Hyperbolic Circle by Frans van Schooten

Hyperbolic Circle by Frans van Schooten

The Hyperbelzirkel of Frans van Schooten is a mechanism of the shape of a hyperbola generated. In 1657 Frans van Schooten published a hyperbolic circle in his work Exercitationum mathematicarum libri quinque in LIBER IV . It is similar to Frans van Schooten's elliptical circle .

The hyperbola circle essentially consists of three parts:

1. a diamond with the points of articulation with the first circular needle as a fixed point of the diamond, ${\displaystyle DLFM,}$ ${\displaystyle F,}$
2. a diagonal rail made of two rods connected at the ends, with the pen in ${\displaystyle |NO|}$ ${\displaystyle e,}$
3. a guide rail with the second circular needle as a fixed point, which is slotted after the hinge point and runs through the point . ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle e}$

Three so-called sliding blocks enable linearly movable connections. One of them guides the articulation point of the diamond, the second the pen at the point and the third sliding block guides the articulation point of the diamond. The diagonal rail is not mounted in a hinge point or the rhombus, it can therefore be moved in the longitudinal direction from to and from to . In comparison, in Frans van Schooten's elliptical circle, the diagonal bar is mounted at the hinge point of the diamond. ${\displaystyle L}$ ${\displaystyle e}$ ${\displaystyle M}$ ${\displaystyle |NO|}$ ${\displaystyle |M|}$ ${\displaystyle |L|}$ ${\displaystyle O}$ ${\displaystyle L}$ ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle |OQ|}$ ${\displaystyle O}$

The drawing of a hyperbola with the help of the pen is intended to clarify the hand. After inserting the compass into the focal points and moving with one hand using a handle in the point of the hyperbolic compass . The guide rail ( through and ) together with the diagonal rail forces the pen into a hyperbolic path. ${\displaystyle e}$ ${\displaystyle C}$ ${\displaystyle F,}$ ${\displaystyle e}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle e,}$ ${\displaystyle |NO,|}$ ${\displaystyle e}$

Based on the illustration, it can be assumed that the hyperbolic circle in certain unstable positions (including when focal points and pen are on a line) with the other hand, at the point or needs support. ${\displaystyle M}$ ${\displaystyle L,}$

A possible reason why the curves drawn with Frans van Schooten's hyperbola circle are exact hyperbolas is described in the following section, Geometric Consideration .

Geometric view

To illustrate why the the Hyperbelzirkel generated curves exact hyperboles are a Hyperbelpunkt is below first in a basic construction of definition with Leitkreis determined and in principle incorporated subsequently the Hyperbelzirkel. The sequence of movements is explained in Draw hyperbola . The designations of the points are taken from the above original illustration of hyperbolic circles by Frans van Schooten . ${\displaystyle e}$

Hyperbola point as defined with a guide circle

Fig. 1: Hyperbolic point as defined with guide circle or basic construction for hyperbolic circle by Frans van Schooten ${\displaystyle k_{L}}$
${\displaystyle |ek_{L}|=|eF|}$ ${\displaystyle |eD|=|eF|}$

With the designations of the points used in Figure 1, a decisive statement of the definition with guide circle , based on the right branch of the hyperbola , reads :

"Is the circle of radius , so does the circle the same distance as the focal point : It is called to be associated Leitkreis of hyperbole." ${\displaystyle k_{L}}$ ${\displaystyle C}$ ${\displaystyle 2a}$ ${\displaystyle e}$ ${\displaystyle k_{L}}$ ${\displaystyle F}$ ${\displaystyle |ek_{L}|=|eF|\ .}$ ${\displaystyle k_{L}}$ ${\displaystyle F}$

It begins with the drawing of a straight line, the central axis of the later hyperbola. The first vertex is then marked as desired and then the second vertex is defined with a freely selectable distance . Thus, the distance between the vertices is determined to be equivalent to the radius of the guide circle . Now, with an estimated but the same distance to or to the outside, the focal points and with the selected focal points and as well as one of the two vertices or (three known points) the hyperbola is already mathematically determined. The hyperbola (green) can e.g. B. can be entered using a dynamic geometry software (DGS) . ${\displaystyle E}$ ${\displaystyle K}$ ${\displaystyle |EK|}$ ${\displaystyle 2a}$ ${\displaystyle k_{L},}$ ${\displaystyle E}$ ${\displaystyle K,}$ ${\displaystyle C}$ ${\displaystyle F.}$ ${\displaystyle C}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle K}$

It continues with the pulling of the Leitkreises to the radius of the right branch of the hyperbola; it results in the auxiliary point To find the point that has the same distance to the focal point as to the guide circle, one draws with any radius, but with the same compass opening around the circle and around the circle . The second auxiliary point is now a circle with the radius To be drawn in, the radius of the circle is added to the radius of the guide circle , so to speak . The intersection of the circle with is the point you are looking for ${\displaystyle k_{L}}$ ${\displaystyle C}$ ${\displaystyle |EK|=2a}$ ${\displaystyle A.}$ ${\displaystyle e}$ ${\displaystyle F}$ ${\displaystyle k_{L},}$ ${\displaystyle F}$ ${\displaystyle k_{1}}$ ${\displaystyle A}$ ${\displaystyle k_{2};}$ ${\displaystyle B.}$ ${\displaystyle k_{3}}$ ${\displaystyle |CB|}$ ${\displaystyle C}$ ${\displaystyle |AB|}$ ${\displaystyle k_{2}}$ ${\displaystyle |CA|}$ ${\displaystyle k_{L}}$ ${\displaystyle k_{3}}$ ${\displaystyle k_{1}}$ ${\displaystyle e.}$

