is homothetic to .
If is the homothety center of two polygons and , then for any pair of corresponding points under the homothety (that is, passes through ), the ratio
is known as homothety ratio, and it coincides with the similarity ratio. It is customary to work with directed segments when talking of homothety, so the homothety ratio is a signed number.
When is positive we speak of direct homothety and not only similarity is preserved, but pictures also have the same orientation (corresponding points to the upper part of one figure are on the upper part of the other). However, if is negative we have reverse homothety, and corresponding lines remain parallel but all orientations are reversed, so it looks like one picture is upside-down when compared to the other.
|Direct homothety||Reverse homothethy|
An alternate way of defining homothety is requiring the existence of some point such that all the lines where is a point on a figure, intersect the other figure on a point in such way that all the ratios are the same.
Although both definitions are equivalent for polygons, the later has the benefit that it can be applied to arbitrary figures, and so we can talk about homothethy between arbitrary figures. For instance, it can be proved that any two circles are homotopic, in both direct and inverse ways, and the homothety centers can be located intersecting appropiated circle tangents when they exist.
|Date of creation||2013-03-22 15:04:06|
|Last modified on||2013-03-22 15:04:06|
|Last modified by||drini (3)|