The concept of midpointMathworldPlanetmathPlanetmathPlanetmath of line segmentMathworldPlanetmath (http://planetmath.org/Midpoint) is a special case of the midpoint of a curve or arbitrary figure in 2 or 3.

A point T is a midpoint of the figure f, if for each point A of f there is a point B of f such that T is the midpoint of the line segment AB.  One says also that f is symmetric about the point T.

Given the equation of a curve in 2 or of a surface f in 3, one can, if , take a new point T for the origin by using the linear substitutions of the form


Thus one may test whether the origin is the midpoint of f by checking whether f always contains along with any point  (x,y,z)  also the point  (-x,-y,-z).

It is easily verified the

Theorem.  If the origin is the midpoint of a quadratic curveMathworldPlanetmath or a quadratic surface, then its equation has no terms of degree (http://planetmath.org/BasicPolynomial) 1.

Similarly one can verify the generalisation, that if the origin is the midpoint of an algebraic curveMathworldPlanetmath or surface of degree n, the equation has no terms of degree n-1,  n-3  and so on.

Note.  Some curves and surfaces have infinitely many midpoints (see quadratic surfaces (http://planetmath.org/QuadraticSurfaces)).


  • 1 Felix Iversen: Analyyttisen geometrian oppikirja. Tiedekirjasto Nr. 19.  Second edition.  Kustannusosakeyhtiö Otava, Helsinki (1963).
Title midpoint
Canonical name Midpoint1
Date of creation 2015-04-25 17:39:00
Last modified on 2015-04-25 17:39:00
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 51M15
Classification msc 51-00
Synonym centre
Synonym center