isometry

Thân thiện

An isometry preserves the distance between any two points on a geometric shape.

Definition

Noun: 1. Equality of measure or dimension: A state of having equal measurements, dimensions, or rates of growth. This can refer to equality in height, loudness, or other measurable properties. 2. Equality of elevation: Specifically, the state of having the same height above a fixed reference point, such as sea level. 3. A distance-preserving transformation: In mathematics, a mapping or function between two metric spaces that preserves the exact distances between every pair of points. It is a rigid transformation that can include rotations, reflections, and translations.

Usage Examples

• Equality of measure:
  • The isometry of sound pressure levels was crucial for the hearing test.
  • The architect ensured the isometry of the columns' heights.
• Equality of elevation:
  • The contour lines on the map connect points of isometry.
  • These two locations share an isometry of 500 meters above sea level.
• Mathematical transformation:
  • A rotation is a classic example of an isometry in Euclidean geometry.
  • The proof relied on finding an isometry that mapped one geometric shape onto the other.

Advanced Usage

• "To be an isometry": Used to describe a function that satisfies the distance-preserving property.
  • For the function f to be an isometry, the distance between f(x) and f(y) must equal the distance between x and y for all points.

Variants and Related Words

• Isometric (adjective):
  • Of or relating to equality of measure.

    • The map shows isometric lines of atmospheric pressure.
  • Of or relating to a mathematical isometry.

    • The two shapes are isometric; one can be transformed into the other without stretching.
  • (In physiology) Of or involving muscular contraction against resistance without movement of the involved body part.

    • He held an isometric pose to build strength.

Synonyms

• Congruence (specifically in geometry for shapes).
• Equidistance (for the preservation of distance).
• Equality of measure.

Related Concepts (Not Phrasal Verbs)

• Rigid Motion: A term often used interchangeably with isometry in geometry.
• Metric Preserving Map: A more technical synonym for the mathematical definition.
• Contour Line: A line on a map connecting points of equal elevation (a practical example of isometry in geography).

Học thuật

Noun

1. equality of measure (e.g., equality of height above sea level or equality of loudness etc.)
2. equality of elevation above sea level
3. a one-to-one mapping of one metric space into another metric space that preserves the distances between each pair of points
   • the isometries of the cube
4. the growth rates in different parts of a growing organism are the same