View source for Strophoid

{{Short description|Geometric curve constructed from another curve and two points}}
[[File:Allgemeine strophoide5.svg|thumb|right|upright=1.25|'''Construction of a strophoid.'''
{{legend-line|solid #333333|Given curve {{mvar|C}}}}
{{legend-line|solid blue|Variable line {{mvar|L}} which rotates about the pole {{mvar|O}}; intersects {{mvar|C}} at point {{mvar|K}}}}
{{legend-line|dashed red|Circle centered at {{mvar|K}} whose size is constrained by the fixed point {{mvar|A}}; intersects {{mvar|L}} at {{math|''P''{{sub|1}}}} and {{math|''P''{{sub|2}}}}}}
{{legend-line|solid orange|Inner portion of the strophoid curve, traced by {{math|''P''{{sub|1}}}} as {{mvar|L}} rotates}}
{{legend-line|solid #FF1493|Outer portion of the strophoid curve, traced by {{math|''P''{{sub|2}}}} as {{mvar|L}} rotates}}
]]

In [[geometry]], a '''strophoid''' is a curve generated from a given curve {{mvar|C}} and points {{mvar|A}} (the '''fixed point''') and {{mvar|O}} (the '''pole''') as follows: Let {{mvar|L}} be a variable [[Line (geometry)|line]] passing through {{mvar|O}} and [[Intersection (geometry)|intersecting]] {{mvar|C}} at {{mvar|K}}. Now let {{math|''P''{{sub|1}}}} and {{math|''P''{{sub|2}}}} be the two points on {{mvar|L}} whose distance from {{mvar|K}} is the same as the distance from {{mvar|A}} to {{mvar|K}} (i.e. {{math|1={{overline|''KP''}}{{sub|1}} = {{overline|''KP''}}{{sub|2}} = {{overline|''AK''}}}}). The [[locus (mathematics)|locus]] of such points {{math|''P''{{sub|1}}}} and {{math|''P''{{sub|2}}}} is then the strophoid of {{mvar|C}} with respect to the pole {{mvar|O}} and fixed point {{mvar|A}}. Note that {{math|{{overline|''AP''}}{{sub|1}}}} and {{math|{{overline|''AP''}}{{sub|2}}}} are at right angles in this construction.

In the special case where {{mvar|C}} is a line, {{mvar|A}} lies on {{mvar|C}}, and {{mvar|O}} is not on {{mvar|C}}, then the curve is called an '''oblique strophoid'''. If, in addition, {{mvar|{{overline|OA}}}} is [[perpendicular]] to {{mvar|C}} then the curve is called a '''right strophoid''', or simply ''strophoid'' by some authors. The right strophoid is also called the '''logocyclic curve''' or '''foliate'''.

==Equations==

===Polar coordinates===
Let the curve {{mvar|C}} be given by <math>r = f(\theta),</math> where the origin is taken to be {{mvar|O}}. Let {{mvar|A}} be the point {{math|(''a'', ''b'')}}. If <math>K = (r \cos\theta,\ r \sin\theta)</math> is a point on the curve the distance from {{mvar|K}} to {{mvar|A}} is 
:<math>d = \sqrt{(r \cos\theta - a)^2 + (r \sin\theta - b)^2} = \sqrt{(f(\theta) \cos\theta - a)^2 + (f(\theta) \sin\theta - b)^2}.</math>
The points on the line {{mvar|{{overline|OK}}}} have polar angle {{mvar|θ}}, and the points at distance {{mvar|d}} from {{mvar|K}} on this line are distance <math>f(\theta) \pm d</math> from the origin. Therefore, the equation of the strophoid is given by
:<math>r = f(\theta) \pm \sqrt{(f(\theta) \cos\theta - a)^2 + (f(\theta) \sin\theta - b)^2}</math>

===Cartesian coordinates===
Let {{mvar|C}} be given parametrically by {{math|(''x''(''t''), ''y''(''t''))}}. Let {{mvar|A}} be the point {{math|(''a'', ''b'')}} and let {{mvar|O}} be the point {{math|(''p'', ''q'')}}. Then, by a straightforward application of the polar formula, the strophoid is given parametrically by:
:<math>u(t) = p + (x(t)-p)(1 \pm n(t)),\ v(t) = q + (y(t)-q)(1 \pm n(t)),</math>
where
:<math>n(t) = \sqrt{\frac{(x(t)-a)^2+(y(t)-b)^2}{(x(t)-p)^2+(y(t)-q)^2}}.</math>

===An alternative polar formula===
The complex nature of the formulas given above limits their usefulness in specific cases. There is an alternative form which is sometimes simpler to apply. This is particularly useful when {{mvar|C}} is a [[sectrix of Maclaurin]] with poles {{mvar|O}} and {{mvar|A}}.

