Definition

1. Noun:
   • The circle that touches a curve at a given point and has the same curvature at that point: An osculating circle is the circle that best approximates a curve at a specific point. It "kisses" the curve, sharing the same tangent and curvature at the point of contact, lying on the concave side of the curve. Its radius is the radius of curvature.

Usage Examples

• Noun:
  • For any point on a smooth curve, you can calculate its osculating circle.
  • The path of the planet was approximated by a series of osculating circles at different points in its orbit.
  • In the diagram, the osculating circle is drawn inside the bend of the curve.

Advanced Usage

• "To find the osculating circle": A common phrase in differential geometry describing the calculation process.
  • The first step in analyzing the bend is to find the osculating circle at the vertex.
• Conceptual Use: Used to describe a perfect local approximation.
  • The trajectory, over a very short interval, is virtually identical to its osculating circle.

Variants and Related Words

• Osculate (verb): To touch or kiss; in mathematics, for two curves or surfaces to have contact of at least second order (sharing a common tangent and curvature at a point).
• Osculating plane (noun): The plane that contains the osculating circle at a point on a space curve.
• Radius of curvature (noun): The radius of the osculating circle.

Synonyms

• Circle of curvature: A direct synonym for osculating circle.
• Kissing circle: An informal, descriptive synonym.

Related Phrases

• Center of curvature: The center point of the osculating circle.

Noun

1. the circle that touches a curve (on the concave side) and whose radius is the radius of curvature