traces a gear-mounted pen, producing roses, stars, and dense fills.

Frequently asked questions

When does the pattern close and repeat to form a finite shape?: The curve closes and repeats, forming a finite rosette, when the ratio R/r is a rational number (a fraction of two integers). The number of times the rolling circle must complete a full revolution before the pen returns to its starting point determines the number of lobes or points in the final design. The simulator often provides a hint about this period.
What is the difference between a hypotrochoid and an epitrochoid?: A hypotrochoid is generated when the rolling circle moves along the *inside* of the fixed circle, like a coin rolling inside a hula hoop. An epitrochoid is generated when the rolling circle moves along the *outside* of the fixed circle, like a planet's epicycle. The parametric equations and the resulting families of shapes are distinct for each case.
What happens if the pen distance 'd' is equal to the rolling radius 'r'?: When d = r, the pen is located on the circumference of the rolling circle. In this special case, a hypotrochoid becomes a hypocycloid and an epitrochoid becomes an epicycloid. These curves have sharp cusps instead of loops or smoothed corners, as the pen touches the fixed circle during its motion.
Are these curves just for toys, or do they appear in real-world applications?: Absolutely. Trochoidal shapes are fundamental in engineering and nature. They describe the motion of gears (forming the shape of gear teeth), the path of a piston in a Wankel rotary engine, and even the orbits of celestial bodies in certain historical astronomical models. The Spirograph is an accessible introduction to this important geometry.

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Frequently asked questions

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