midpoint

The concept of midpoint of line segment (http://planetmath.org/Midpoint) is a special case of the midpoint of a curve or arbitrary figure in ${ℝ}^2$ or ${ℝ}^3$.

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 ${ℝ}^2$ or of a surface $f$ in ${ℝ}^3$, one can, if , take a new point $T$ for the origin by using the linear substitutions of the form

$$x:=x^{\prime }+a,y:=y^{\prime }+b\mathit{\hspace{1em}}\text{etc.}$$

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 curve 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 curve 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)).

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