Let {{mvar|O}} be the origin and {{mvar|A}} be the point {{math|(''a'', 0)}}. Let {{mvar|K}} be a point on the curve, {{mvar|θ}} the angle between {{mvar|{{overline|OK}}}} and the {{mvar|x}}-axis, and {{tmath|\vartheta}} the angle between {{mvar|{{overline|AK}}}} and the {{mvar|x}}-axis. Suppose {{tmath|\vartheta}} can be given as a function {{mvar|θ}}, say <math>\vartheta = l(\theta).</math> Let {{mvar|ψ}} be the angle at {{mvar|K}} so <math>\psi = \vartheta - \theta.</math> We can determine {{mvar|r}} in terms of {{mvar|l}} using the [[law of sines]]. Since
:<math>{r \over \sin \vartheta} = {a \over \sin \psi},\ r = a \frac {\sin \vartheta}{\sin \psi} = a \frac {\sin l(\theta)}{\sin (l(\theta) - \theta)}.</math>

Let {{math|''P''{{sub|1}}}} and {{math|''P''{{sub|2}}}} be the points on {{mvar|{{overline|OK}}}} that are distance {{mvar|{{overline|AK}}}} from {{mvar|K}}, numbering so that <math>\psi = \angle P_1KA</math> and <math>\pi-\psi = \angle AKP_2.</math> {{math|△''P''{{sub|1}}''KA''}} is isosceles with vertex angle {{mvar|ψ}}, so the remaining angles, {{tmath|\angle AP_1K}} and {{tmath|\angle KAP_1,}} are <math>\tfrac{\pi-\psi}{2}.</math> The angle between {{math|{{overline|''AP''}}{{sub|1}}}} and the {{mvar|x}}-axis is then 
:<math>l_1(\theta) = \vartheta + \angle KAP_1 = \vartheta + (\pi-\psi)/2 = \vartheta + (\pi - \vartheta + \theta)/2 = (\vartheta+\theta+\pi)/2.</math>

By a similar argument, or simply using the fact that {{math|{{overline|''AP''}}{{sub|1}}}} and {{math|{{overline|''AP''}}{{sub|2}}}} are at right angles, the angle between {{math|{{overline|''AP''}}{{sub|2}}}} and the {{mvar|x}}-axis is then 
:<math>l_2(\theta) = (\vartheta+\theta)/2.</math>

The polar equation for the strophoid can now be derived from {{math|''l''{{sub|1}}}} and {{math|''l''{{sub|2}}}} from the formula above:
:<math>\begin{align}
& r_1=a \frac {\sin l_1(\theta)}{\sin (l_1(\theta) - \theta)} = a \frac {\sin ((l(\theta)+\theta+\pi)/2)}{\sin ((l(\theta)+\theta+\pi)/2 - \theta)} = a \frac{\cos ((l(\theta)+\theta)/2)}{\cos ((l(\theta)-\theta)/2)} \\

& r_2=a \frac {\sin l_2(\theta)}{\sin (l_2(\theta) - \theta)} = a \frac {\sin ((l(\theta)+\theta)/2)}{\sin ((l(\theta)+\theta)/2 - \theta)} = a \frac{\sin((l(\theta)+\theta)/2)}{\sin((l(\theta)-\theta)/2)}
\end{align}</math>

{{mvar|C}} is a sectrix of Maclaurin with poles {{mvar|O}} and {{mvar|A}} when {{mvar|l}} is of the form <math>q \theta + \theta_0,</math> in that case {{math|''l''{{sub|1}}}} and {{math|''l''{{sub|2}}}} will have the same form so the strophoid is either another sectrix of Maclaurin or a pair of such curves. In this case there is also a simple polar equation for the polar equation if the origin is shifted to the right by {{mvar|a}}.

==Specific cases==
Strophoids of lines are actually expressible as singular cubics in the projective plane.
===Oblique strophoids===
Let {{mvar|C}} be a line through {{mvar|A}}. Then, in the notation used above, <math>l(\theta) = \alpha</math> where {{mvar|α}} is a constant. Then <math>l_1(\theta) = (\theta + \alpha + \pi)/2</math> and <math>l_2(\theta) = (\theta + \alpha)/2.</math> The polar equations of the resulting strophoid, called an oblique strphoid, with the origin at {{mvar|O}} are then
:<math>r = a \frac{\cos ((\alpha+\theta)/2)}{\cos ((\alpha-\theta)/2)}</math>
and
:<math>r = a \frac{\sin ((\alpha+\theta)/2)}{\sin ((\alpha-\theta)/2)}.</math>
It's easy to check that these equations describe the same curve.

