Editing Ellipsoid (section)

== Pins-and-string construction ==
[[File:Ellipse-gaertner-k.svg|upright=1|thumb|Pins-and-string construction of an ellipse:<br>
{{math|{{abs|''S''<sub>1</sub> ''S''<sub>2</sub>}}}}, length of the string (red)]]
[[File:Fokalks-ellipsoid.svg|thumb|upright=1.2|Pins-and-string construction of an ellipsoid, blue: focal conics]]
[[File:Fokalks-ellipsoid-xyz.svg|thumb|upright=1.2|Determination of the semi axis of the ellipsoid]]
The pins-and-string construction of an ellipsoid is a transfer of the idea constructing an ellipse using two [[ellipse#Pins-and-string method|pins and a string]] (see diagram).

A pins-and-string construction of an [[ellipsoid of revolution]] is given by the pins-and-string construction of the rotated ellipse. 

The construction of points of a ''triaxial ellipsoid'' is more complicated. First ideas are due to the Scottish physicist [[James Clerk Maxwell|J. C. Maxwell]] (1868).<ref> W. Böhm: ''Die FadenKonstruktion der Flächen zweiter Ordnung'', Mathemat. Nachrichten 13, 1955, S. 151</ref> Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898.<ref>Staude, O.: ''Ueber Fadenconstructionen des Ellipsoides''. Math. Ann. 20, 147–184 (1882)</ref><ref> Staude, O.: ''Ueber neue Focaleigenschaften der Flächen 2. Grades.'' Math. Ann. 27, 253–271 (1886).</ref><ref> Staude, O.: ''Die algebraischen Grundlagen der Focaleigenschaften der Flächen 2. Ordnung'' Math. Ann. 50, 398 - 428 (1898).</ref> The description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the book ''Geometry and the imagination'' written by [[David Hilbert|D. Hilbert]] & S. Vossen,<ref>D. Hilbert & S Cohn-Vossen: ''Geometry and the imagination'', Chelsea New York, 1952, {{ISBN|0-8284-1087-9}}, p. 20 .</ref> too.

=== Steps of the construction ===
# Choose an ''ellipse'' {{mvar|E}} and a ''hyperbola'' {{mvar|H}}, which are a pair of [[focal conics]]: <math display="block">\begin{align}
E(\varphi) &= (a\cos\varphi, b\sin\varphi, 0) \\
H(\psi) &= (c\cosh\psi, 0, b\sinh\psi),\quad c^2 = a^2 - b^2
\end{align} </math> with the vertices and foci of the ellipse <math display="block">S_1 = (a, 0, 0),\quad F_1 = (c, 0, 0),\quad F_2 = (-c, 0, 0),\quad S_2 = (-a, 0, 0)</math> and a ''string'' (in diagram red) of length {{mvar|l}}.
# Pin one end of the string to [[vertex (curve)|vertex]] {{math|''S''<sub>1</sub>}} and the other to focus {{math|''F''<sub>2</sub>}}. The string is kept tight at a point {{mvar|P}} with positive {{mvar|y}}- and {{mvar|z}}-coordinates, such that the string runs from {{math|''S''<sub>1</sub>}} to {{mvar|P}} behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from {{mvar|P}} to {{math|''F''<sub>2</sub>}} runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance {{math|{{abs|''S''<sub>1</sub> ''P''}}}} over any hyperbola point is at a minimum. The analogous statement on the second part of the string and the ellipse has to be true, too.
# Then: {{mvar|P}} is a point of the ellipsoid with equation  <math display="block">\begin{align}
 &\frac{x^2}{r_x^2} + \frac{y^2}{r_y^2} + \frac{z^2}{r_z^2} = 1 \\
 &r_x = \tfrac{1}{2}(l - a + c), \quad
  r_y = {\textstyle \sqrt{r^2_x - c^2}}, \quad
  r_z = {\textstyle \sqrt{r^2_x - a^2}}.
\end{align}</math>
# The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.

=== Semi-axes ===
Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point {{mvar|P}}:
:<math>Y = (0, r_y, 0),\quad Z = (0, 0, r_z).</math>

The lower part of the diagram shows that {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}} are the foci of the ellipse in the {{mvar|xy}}-plane, too. Hence, it is [[Confocal conic sections|confocal]] to the given ellipse and the length of the string is {{math|''l'' {{=}} 2''r<sub>x</sub>'' + (''a'' − ''c'')}}. Solving for {{mvar|r<sub>x</sub>}} yields {{math|''r<sub>x</sub>'' {{=}} {{sfrac|1|2}}(''l'' − ''a'' + ''c'')}}; furthermore {{math|''r''{{su|p=2|b=''y''}} {{=}} ''r''{{su|p=2|b=''x''}} − ''c''<sup>2</sup>}}.

