Strophoids

straight strophoids

The strophoid ( adjective made-up word from Greek στροφή, strofí - the stanza , turn, curve, twist, bend ), more precisely the straight strophoid , is a special plane algebraic curve of the 3rd order.

Equations of the straight strophoids

The following is a positive real number . In the graph on the right edge of the strophoid is as designated. In Cartesian coordinates the strophoid is defined by ${\displaystyle a}$ ${\displaystyle (-a,0)}$ ${\displaystyle S}$

${\displaystyle (a+x)x^{2}-(a-x)y^{2}=0\,.}$

A parametric plot of this curve is

${\displaystyle x={\frac {a(t^{2}-1)}{1+t^{2}}};\qquad y={\frac {at(t^{2}-1)}{1+t^{2}}}\,.}$

Looking at the strophoids in polar coordinates , their defining equation is

${\displaystyle r=-{\frac {a\cos(2\varphi )}{\cos \varphi }}\,.}$

Properties of the straight strophoids

In the following it is assumed that the coordinate axes are as shown in the sketch.

• The points of the straight strophoids are characterized by the following geometric property: Let S be the apex of the curve and P be any point on the curve that is different from S. If the point of intersection of the straight line SP and the curve, which is different from S and P, is designated as Q and the point of intersection with the y-axis as R, then R is equidistant from P and Q and from the origin O
• The straight strophoid is axially symmetrical with respect to the x-axis. Exactly two points of the curve lie on the symmetry axis, namely the origin and the vertex S with the coordinates . ${\displaystyle (-a;0)}$
• The origin of the coordinate system is a colon on the curve; i.e. it is run through twice. The two bisectors of the quadrants of the coordinate system coincide with the two tangents at the origin.
• The straight line with the equation (dashed in the sketch) is the asymptote of the curve. ${\displaystyle x=a}$
• The loop of the straight strophoids encloses a surface with the content . ${\displaystyle a^{2}\cdot \left(2-{\tfrac {\pi }{2}}\right)}$
• The area that is bounded by the curve and the asymptote and that extends to infinity has the area . ${\displaystyle a^{2}\cdot \left(2+{\tfrac {\pi }{2}}\right)}$
• The strophoide is also known under the names Ala , Focal , harmonic curve (after Werth ), Kukumaide and Pteroides torricellana .

generalization

general strophoids: orange + pink curve

A strophoid in the general sense can be defined as follows using a given curve C , a fixed point A and another point O ( Pol ): Let L be a variable straight line through O that intersects the given curve C at point K. P ₁ and P ₂ are the two points on L whose distance from K with the distance between A and K match. The locus of such points P ₁ and P ₂ is then the strophoid of C with respect to the pole O and the fixed point A . Note that AP ₁ and AP ₂ enclose a right angle in this construction ( Thales circle ).

In the special case in which C is a straight line, A lies on C and O outside of C , one speaks of a crooked strophoid . If OA is also perpendicular to C , the curve is called a straight strophoid , often just a strophoid (see above).

Equation in polar coordinates

The curve C is given by , where the point O is chosen as the origin. In addition, let A be the point ( a , b ). If there is a point on the curve, the distance between K and A is ${\displaystyle r=f(\varphi )}$ ${\displaystyle K=(r\cos \varphi ,\ r\sin \varphi )}$

${\displaystyle d={\sqrt {(r\cos \varphi -a)^{2}+(r\sin \varphi -b)^{2}}}={\sqrt {(f(\varphi )\cos \varphi -a)^{2}+(f(\varphi )\sin \varphi -b)^{2}}}}$.

The point on the straight line OK has the polar angle , and the points at distance d from K on this straight line have the distance from the origin. Hence the equation of the strophoids is given by ${\displaystyle \varphi }$ ${\displaystyle f(\varphi )\pm d}$

${\displaystyle r=f(\varphi )\pm {\sqrt {(f(\varphi )\cos \varphi -a)^{2}+(f(\varphi )\sin \varphi -b)^{2}}}}$

Equation in Cartesian coordinates

Let C be given by the parametric representation ( x ( t ), y ( t )). In addition, let A be the point ( a , b ) and O the point ( p , q ). Using the polar coordinate representation one immediately obtains:

${\displaystyle u(t)=p+(x(t)-p)(1\pm n(t)),\ v(t)=q+(y(t)-q)(1\pm n(t))}$,

in which

${\displaystyle n(t)={\sqrt {\frac {(x(t)-a)^{2}+(y(t)-b)^{2}}{(x(t)-p)^{2}+(y(t)-q)^{2}}}}}$.

Web links

Commons : Strophoid  - collection of pictures, videos and audio files

• Eric W. Weisstein : Strophoid . In: MathWorld (English).
• Eric W. Weisstein : Right Strophoid . In: MathWorld (English).
• John J. O'Connor, Edmund F. Robertson :  Right Strophoid. In: MacTutor History of Mathematics archive .

Individual evidence

1. ↑ Strophoids . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
2. ↑ Dörte Haftendorn: Exploring and Understanding Curves , Spektrum Akademischer Verlag 2016, p. 58