cardioid

Thân thiện

A child draws a cardioid shape on a piece of paper.

Definition

Noun: A heart-shaped plane curve that is the path traced by a point on the circumference of a circle as it rolls, without slipping, around another fixed circle of equal radius. It is a specific type of epicycloid.

Usage

The term is used primarily in mathematics, specifically in geometry and calculus, to describe this particular curve. It is also relevant in physics and engineering contexts involving wave patterns or directional sensitivity (e.g., a cardioid microphone pickup pattern).

Examples

• The polar equation ( r = a(1 + \cos \theta) ) generates a cardioid.
• The unique shape of a cardioid can be seen in the coffee cup caustic created by light reflecting inside a circular cup.
• A unidirectional microphone with a cardioid pickup pattern is most sensitive to sound arriving from the front.

Advanced Usage

• Cardioid as a caustic: In optics, a cardioid can appear as a caustic curve—a pattern of light reflected or refracted by a curved surface.
• Cardioid in complex analysis: The transformation ( w = z^2 ) maps a circle through the origin in the complex plane to a cardioid.

Variants and Related Words

• Cardioid (Adjective): Describing something having the shape of a cardioid.
  • Example: The microphone has a cardioid polar pattern.
• Epicycloid (Noun): The general class of curves to which the cardioid belongs, traced by a point on a circle rolling around the outside of another circle.
• Limacon (Noun): A more general family of polar curves (( r = a + b\cos \theta )) that includes the cardioid as a special case when ( a = b ).

Synonyms

• Heart-shaped curve (descriptive, non-technical)

Notes on Meaning

The cardioid is distinguished from other heart-shaped curves by its precise mathematical definition as an epicycloid with equal circles. While it resembles a heart symbol, it is a specific geometric construct.

Học thuật

Noun

1. an epicycloid in which the rolling circle equals the fixed circle