From the upper diagram we see that {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}} are the foci of the ellipse section of the ellipsoid in the {{mvar|xz}}-plane and that {{math|''r''{{su|p=2|b=''z''}} {{=}} ''r''{{su|p=2|b=''x''}} − ''a''<sup>2</sup>}}.

=== Converse ===
If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters {{mvar|a}}, {{mvar|b}}, {{mvar|l}} for a pins-and-string construction. 

=== Confocal ellipsoids ===
If {{overline|{{mathcal|E}}}} is an ellipsoid [[Confocal quadrics|confocal]] to {{mathcal|E}} with the squares of its semi-axes 
: <math>\overline r_x^2 = r_x^2 - \lambda, \quad
  \overline r_y^2 = r_y^2 - \lambda, \quad
  \overline r_z^2 = r_z^2 - \lambda</math>

then from the equations of {{mathcal|E}}
: <math>  r_x^2 - r_y^2 = c^2, \quad
  r_x^2 - r_z^2 = a^2, \quad
  r_y^2 - r_z^2 = a^2 - c^2 = b^2</math>

one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes {{math|''a'', ''b'', ''c''}} as ellipsoid {{mathcal|E}}. Therefore (analogously to the foci of an ellipse) one considers the focal conics of a triaxial ellipsoid as the (infinite many) foci and calls them the '''focal curves''' of the ellipsoid.<ref>O. Hesse: ''Analytische Geometrie des Raumes'', Teubner, Leipzig 1861, p.&nbsp;287</ref>

The converse statement is true, too: if one chooses a second string of length {{math|{{overline|''l''}}}} and defines
:<math>\lambda = r^2_x - \overline r^2_x</math>
then the equations
:<math>\overline r_y^2 = r_y^2 - \lambda,\quad \overline r_z^2 = r_z^2 - \lambda</math>
are valid, which means the two ellipsoids are confocal.

=== Limit case, ellipsoid of revolution ===
In case of {{math|''a'' {{=}} ''c''}} (a [[spheroid]]) one gets {{math|''S''<sub>1</sub> {{=}} ''F''<sub>1</sub>}} and {{math|''S''<sub>2</sub> {{=}} ''F''<sub>2</sub>}}, which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the {{mvar|x}}-axis. The ellipsoid is [[Rotational symmetry|rotationally symmetric]] around the {{mvar|x}}-axis and
:<math>r_x = \tfrac12l,\quad r_y = r_z = {\textstyle \sqrt{r^2_x - c^2}}</math>.

=== Properties of the focal hyperbola ===
[[File:Ellipsoid-pk-zk.svg|thumb|upright=1.5|'''Top:''' 3-axial Ellipsoid with its focal hyperbola.<br>
'''Bottom:''' parallel and central projection of the ellipsoid such that it looks like a sphere, i.e. its apparent shape is a circle]]
; True curve
: If one views an ellipsoid from an external point {{mvar|V}} of its focal hyperbola, then it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing point {{mvar|V}} are the lines of a circular [[cone]], whose axis of rotation is the [[Tangent (geometry)|tangent line]] of the hyperbola at {{mvar|V}}.<ref>D. Hilbert & S Cohn-Vossen: ''Geometry and the Imagination'', p. 24</ref><ref>O. Hesse: ''Analytische Geometrie des Raumes'', p.&nbsp;301</ref> If one allows the center {{mvar|V}} to disappear into infinity, one gets an [[Orthogonal projection|orthogonal]] [[parallel projection]] with the corresponding [[asymptote]] of the focal hyperbola as its direction. The ''true curve of shape'' (tangent points) on the ellipsoid is not a circle.{{paragraph}} The lower part of the diagram shows on the left a parallel projection of an ellipsoid (with semi-axes 60, 40, 30) along an asymptote and on the right a central projection with center {{mvar|V}} and main point {{mvar|H}} on the tangent of the hyperbola at point {{mvar|V}}. ({{mvar|H}} is the foot of the perpendicular from {{mvar|V}} onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin {{mvar|O}} is the circle's center; in the central case main point {{mvar|H}} is the center.
; Umbilical points
: The focal hyperbola intersects the ellipsoid at its four [[umbilical point]]s.<ref>W. Blaschke: ''Analytische Geometrie'', p. 125</ref>

=== Property of the focal ellipse ===
The focal ellipse together with its inner part can be considered as the limit surface (an infinitely thin ellipsoid) of the [[Pencil (mathematics)|pencil]] of confocal ellipsoids determined by {{math|''a'', ''b''}} for {{math|''r<sub>z</sub>'' → 0}}. For the limit case one gets
:<math>r_x = a,\quad r_y = b,\quad l = 3a - c.</math>