Definition

1. Noun:
   • A plane that touches a surface at a single point: In geometry, a tangent plane is a flat, two-dimensional surface that just touches a curved surface at one specific point without cutting through it. At that point of contact, the tangent plane has the same direction or slope as the curved surface.

Usage

• The tangent plane provides a linear approximation to a surface near the point of tangency.
• It is used in calculus and differential geometry to analyze the properties of surfaces.
• It is defined for smooth surfaces at points where the surface is not pointed or folded.

Examples

• Noun:
  • To find the linear approximation of the function at (1,1), we first calculated the equation of the tangent plane.
  • The geometer explained that the tangent plane at the vertex of a cone is not uniquely defined.
  • The sculpture was designed so that a glass sheet could lie perfectly flush as a tangent plane to the curved metal at one central point.

Advanced Usage

• "To be tangent to a surface": Describes the relationship where a plane touches a surface at a point.
  • The calculated plane was tangent to the paraboloid at the given coordinates.
• "Equation of the tangent plane": A mathematical formula (often involving partial derivatives) that defines the specific tangent plane at a point.
  • Finding the equation of the tangent plane is a common exercise in multivariable calculus.

Variants and Related Words

• Tangent (adj/n): Touching at a single point without intersecting. As a noun, it can refer to a tangent line in two dimensions.
  • The line is tangent to the circle.
• Tangency (n): The state of being tangent; the point of contact.
  • The point of tangency was clearly marked on the diagram.
• Tangent space (n): A more abstract generalization of the tangent plane concept in higher mathematics.

Synonyms

• Touching plane: (Descriptive synonym) A plane that touches a surface.
• Linearizing plane: (Functional synonym) A plane that linearizes a surface locally.

Related Phrases

• Plane of tangency: Another way to refer to the tangent plane.
  • The plane of tangency approximated the hill's slope at the hiker's location.

Noun

1. the plane that contains all the lines tangent to a specific point on a surface