The ensuing half-line starting by cutting the Leitkreis in and delivers the isosceles triangle (pink) with the two equally long side lengths and conclusion, which is still Mittelsenkrechte the distance registered; because of the isosceles triangle it runs through the point From this it follows: ${\displaystyle H_{G}}$ ${\displaystyle C}$ ${\displaystyle e}$ ${\displaystyle k_{L}}$ ${\displaystyle D}$ ${\displaystyle DFe}$ ${\displaystyle {\overline {eD}}}$ ${\displaystyle {\overline {eF}}.}$ ${\displaystyle M_{S}}$ ${\displaystyle |DF|}$ ${\displaystyle DFe}$ ${\displaystyle e.}$

the isosceles triangle with ${\displaystyle DFe}$

${\displaystyle |eD|=|eF|,}$

is a half diamond in which the perpendicular and the bisector of the angle are tangents of the hyperbola . Thus the constructed point is a hyperbolic point . ${\displaystyle M_{S}}$ ${\displaystyle DeF}$ ${\displaystyle e}$

Construction of the hyperbola circle

• In order for the hyperbolic circle to be able to draw a complete hyperbolic line, it is necessary that the guide bar (from through ) lies above the diamond (circle needle in ). It should be noted that in the above original illustration of the hyperbolic circle by Frans van Schooten , the guide rail is below the diamond. With this position of the guide rail, the hyperbolic line cannot be drawn through the apex , but only, for. B. counterclockwise until the guide rail rests against the circular needle in the diamond. ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle DLFM}$ ${\displaystyle F}$ ${\displaystyle K}$ ${\displaystyle F}$

The basic sketch (Fig. 2) is a continuation of the basic construction of the hyperbola point as defined with a guide circle ${\displaystyle k_{L}}$ (Fig. 1). For a better overview, the irrelevant circles, points etc. have been hidden. The Leitkreis as well as u. a. the points and are already determined in the previous construction (Fig. 1), so all that is required is a simple incorporation of the essential parts of the hyperbola circle described above. ${\displaystyle k_{L}}$ ${\displaystyle D}$ ${\displaystyle F}$

First, the two side lengths and the rhombus, with the estimated compass opening, clearly larger than the distance , are determined on the vertical line . The connection of the points of articulation with and with follows and completes the diamond with the isosceles triangle (light blue). This is followed by drawing in the diagonal rail, the length of which is greater than the diagonal of the diamond. Finally, the guide rail is drawn in through . It cuts the diagonal rail , as indicated, also in the hyperbola point of the isosceles triangle (pink). ${\displaystyle {\overline {FM}}}$ ${\displaystyle {\overline {FL}}}$ ${\displaystyle |Fe|}$ ${\displaystyle M_{S}}$ ${\displaystyle D}$ ${\displaystyle M}$ ${\displaystyle D}$ ${\displaystyle L}$ ${\displaystyle FMDL}$ ${\displaystyle FDL}$ ${\displaystyle {\overline {NO}}}$ ${\displaystyle |LM|}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle |NO|}$ ${\displaystyle e}$ ${\displaystyle DFe}$

Image 2: Outline sketch, guide rail (from through ) is above the diamond, see animation ${\displaystyle C}$ ${\displaystyle D}$
${\displaystyle DLFM,}$

Draw hyperbola

If the hyperbolic compass is moved by hand as described above, the point runs on the guide circle and the pen ( ) runs in the gap of the diagonal rail The guide rail ( through and ) forces the diagonal rail to act as a constant center line of the continuously changing isosceles triangles and ,. From this it follows: In every rotated position of the parabolic circle applies ${\displaystyle D}$ ${\displaystyle k_{L}}$ ${\displaystyle e}$ ${\displaystyle |NO|.}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle e}$ ${\displaystyle |NO|}$ ${\displaystyle M_{S},}$ ${\displaystyle FDL}$ ${\displaystyle DFe}$

${\displaystyle |eD|=|eF|}$

This shows: The curves drawn with the hyperbola are exact hyperbolas.

See also

Thread construction of a hyperbola

literature

• C. Edward Sandifer: Van Schooten's Ruler Constructions . In: Convergence August 2010 . August 2010, doi : 10.4169 / convergence20141101 (English, online version from November 2014).

Individual evidence

1. ^ Frans van Schooten: Exercitationum mathematicarum libri quinque. Lugdunum Batavorum (= Johannes Elsevir, Leiden 1657, table of contents, books.google.de ).
2. ^ Frans van Schooten: Exercitationum mathematicarum libri V. Book IV. Johannes Elsevir, Leiden 1657 p. 293 ( books.google.de ).
3. ^ Frans van Schooten: Exercitationum mathematicarum libri V. Volume IV. Johannes Elsevir, Leiden 1657, p. 349 ( books.google.de ).