Ruler geometry

The ruler geometry describes the restriction of construction tasks of the Euclidean geometry , in which the compass may not be used (and therefore also no angles or other drawing devices ). Only the ruler (without scale) may be used. Sometimes, for example, the use of a single circle is also allowed, but the rest of the construction can only be done with the ruler. The name comes from Johann Heinrich Lambert (in his book Freye Perspective , Zurich 1759, 1774). In addition to August Ferdinand Möbius , the ruler geometry was developed primarily by Jakob Steiner and Karl Georg Christian von Staudt . Steiner and Jean Victor Poncelet came up with the sentence that constructions with compasses and ruler can also be carried out with ruler and a given circle.

Examples

Tangents to circle

Figure 1: Tangents on a circle

Given a point and a circle ( center not known). Find the two tangents of to the circle (see Figure 1). In the ruler geometry the solution is obtained as follows: You draw two secants through the circle from and get the points up to The connections of the points with and with as well as the half-straight lines down through and down through this result in the intersection points or you now draw a line from through to the circle, you get the two tangent points and ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{4}.}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{4}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{4}}$ ${\displaystyle P_{2};}$ ${\displaystyle X}$ ${\displaystyle P.}$ ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle T_{1}}$ ${\displaystyle T_{2}.}$

Parallel to a straight line

It is not possible to draw a parallel to a given straight line with the ruler alone . However, if there is a line segment and its bisection point - as shown in Figure 2 - you can construct a parallel to the line.

Let it be a segment on a straight line, the bisecting point of and a point through which the parallel searched for should run. ${\displaystyle {\overline {FG}}}$ ${\displaystyle D}$ ${\displaystyle {\overline {FG}}}$ ${\displaystyle H}$ ${\displaystyle {\overline {FG}}}$

You start with a half-line from through and to any fixed point is followed by the compounds of the points with with and with ; this results in the intersection point Now draw a straight line through until it intersects the line in . The final straight line through and is the parallel you are looking for. ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle A.}$ ${\displaystyle F}$ ${\displaystyle H,}$ ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle D}$ ${\displaystyle A}$ ${\displaystyle C.}$ ${\displaystyle G}$ ${\displaystyle C}$ ${\displaystyle {\overline {FA}}}$ ${\displaystyle J}$ ${\displaystyle H}$ ${\displaystyle J}$

Fig. 2: Parallel through the given route to Steiner ${\displaystyle H}$ ${\displaystyle {\overline {FA}},}$

Midpoint of a line

It is not possible to cut a distance in half with the ruler alone. However, if there is a parallel to a line - as shown in Figure 3 - you can construct the bisection of the line.

Given a line and a parallel to Wanted is the bisection of the line . ${\displaystyle {\overline {FG}}}$ ${\displaystyle {\overline {FG}}.}$ ${\displaystyle {\overline {FG}}}$

You begin by connecting any point you have defined with the points and ; this creates the intersection points or the connections between the points with and with ; the result is the intersection. Finally, a straight line is drawn through to the line and thus the desired bisection point of the line is obtained ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle F}$ ${\displaystyle H}$ ${\displaystyle J.}$ ${\displaystyle F}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle J}$ ${\displaystyle C.}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle {\overline {FG}},}$ ${\displaystyle D}$ ${\displaystyle {\overline {FG}}.}$

Image 3: The bisection of the route to Steiner ${\displaystyle D}$ ${\displaystyle {\overline {FG}},}$

See also

• Poncelet-Steiner's theorem
• Construction of the 4th harmonic point

source

• Ruler geometry . In: Meyers Großes Konversations-Lexikon, Volume 12. Leipzig 1908, p. 572.

Individual evidence

1. ↑ Möbius From the metrical relations in the field of ruler geometry , Journal for pure and applied mathematics, Volume 4, 1829.
2. ↑ Jakob Steiner: The geometric constructions, carried out by means of the straight line and a solid circle, as a subject of instruction in higher educational institutions and for practical use . Ed .: Ferdinand Dümmler. Berlin 1833 ( title view [accessed January 26, 2020]).
3. ↑ ^{a b} J. Sommer: Elementary Geometry from the Standpoint of Modern Analysis , Encyclopedia of Mathematical Sciences, Volume III, 1,2, P. 790 ff., 7. Constructions with the ruler . SUB Göttingen Digitization Center, accessed on January 26, 2020 . 
4. ↑ ^{a b} Jakob Steiner: The geometrical constructions, carried out by means of the straight line and a solid circle, as a subject of teaching in higher educational institutions and for practical use . Ed .: Ferdinand Dümmler. Berlin 1833 ( ETH Library, ⅠⅠ Constructions using a ruler under certain conditions [page 14, § 6.], page 15, task Ⅰ. See also panel Ⅰ, Fig. 3 [accessed on January 26, 2020]).