Editing Ellipse (section)

=== 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]]