# 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 ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$.

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 ${\mathbb{R}}^{2}$ or of a surface $f$ in ${\mathbb{R}}^{3}$, 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)).

## References

- 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 |