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.
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.
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.
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.
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3P-point
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4P-point
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5P-point
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6P-point
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nP-point
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X(2)
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4P-n-P1 = QA-P1
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5P-n-P1
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6P-n-P1
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Etc.
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X(3)
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4P-n-P2 = QA-P32
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5P-n-P2
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6P-n-P2
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Etc.
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X(4)
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4P-n-P3 = QA-P33
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5P-n-P3
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6P-n-P3
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Etc.
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X(5)
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5P-n-P4
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6P-n-P4
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Etc.
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Distance ratios for 3P-/4P-/5P-/6P-Eulerline points, etc., are preserved at the different n-levels.
X(2)=Centroid, X3=Circumcenter, X(4)=Orthocenter, X(5)=Nine-point Center.