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Complement


In general, the word "complement" refers to that subset F^' of some set S which excludes a given subset F. Taking F and its complement F^' together then gives the whole of the original set. The notations F^' and F^_ are commonly used to denote the complement of a set F.

This concept is commonly used and made precise in the particular cases of a complement point, graph complement, knot complement, and complement set. The word "complementary" is also used in the same way, so combining an angle and its complementary angle gives a right angle and a complementary error function erfc and the usual error function erf give unity when added together,

 erfc(x)+erf(x)=1.
(1)

The complement point of a point P with respect to a reference triangle DeltaABC, also called the inferior point, subordinate point, or medial image, is the point P^' such that

 PG^->=2GP^'^->,
(2)

where G is the triangle centroid.

The complement point of a point with trilinear coordinates alpha:beta:gamma is therefore given by

 (bbeta+cgamma)/a:(aalpha+cgamma)/b:(aalpha+bbeta)/c.
(3)

The following table lists the complements of some named circles.

The complement of a line

 lalpha+mbeta+ngamma=0
(4)

is given by the line

 a(-bcl+acm+abn)alpha+b(bcl-acm+abn)beta 
 +(bcl+acm-abn)gamma=0.
(5)

The following table summarizes the complements of a number of named lines.

The following table summarizes the complements of several common triangle centers.


See also

Anticomplement, Complement Set, Complementary Angles, Erfc, Graph Complement, Homothecy, Homothetic Center, Knot Complement, Similitude Ratio, Triangle Centroid

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References

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 23, 1984.

Referenced on Wolfram|Alpha

Complement

Cite this as:

Weisstein, Eric W. "Complement." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Complement.html

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