Chord Length of Hypocycloid Calculator

Chord Length of Hypocycloid Formula:

\[ l_c = 2 \times \sin\left(\frac{\pi}{N_{cusps}}\right) \times r_{Large} \]

Chord Length of Hypocycloid:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Chord Length of Hypocycloid?

The chord length of a hypocycloid is the linear distance between any two adjacent cusps of the hypocycloid. A hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle.

2. How Does the Calculator Work?

The calculator uses the chord length formula:

\[ l_c = 2 \times \sin\left(\frac{\pi}{N_{cusps}}\right) \times r_{Large} \]

Where:

\( l_c \) — Chord Length of Hypocycloid (meters)
\( N_{cusps} \) — Number of Cusps of Hypocycloid
\( r_{Large} \) — Larger Radius of Hypocycloid (meters)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function

Explanation: The formula calculates the straight-line distance between two adjacent cusps of a hypocycloid based on the number of cusps and the radius of the larger circle.

3. Importance of Chord Length Calculation

Details: Calculating the chord length is essential in geometry, mechanical engineering, and design applications where hypocycloid shapes are used, such as in gear design, decorative patterns, and mathematical modeling.

4. Using the Calculator

Tips: Enter the number of cusps (must be at least 3) and the larger radius of the hypocycloid. The calculator will compute the chord length between adjacent cusps.

5. Frequently Asked Questions (FAQ)

Q1: What is a hypocycloid?
A: A hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls without slipping inside a larger circle.

Q2: What are cusps in a hypocycloid?
A: Cusps are the sharp points or corners of the hypocycloid where the curve changes direction abruptly.

Q3: What is the minimum number of cusps a hypocycloid can have?
A: A hypocycloid must have at least 3 cusps to form a closed curve with distinct points.

Q4: How does the number of cusps affect the chord length?
A: As the number of cusps increases, the chord length decreases for a fixed larger radius, since the angle between cusps becomes smaller.

Q5: What are some practical applications of hypocycloids?
A: Hypocycloids are used in gear design (hypocycloidal gear mechanisms), decorative art patterns, and mathematical modeling of various physical phenomena.