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You are given a pen, pencil, pair of compasses, scissors, a ruler and a piece of paper.

I require a natural way to draw an ellipse. This can be simple, from using a circle piece of paper, to a complex algorithm, just as long as it gives a perfect ellipse. How would I go about doing this?

$a$ is the length of the minor axis

$b$ is the length of the major axis

$a,b \neq 0$

$a\neq b$

For fleablood, it is required to make an ellipse of specific proportions of $a$ and $b$

Well, you are given these equipment. Why can't you bend the usage of it? Just as long as it creates a perfect eclipse

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    $\begingroup$ If you're allowed to add a piece of string and a couple thumb tacks, then pin the string at two points and use the pencil to draw the string taut. $\endgroup$ – Daniel Schepler Jun 30 '17 at 18:42
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    $\begingroup$ Take a look at (britton.disted.camosun.bc.ca/folding_conic_sections.pdf) (no scissors, no rule, no compass...) $\endgroup$ – Jean Marie Jun 30 '17 at 18:43
  • $\begingroup$ Or, if you can add a point light source (or something close to that), then take the circular piece of paper and rotate it a bit. The shadow will form a conic section. $\endgroup$ – Daniel Schepler Jun 30 '17 at 18:45
  • $\begingroup$ Cheating. A circle is an ellipse. You never claimed anything about the dimensions of the ellipse. Can it be any focal length (but non zero) and minor axis length (but not zero). Must you be able to construct all arbitrary ellipses or merely any incidental one. $\endgroup$ – fleablood Jun 30 '17 at 18:45
  • $\begingroup$ I think we can assume string, light, and freehand tracing are not allowed. $\endgroup$ – fleablood Jun 30 '17 at 18:46
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Draw points like this, as many as you wish:

ellipse

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Draw the axes of the ellipse, perpendicular between them.

Mark on the edge of your ruler three points $F$, $P$, $E$ (in that order), such that $PF=a$ and $PE=b$. Place the ruler with $F$ at any point on an axis and $E$ on the other one: then $P$ will be at a point on the required ellipse.

Marking many positions of point $P$ you can draw as many points of the ellipse as you want.

This animation shows three different ellipses having the same $a+b$.

animation

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You can't with the objects that you described, as far as I know, (Just think that you can't draw a curve without constant curvature with the objects you mentioned) but if you generalize to a 3D problem, then select the plane and the two points that are the centre of the ellipse, $r$ radius of the ellipse, and act as follows:

Choose two reals $r_1, r_2 $ such that $r_1 + r_2 = r$, rather easy geometry problem with pen and ruler assuming you can just choose a point in a line, don't want to go into details on this. With your 3D compass draw two spheres tangent to the plane in the given points with the given radius (can you draw 3D perpendiculars? Of course you can, start by picking a perpendicular plane by intersecting two spheres).

Draw the cone that is tangent to both spheres (simple in 2D, no complication arises in 3D) and intersect such cone with the original plane.

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If this was a straightedge and compass problem then I would agree with PenasRaul that it's not possible to make an ellipse. However we are given a few more things to work with. 1) a ruler and 2) a pair of scissors either one can be used to produce an ellipse. I say ellipse because trying to make a perfect ellipse in real life is like trying to make a perfect rectangle in real life (the existence of which could break both euclidean and non euclidean geometry). If you want a perfect ellipse use a computer. Otherwise, please continue to read.

As there is already an answer to how to use a ruler to make one, so I shall go through a step by step process with the use of the scissors. There are a few way to do this as well but I like this one the best. (Oh And I will only use the ruler as a straightedge btw.)

1) Use the ruler and scissors to cut a strip of paper along the longest edge of the paper. This doesn't have to be exact, but it should be about as wide as the diameter of the pen or pencil.

2) (optional) roll the strip into a cylinder. This will make things easier to use, but it is not as necessary for what we're going to use it for. you can also fold it in half or fourths (if you don't want any frayed edges).

3) At one end make a loop and thread the other end through the loop and looping it back through to make a square knot. Secure the knot, doesn't have to be tight.

4) Set both compasses to be at right angles. This can be done by constructing a perpendicular bisector with the straightedge and a compass. Once that is done, hold onto both of the compasses in one hand such that the needles are jutting out, making sure that the paper loop can fit over it. This will act as foci.

5) Place the paper loop made in step 3) around the two compasses and place the needles on the paper.

6) Use either the pen or the pencil to trace the ellipse using the paper loop as a tracing guide.

While the paper loop is a makeshift string, it was constructed using the materials at hand. So No magical can opener.

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Draw a circle inscribed in a square $a$ by $a$.

Mark as many points on a circle as you like, and project them orthogonally to the square's sides.

Scale one $a$ side to a length of $b$ (e.g. by a central projection to a parallel line).

Build a rectangle $a$ by $b$ with copies of those projected points on its sides.

Draw a pair of lines paralel to the rectangle's sides through each pair of corresponding points on rectangle's sides. Intersection of each pair defines a point on the ellipse.

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