Hypotrochoid Curve Generator

Create beautiful mathematical curves like spirograph drawings. Adjust parameters in real-time and explore geometric patterns.

Parametric Equations:

x(θ) = (R - r) cos θ + d cos((R - r)/r θ)

y(θ) = (R - r) sin θ - d sin((R - r)/r θ)

Where: R = fixed circle radius, r = rolling circle radius, d = distance from rolling circle center

5
Fixed Radius (R)
3
Rolling Radius (r)
5
Distance (d)
10
Periods

Curve Parameters

Fixed Circle Radius (R): 5
Radius of the fixed outer circle
Rolling Circle Radius (r): 3
Radius of the rolling inner circle
Distance from Center (d): 5
Distance of the tracing point from rolling circle center
Animation Speed: 5
Speed of the drawing animation
Number of times the rolling circle completes a revolution
Higher density = smoother curve but slower rendering

Quick Presets:

Deltoid (R=3, r=1, d=2)
Astroid (R=4, r=1, d=3)
Spirograph 1 (R=5, r=3, d=5)
Spirograph 2 (R=8, r=3, d=7)
Ellipse-like (R=5, r=2, d=3)
5-point Star (R=5, r=2, d=4)

Curve Properties

Curve Type
Hypotrochoid
Symmetry
Rotational
Periodicity
Periodic
Loops
Multiple
R/r Ratio
1.67
d/r Ratio
1.67

Understanding Hypotrochoids

A hypotrochoid is a curve traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is at a distance d from the center of the interior circle.

Parametric Equations:

x(θ) = (R - r) cos θ + d cos((R - r)/r θ)

y(θ) = (R - r) sin θ - d sin((R - r)/r θ)

Where θ is the angle of rotation of the rolling circle around the fixed circle.

Special Cases

Hypocycloid (d = r):

When d = r, the point lies on the circumference of the rolling circle, producing a hypocycloid.

x(θ) = (R - r) cos θ + r cos((R - r)/r θ)

y(θ) = (R - r) sin θ - r sin((R - r)/r θ)

Ellipse (r = R/2, d small):

When r = R/2 and d is small relative to r, the curve approximates an ellipse.

For R = 2r, the equations simplify to:

x(θ) = r cos θ + d cos θ = (r + d) cos θ

y(θ) = r sin θ - d sin θ = (r - d) sin θ

Geometric Interpretation

1

Fixed Circle: The outer circle with radius R remains stationary. The rolling circle moves along its interior circumference.

2

Rolling Circle: A circle of radius r rolls without slipping along the inside of the fixed circle.

3

Tracing Point: A point at distance d from the center of the rolling circle traces the hypotrochoid curve.

Applications of Hypotrochoids

  • Spirograph Toys: The classic drawing toy creates hypotrochoid patterns
  • Gear Design: Some gear tooth profiles use hypotrochoidal curves
  • Art and Design: Creating decorative patterns and mandalas
  • Mathematics Education: Visualizing parametric equations and periodic functions
  • Computer Graphics: Generating interesting visual patterns and animations

Calculator Features:

  • Real-time parametric curve plotting with smooth animation
  • Interactive sliders to adjust R, r, and d parameters
  • Preset examples of common hypotrochoid patterns
  • Color customization for the curve visualization
  • Export functionality to save your creations as images

Frequently Asked Questions

A hypotrochoid is generated when a circle rolls inside another circle. An epitrochoid is generated when a circle rolls outside another circle. The parametric equations are similar but with some sign differences.

A hypotrochoid is closed when R/r is a rational number. The curve will repeat after a finite number of rotations. If R/r is irrational, the curve never closes and continues indefinitely, creating a dense pattern.

When d > r, the tracing point is outside the rolling circle. This creates more complex curves with loops that extend beyond the circumference of the rolling circle. The resulting curves often have interesting intersecting patterns.

Some well-known hypotrochoids include:
  • Deltoid: R=3r, d=r (a 3-cusped hypocycloid)
  • Astroid: R=4r, d=r (a 4-cusped hypocycloid)
  • Ellipse: R=2r, d<r (approximates an ellipse)
  • Cardioid-like: R=2r, d≈r (creates heart-shaped patterns)

While this calculator generates 2D hypotrochoids, 3D hypotrochoids (or spherical trochoids) can be created by rolling a circle inside a sphere. These are more complex and require 3D visualization tools. Some 3D printing and CAD software can generate such patterns.