Moving the origin to {{mvar|A}} (again, see [[Sectrix of Maclaurin]]) and replacing {{mvar|−a}} with {{mvar|a}} produces
:<math>r=a\frac{\sin(2\theta-\alpha)}{\sin(\theta-\alpha)},</math>
and rotating by <math>\alpha</math> in turn produces
:<math>r=a\frac{\sin(2\theta+\alpha)}{\sin(\theta)}.</math>

In rectangular coordinates, with a change of constant parameters, this is
:<math>y(x^2+y^2)=b(x^2-y^2)+2cxy.</math>
This is a cubic curve and, by the expression in polar coordinates it is rational. It has a [[crunode]] at {{math|(0, 0)}} and the line {{math|1=''y'' = ''b''}} is an asymptote.

===The right strophoid===
[[File:Gerade strophoide rechts.svg|thumb|upright=1.0|A right strophoid]]
Putting <math>\alpha = \pi/2</math> in
:<math>r=a\frac{\sin(2\theta-\alpha)}{\sin(\theta-\alpha)}</math>
gives
:<math>r=a\frac{\cos 2\theta}{\cos \theta} = a(2\cos\theta-\sec\theta).</math>
This is called the '''right strophoid''' and corresponds to the case where {{mvar|C}} is the {{mvar|y}}-axis, {{mvar|A}} is the origin, and {{mvar|O}} is the point {{math|(''a'', 0)}}.

The [[Cartesian coordinate system|Cartesian]] equation is 
:<math>y^2 = x^2(a-x)/(a+x).</math>

The curve resembles the [[Folium of Descartes]]<ref>{{cite EB1911 |wstitle=Logocyclic Curve |display=Logocyclic Curve, Strophoid or Foliate|volume=16 |page=919}}</ref> and the line {{math|1=''x'' = –''a''}} is an [[asymptote]] to two branches. The curve has two more asymptotes, in the plane with complex coordinates, given by
:<math>x\pm iy = -a.</math>

This curve passes through the two [[circular points at infinity]] and is a special case of a focal circular Van Rees cubic.

===Circles===
Let {{mvar|C}} be a circle through {{mvar|O}} and {{mvar|A}}, where {{mvar|O}} is the origin and {{mvar|A}} is the point {{math|(''a'', 0)}}. Then, in the notation used above, <math>l(\theta) = \alpha+\theta</math> where <math>\alpha</math> is a constant. Then <math>l_1(\theta) = \theta + (\alpha + \pi)/2</math> and <math>l_2(\theta) = \theta + \alpha/2.</math> The polar equations of the resulting strophoid, called an oblique strophoid, with the origin at {{mvar|O}} are then
:<math>r = a \frac{\cos (\theta+\alpha/2)}{\cos (\alpha/2)}</math>
and
:<math>r = a \frac{\sin (\theta+\alpha/2)}{\sin (\alpha/2)}.</math>
These are the equations of the two circles which also pass through {{mvar|O}} and {{mvar|A}} and form angles of <math>\pi/4</math> with {{mvar|C}} at these points.

==See also==
*[[Conchoid (mathematics)|Conchoid]]
*[[Cissoid]]

==References==
{{Reflist}}
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogofspecial00lawr/page/51 51–53,95,100–104,175] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/51 }}
* {{cite book | author=E. H. Lockwood | title=A Book of Curves | publisher=Cambridge University Press | year=1961 | pages=134–137 | chapter = Strophoids |location = Cambridge, England | isbn=0-521-05585-7}}
* {{cite book | author=R. C. Yates| title=A Handbook on Curves and Their Properties | publisher=J. W. Edwards | year=1952| pages=217–220 | chapter = Strophoids |location = Ann Arbor, MI}}
* {{MathWorld|title=Strophoid|urlname=Strophoid}}
* {{MathWorld|title=Right Strophoid|urlname=RightStrophoid}}
* {{springer| title=Strophoid | id=S/s090630 | last=Sokolov | first=D.D.}}
* {{MacTutor|class=Curves|id=Right|title=Right Strophoid}}

==External links==
{{commonscat-inline|Strophoid}}

{{Differential transforms of plane curves}}


[[Category:Curves]]