Partial ratio

Definition of the partial ratio and special cases

In geometry, in the simplest case, the partial ratio is the ratio of two partial routes of a given route. Is z. B. the line through a point in two sections and divided (s. A first example), the number is the associated division ratio. However, one could also explain the reciprocal value that arises from swapping and as a partial ratio. When dealing with partial relationships, it is essential to pay attention to the designation of the points. ${\displaystyle \left[AB\right]}$ ${\displaystyle T}$ ${\displaystyle \left[AT\right]}$ ${\displaystyle \left[TB\right]}$ ${\displaystyle \lambda ={\tfrac {|AT|}{|TB|}}}$ ${\displaystyle A}$ ${\displaystyle B}$

The division ratio is of great importance through the generalization to any subpoints on the straight line . ${\displaystyle T}$ ${\displaystyle A,B}$

The great importance of the partial ratio lies in its invariance under affine maps (linear maps and translations) and parallel projections . In the case of projective images and central projections, the partial ratio generally does not remain invariant, but the so-called double ratio .

definition

In the literature one finds the following definition for three points in the Euclidean plane :

For three different collinear points the number is called with the property ${\displaystyle A,B,T}$ ${\displaystyle \lambda }$

${\displaystyle {\overrightarrow {AT}}=\lambda \;{\overrightarrow {TB}}\ }$

the division ratio in which the point divides the pair of points and denotes it with or . ${\displaystyle T}$ ${\displaystyle A,B}$ ${\displaystyle (A,B;T)}$ ${\displaystyle TV(ABT)}$

The case can be included and delivers . The division ratio can take on any real number except −1 (see below). ${\displaystyle A=T}$ ${\displaystyle \lambda =0}$

The word “divides” must not be taken too literally after it has been extended to any point , because the route divides only if there is between . ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle A,B}$ ${\displaystyle T}$ ${\displaystyle \left[AB\right]}$

The following applies:

• If it lies between and , it is and one speaks of an inner division . ${\displaystyle T}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \lambda >0}$
• If it is outside, then it is and one speaks of an external division . If outside is on the side of , then is . If is on the side of , then applies . ${\displaystyle T}$ ${\displaystyle \lambda <0}$ ${\displaystyle T}$ ${\displaystyle B}$ ${\displaystyle \lambda <-1}$ ${\displaystyle T}$ ${\displaystyle A}$ ${\displaystyle -1<\lambda <0}$
• Approaching from inside , strives against , in the other case (from outside) goes against . ${\displaystyle T}$ ${\displaystyle B}$ ${\displaystyle \lambda }$ ${\displaystyle \infty }$ ${\displaystyle \lambda }$ ${\displaystyle -\infty }$
• If the midpoint of the line is, it results . ${\displaystyle T}$ ${\displaystyle \left[AB\right]}$ ${\displaystyle \lambda =1}$

Note that an exchange of changes (inverts) the division ratio, except in the case that the midpoint is the segment. ${\displaystyle A,B}$ ${\displaystyle T}$

Calculation of the partial ratio or the partial point

Vectors for calculating the division ratio

Split ratio depending on the parameter t: ${\displaystyle \lambda ={\tfrac {t}{1-t}}}$

The point of the straight line through the points and lets through ${\displaystyle T}$ ${\displaystyle A}$ ${\displaystyle B}$

• ${\displaystyle {\overrightarrow {OT}}={\overrightarrow {OA}}+t{\overrightarrow {AB}}}$describe with a parameter . ${\displaystyle t}$

From and we get the equation and finally ${\displaystyle {\overrightarrow {AT}}=t{\overrightarrow {AB}}}$ ${\displaystyle {\overrightarrow {TB}}=(1-t){\overrightarrow {AB}}}$ ${\displaystyle t{\overrightarrow {AB}}=\lambda \;(1-t){\overrightarrow {AB}}}$

• ${\displaystyle \lambda ={\frac {t}{1-t}}}$.

