Functions of Leibniz

In Géométrie refines or Euclidean, the vector and scalar functions of Leibniz is functions which, with points, associate vectors (vector function) or numbers (scalar function). These functions are very closely related to the concept of barycentre which makes it possible to give a simplified form of it.

Vector function of Leibniz

One places oneself in a space refines E associated with a vector space V. Are $(A\_i)\; \_\; \{i=1\; \backslash \; cdots\; N\}$ a family of N points and $(a\_i)\; \_\; \{i=1\; \backslash \; cdots\; N\}$ a family of N scalar, one calls vector function of Leibniz associated with the system $\backslash \; left\; \backslash \; \{\backslash \; left\; (A\_i,\; a\_i\; \backslash \; right)\; \_\; \{i=1\; \backslash \; cdots\; N\}\; \backslash \; right\; \backslash \}$, the application of E in V which, with the point M associates the vector $\backslash \; vec\; F\; (M)\; =\; \backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\; \backslash overrightarrow\{MA\_i\}$

If the sum of the coefficients $\backslash \; sum\_\; \{i=1\}\; ^n\; a\_i$ is null, this function is constant. If one of the coefficients is nonnull (for example $a\_1$), this constant is equal to $a\_1\; \backslash \; overrightarrow\; \{G\_1A\_1\}$ where $G\_1$ is the barycentre system $\backslash \; left\; \backslash \; \{\backslash \; left\; (A\_i,\; a\_i\; \backslash \; right)\; \_\; \{i=2\; \backslash \; cdots\; N\}\; \backslash \; right\; \backslash \}$

If the sum of the coefficients $\backslash \; sum\_\; \{i=1\}\; ^n\; a\_i$ is nonnull, this function is simplified in

$\backslash \; vec\; F\; (M)\; =\; \backslash \; left\; (\backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\; \backslash \; right)\; \backslash \; overrightarrow\; \{MG\}$

This property makes it possible to reduce a linear Combinaison several vectors in only one vector thanks to a barycentre. It also makes it possible to give the coordinates of the barycentre when space is of finished size.

Indeed $\backslash \; overrightarrow\; \{OG\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\}\; \backslash \; vec\; F\; (O)\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\}\; \backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\; \backslash \; overrightarrow\; \{OA\_i\}$.

What is translated into term of coordinates by

$x\_\; \{G,\; K\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\}\; \backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\; x\_\; \{A\_i,\; K\}$

Scalar function of Leibniz

One places oneself in a space refines Euclidean on a body $\backslash \; mathbb\; K$. Are $(A\_i)\; \_\; \{i=1\; \backslash \; cdots\; N\}$ a family of N points and $(a\_i)\; \_\; \{i=1\; \backslash \; cdots\; N\}$ a family of N scalar, one calls scalar function of Leibniz associated with the system $\backslash \; left\; \backslash \; \{\backslash \; left\; (A\_i,\; a\_i\; \backslash \; right)\; \_\; \{i=1\; \backslash \; cdots\; N\}\; \backslash \; right\; \backslash \}$, the application of E in $\backslash \; mathbb\; K$ which, with the point M associates the scalar $f\; (M)\; =\; \backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\; MA\_i^2$

If the sum of the coefficients is null, this function is simplified in

$f\; (M)\; =\; F\; (O)\; +\; 2\; \backslash \; overrightarrow\; \{MO\}\; \backslash \; cdot\; \backslash \; vec\; u$

where $\backslash \; vec\; u$ is the constant equal to the vector function of Leibniz associated with the system and where O is an arbitrarily fixed point.

If the sum of the coefficients is nonnull, this function is simplified in

$f\; (M)\; =\; F\; (G)\; +\; \backslash \; left\; (\backslash \; sum\_\; \{i=1\}\; ^n\; a\_i\; \backslash \; right)\; MG^2$

where G is the barycentre of the system $\backslash \; left\; \backslash \; \{\backslash \; left\; (A\_i,\; a\_i\; \backslash \; right)\; \_\; \{i=1\; \backslash \; cdots\; N\}\; \backslash \; right\; \backslash \}$

This reduction makes it possible to solve more simply of the problems of places of points (see Théorème of Leibniz)

Example: in dimension two, the whole of the points M such as F (M) = K is

• if the sum of the coefficients is null
• an orthogonal line with $\backslash \; vec\; u$ if $\backslash \; vec\; u$ is nonnull
• all the plan or the empty set (according to the values of K) if $\backslash \; vec\; u$ is null
• if the sum of the coefficients is nonnull
• a circle of center G, the point G or the empty set (according to the values of K)

See too

• Theorem of Leibniz
• Barycentre (geometry refines)

Category: Geometry refines Category: Euclidean geometry

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