Definition

1. Noun:
   • A type of smooth, open curve in mathematics: A hyperbola is a symmetrical open curve formed by the intersection of a plane with both halves of a double cone (a right circular cone). It consists of two separate, mirror-image branches that are unbounded.
   • A conic section: It is one of the four classic conic sections, along with the circle, ellipse, and parabola. It is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant.

Examples of Usage

• Noun:

Advanced Usage

• Rectangular hyperbola: A specific type of hyperbola where its asymptotes are perpendicular to each other.
• Conjugate hyperbola: Refers to a pair of hyperbolas that share the same asymptotes but have transverse and conjugate axes swapped.

Variants and Related Words

• Hyperbolic (adj): Relating to or having the form of a hyperbola; also used to describe exaggerated statements (a different, but etymologically related meaning).
• Hyperboloid (n): A three-dimensional surface generated by rotating a hyperbola around one of its axes.

Synonyms

• Conic section (broader category): A curve obtained as the intersection of a cone with a plane.
• There are no direct single-word synonyms for this specific geometric figure.

Related Phrases

• Foci of a hyperbola: The two fixed points used in the geometric definition.
• Asymptotes of a hyperbola: The lines that the branches of the hyperbola approach but never touch.

Related Idioms

• There are no common idioms based directly on the geometric term "hyperbola." The adjective "hyperbolic" is used in the idiom "hyperbolic statement", meaning an exaggerated claim not meant to be taken literally, which derives from the rhetorical term "hyperbole," sharing a Greek root with "hyperbola."

Noun

1. an open curve formed by a plane that cuts the base of a right circular cone