Editing Ellipse (section)

== Drawing ellipses ==
[[File:Zp-turm-tor.svg|thumb|Central projection of circles (gate)]]
Ellipses appear in [[descriptive geometry]] as images (parallel or central projection) of circles. There exist various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools (''[[ellipsograph]]s'') to draw an ellipse without a computer exist. The principle was known to the 5th century mathematician [[Proclus]], and the tool now known as an [[elliptical trammel]] was invented by [[Leonardo da Vinci]].<ref>{{cite journal |last=Blake |first=E. M. |year=1900 |title=The Ellipsograph of Proclus |journal=American Journal of Mathematics |volume=22 |number=2 |pages=146–153 |doi=10.2307/2369752 |jstor=2369752 |jstor-access=free }}</ref>

If there is no ellipsograph available, one can draw an ellipse using an [[#Approximation by osculating circles|approximation by the four osculating circles at the vertices]].

For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of [[Rytz's construction]] the axes and semi-axes can be retrieved.

=== de La Hire's point construction ===
The following construction of single points of an ellipse is due to [[Philippe de La Hire|de La Hire]].<ref>K. Strubecker: ''Vorlesungen über Darstellende Geometrie.'' Vandenhoeck & Ruprecht, Göttingen 1967, S. 26.</ref>  It is based on the [[#Standard parametric representation|standard parametric representation]] <math>(a\cos t,\, b\sin t)</math> of an ellipse:
# Draw the two ''circles'' centered at the center of the ellipse with radii <math>a,b</math> and the axes of the ellipse.
# Draw a ''line through the center'', which intersects the two circles at point <math>A</math> and <math>B</math>, respectively.
# Draw a ''line'' through <math>A</math> that is parallel to the minor axis and a ''line'' through <math>B</math> that is parallel to the major axis. These lines meet at an ellipse point <math>P</math> (see diagram).
# Repeat steps (2) and (3) with different lines through the center.
<gallery widths="220" heights="220" class="float-left">
Elliko-sk.svg|de La Hire's method
Parametric ellipse.gif|Animation of the method
</gallery>

[[File:Elliko-g.svg|250px|thumb|Ellipse: gardener's method]]

===Pins-and-string method===
The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two [[drawing pin]]s, a length of string, and a pencil. In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string is tied at each end to the two pins; its length after tying is <math>2a</math>. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the ''gardener's ellipse''. The Byzantine architect [[Anthemius of Tralles]] ({{c.|600}}) described how this method could be used to construct an elliptical reflector,<ref>From {{lang|el|Περί παραδόξων μηχανημάτων}} [''Concerning Wondrous Machines'']: "If, then, we stretch a string surrounding the points A, B tightly around the first point from which the rays are to be reflected, the line will be drawn which is part of the so-called ellipse, with respect to which the surface of the mirror must be situated." {{pb}} {{cite book |last=Huxley |first=G. L. |year=1959 |title=Anthemius of Tralles: A Study in Later Greek Geometry |lccn=59-14700 |location=Cambridge, MA |pages=8–9 |url=https://archive.org/details/anthemiusoftrall0000huxl/page/8/ |url-access=limited }}</ref> and it was elaborated in a now-lost 9th-century treatise by [[Al-Ḥasan ibn Mūsā ibn Shākir|Al-Ḥasan ibn Mūsā]].<ref>Al-Ḥasan's work was titled {{transliteration|ar|Kitāb al-shakl al-mudawwar al-mustaṭīl}} [''The Book of the Elongated Circular Figure'']. {{pb}} {{cite book |last1=Rashed |first1=Roshdi |translator-last=Shank |translator-first=Michael H. |title=Classical Mathematics from Al-Khwarizmi to Descartes |date=2014 |publisher=Routledge |location=New York |isbn=978-13176-2-239-0 |page=559 }}</ref>

A similar method for drawing [[Confocal conic sections#Graves's theorem: the construction of confocal ellipses by a string|confocal ellipses]] with a ''closed'' string is due to the Irish bishop [[Charles Graves (bishop)|Charles Graves]].

=== Paper strip methods ===
The two following methods rely on the parametric representation (see ''{{slink||Standard parametric representation}}'', above):
<math display="block">(a\cos t,\, b\sin t)</math>

This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes <math> a,\, b</math> have to be known.

;Method 1
The first method starts with
: a strip of paper of length <math>a + b</math>.

The point, where the semi axes meet is marked by <math>P</math>. If the strip slides with both ends on the axes of the desired ellipse, then point <math>P</math> traces the ellipse. For the proof one shows that point <math>P</math> has the parametric representation <math>(a\cos t,\, b\sin t)</math>, where parameter <math>t</math> is the angle of the slope of the paper strip.

A technical realization of the motion of the paper strip can be achieved by a [[Tusi couple]] (see animation). The device is able to draw any ellipse with a ''fixed'' sum <math>a + b</math>, which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method.

