nP-1: General information about Objects in an n-Point

For quick insight pictures of n-Points in EPG often are represented by figures bounded by n line-segments.

How many (n-1)-Points can be made up from an n-Point?

Many of the recursive constructions are based upon the property that from an n-Point exactly n different (n-1)-Points can be made up. This can easily be deduced by omitting one point from the n-Point. This will leave behind an (n-1)-Point. Since exactly n different Points can be omitted there will be n different (n-1)-Points contained in an n-Point. The (n-1)-Points in an n-Point will be called the Component (n-1)-Points. The remaining point after choosing an (n-1)-Point in an n-Point will be called the omitted point.

In descriptions we say “an n-Point contains n (n-1)-Points” or “an n-Point has n Component (n-1)-Points”. When we want to indicate different objects occurring in (n-1)-Points we say that there are n versions of these (n-1)P-objects.

The n versions of an object often will be noted with a suffix consisting of un underscore and a number 1, …, n, indicating the number of the omitted point. For example a 5-Point contains 5 4-Points and therefore has 5 4P-MVP-Centroids (4P-n-P1). They will be noted as 4P-n-P1_1, 4P-n-P1_2, 4P-n-P1_3, 4P-n-P1_4 and 4P-n-P1_5. The suffix number at the end is the number of the omitted point.

Recursive Eulerline situated points in an n-Point

There is a special way of construction of Eulerline points to a higher n-Point level.

This method is based upon the central property that from any n-Point n versions of (n-1)-Points can be constructed. The centroid of these n versions produce a higher level nP-point.

MVP Points: Multi Vector Points

Distance ratios for MVP Eulerline points are preserved at the different n-levels.

X(2)=Centroid, X(3)=Circumcenter, X(4)=Orthocenter, X(5)=Nine-point Center.

Estimated human page views: 279