Solving the last equation for t one obtains

• ${\displaystyle t={\frac {\lambda }{1+\lambda }}}$and thus the partial point with the given partial ratio ${\displaystyle \lambda }$ ${\displaystyle T}$
• ${\displaystyle {\overrightarrow {OT}}={\overrightarrow {OA}}+{\frac {\lambda }{1+\lambda }}{\overrightarrow {AB}}}$

For is and the midpoint of the route . ${\displaystyle \lambda =1}$ ${\displaystyle t={\frac {1}{2}}}$ ${\displaystyle T}$ ${\displaystyle \left[AB\right]}$

Note:
If the points are given by their parameters with regard to a parametric representation of the underlying straight line, this results in their partial ratio ${\displaystyle A,B,T}$ ${\displaystyle a,b,t}$ ${\displaystyle {\vec {x}}={\vec {p}}+x{\vec {v}}}$

• ${\displaystyle \lambda ={\tfrac {t-a}{b-t}}}$and for the reverse . ${\displaystyle t={\tfrac {b\lambda +a}{1+\lambda }}}$

Drawing determination of the partial point

Division of A, B in the ratio (T, inside) or (S, outside) ${\displaystyle \lambda =5:3}$ ${\displaystyle \lambda =-5:3}$

To find the partial point, a construction according to the second theorem of rays is used : If the segment [AB] is to be divided in the ratio m: n, then one draws two parallel straight lines through A and through B. On the parallel through A one plots the same distance m times, on the parallel through B n times. With an inner division, the removal must take place in different directions, with an outer division in the same direction. The straight line is drawn through the end points of the lines that have been removed. Its intersection with the straight line AB is the subpoint you are looking for (S or T).

Invariance of the partial ratio

Any affine mapping of the real coordinate plane can be represented as follows:

• ${\displaystyle {\vec {x}}\to \phi ({\vec {x}})+{\vec {u}}}$, where is a linear map . ${\displaystyle \phi }$

So will on ${\displaystyle {\overrightarrow {AT}}={\overrightarrow {OT}}-{\overrightarrow {OA}}}$

${\displaystyle {\overrightarrow {A'T'}}=\phi ({\overrightarrow {OT}})+{\vec {u}}-\phi ({\overrightarrow {OA}})-{\vec {u}}=\phi ({\overrightarrow {AT}})=\phi (\lambda {\overrightarrow {TB}})=\lambda \phi ({\overrightarrow {TB}})=\lambda {\overrightarrow {T'B'}}}$pictured. From this it follows

• ${\displaystyle {\overrightarrow {A'T'}}=\lambda {\overrightarrow {T'B'}}}$, the invariance of the partial ratio.

A parallel projection can be represented as an affine image or, with suitable coordination, even as a linear image (see ellipse (descriptive geometry) ). So the partial ratio is invariant even with parallel projection.

generalization

Since only numbers and vectors were used to define the partial ratio, it can literally be extended to an affine coordinate plane over any body (the real numbers are simply replaced as the coordinate range by any body). However, the above statements apply use typical properties of real numbers (" " and " ") no longer. The invariance of the partial ratio also applies in this general case. ${\displaystyle >0}$ ${\displaystyle \to \infty }$

See also

• harmonic division
• Double ratio

literature

• dtv-Atlas zur Mathematik, Volume 1, 1978, ISBN 3-423-03007-0 , p. 157
• Siegfried Krauter, Christine Bescherer: Elementary Geometry Experience , ISBN 978-3-8274-3026-7 , p. 159
• Hermann Schaal, Ekkehart Glässner: Linear Algebra and Analytical Geometry , Volume 1, 255 pages, Vieweg; Braunschweig, Wiesbaden 1976, ISBN 3-528-03056-9
• Uwe Storch, Hartmut Wiebe: Textbook of Mathematics, Volume II: Linear Algebra . BI-Wissenschafts-Verlag, 1990, ISBN 3-411-14101-8