<gallery widths="250" heights="250">
Elliko-pap1.svg|Ellipse construction: paper strip method 1
Tusi couple vs Paper strip plus Ellipses horizontal.gif|Ellipses with Tusi couple. Two examples: red and cyan.
</gallery>

A variation of the paper strip method 1 uses the observation that the midpoint <math>N</math> of the paper strip is moving on the circle with center <math>M</math> (of the ellipse) and radius <math>\tfrac{a + b}{2}</math>. Hence, the paperstrip can be cut at point <math>N</math> into halves, connected again by a joint at <math>N</math> and the sliding end <math>K</math> fixed at the center <math>M</math> (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged.<ref>J. van Mannen: ''Seventeenth century instruments for drawing conic sections.'' In: ''The Mathematical Gazette.'' Vol. 76, 1992, p. 222–230.</ref> This variation requires only one sliding shoe.

<gallery widths="300" heights="200">
Ellipse-papsm-1a.svg|Variation of the paper strip method 1
Ellipses with SliderCrank inner Ellipses.gif|Animation of the variation of the paper strip method 1
</gallery>
[[File:Elliko-pap2.svg|250px|thumb|Ellipse construction: paper strip method 2]]
; Method 2:
The second method starts with
: a strip of paper of length <math>a</math>.

One marks the point, which divides the strip into two substrips of length <math>b</math> and <math>a - b</math>. The strip is positioned onto the axes as described in the diagram. Then the free end of the strip traces an ellipse, while the strip is moved. For the proof, one recognizes that the tracing point can be described parametrically by <math>(a\cos t,\, b\sin t)</math>, where parameter <math>t</math> is the angle of slope of the paper strip.

This method is the base for several ''ellipsographs'' (see section below).

Similar to the variation of the paper strip method 1 a ''variation of the paper strip method 2'' can be established (see diagram) by cutting the part between the axes into halves.
<gallery widths="200" heights="150">
File:Archimedes Trammel.gif|[[Elliptical trammel]] (principle)
File:L-Ellipsenzirkel.png|Ellipsograph due to [[Benjamin Bramer]]
File:Ellipses with SliderCrank Ellipses at Slider Side.gif|Variation of the paper strip method 2
</gallery>Most ellipsograph [[Drafting machine|drafting]] instruments are based on the second paperstrip method.[[File:Elliko-skm.svg|250px|thumb|Approximation of an ellipse with osculating circles]]

=== Approximation by osculating circles ===
From ''Metric properties'' below, one obtains:
* The radius of curvature at the vertices <math>V_1,\, V_2</math> is: <math>\tfrac{b^2}{a}</math>
* The radius of curvature at the co-vertices <math>V_3,\, V_4</math> is: <math>\tfrac{a^2}{b}\ .</math>

The diagram shows an easy way to find the centers of curvature <math>C_1 = \left(a - \tfrac{b^2}{a}, 0\right),\, C_3 = \left(0, b - \tfrac{a^2}{b}\right)</math> at vertex <math>V_1</math> and co-vertex <math>V_3</math>, respectively:
# mark the auxiliary point <math>H = (a,\, b)</math> and draw the line segment <math>V_1 V_3\ ,</math>
# draw the line through <math>H</math>, which is perpendicular to the line <math>V_1 V_3\ ,</math>
# the intersection points of this line with the axes are the centers of the osculating circles.
(proof: simple calculation.)

The centers for the remaining vertices are found by symmetry.

With help of a [[French curve]] one draws a curve, which has smooth contact to the [[osculating circle]]s.

=== Steiner generation ===
[[File:Ellipse-steiner-e.svg|250px|thumb|Ellipse: Steiner generation]]
[[File:Ellipse construction - parallelogram method.gif|200px|thumb|Ellipse: Steiner generation]]
The following method to construct single points of an ellipse relies on the [[Steiner conic|Steiner generation of a conic section]]:
: Given two [[pencil (mathematics)|pencils]] <math>B(U),\, B(V)</math> of lines at two points <math>U,\, V</math> (all lines containing <math>U</math> and <math>V</math>, respectively) and a projective but not perspective mapping <math>\pi</math> of <math>B(U)</math> onto <math>B(V)</math>, then the intersection points of corresponding lines form a non-degenerate projective conic section.

For the generation of points of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> one uses the pencils at the vertices <math>V_1,\, V_2</math>. Let <math>P = (0,\, b)</math> be an upper co-vertex of the ellipse and <math>A = (-a,\, 2b),\, B = (a,\,2b)</math>.

<math>P</math> is the center of the rectangle <math>V_1,\, V_2,\, B,\, A</math>. The side  <math>\overline{AB}</math> of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal <math>AV_2</math> as direction onto the line segment  <math>\overline{V_1B}</math> and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation is part of the projective mapping between the pencils at <math>V_1</math> and <math>V_2</math> needed. The intersection points of any two related lines   <math>V_1 B_i</math> and <math>V_2 A_i</math> are points of the uniquely defined ellipse. With help of the points <math>C_1,\, \dotsc</math> the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.

Steiner generation can also be defined for hyperbolas and parabolas. It is sometimes called a ''parallelogram method'' because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.

=== As hypotrochoid ===
[[File:Ellipse as hypotrochoid.gif|right|300px|thumb|An ellipse (in red) as a special case of the [[hypotrochoid]] with&nbsp;''R''&nbsp;=&nbsp;2''r'']]
The ellipse is a special case of the [[hypotrochoid]] when <math>R = 2r</math>, as shown in the adjacent image. The special case of a moving circle with radius <math>r</math> inside a circle with radius <math>R = 2r</math> is called a [[Tusi couple]].