midpoint (original) (raw)

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 A⁢B. One says also that f is symmetric about the point T.

Given the equation of a curve in ℝ2 or of a surface f in ℝ3, 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)).

References