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2024-01-29 - Constructing a four-point egg

  * ⇐ 2023-12-23 ⫤
  *  
  * ⊨ ☆ ⇒

For reasons beyond the scope of this entry, I have been investigating
elliptical and ovoid shapes. The Wikipedia article for Moss's egg has
a link to a tutorial on Euclidean Eggs by Freyja Hreinsdottir which
(amongst other things) describes how to construct the "four point
egg". I think it is a nicer shape than Moss's egg.

Freyja's construction uses a straight edge and compasses in classical
style, so it lacks dimensions. Below is my version, with the numbers
needed to draw it on a computer.

At first this construction might seem fairly rigid, but some of its
choices are more arbitrary than they might appear to be. I have made
an interactive four-point egg so you can drag the points around and
observe how its shape changes.

In the following, I will measure angles in fractions of a turn ,
clockwise from noon because the egg's pointy end is upwards.

 1. Start with axes and a unit circle.

 2. Imagine two diagonal scaffolding lines: one through the unit
    circle's south and east points; and another through its south and
    west points.

 3. Draw an arc centred on the unit circle's south point and
    equidistant from its north point, from one scaffolding line to
    the other. This forms the big end of the egg.

x: 0 y: -1 radius: 2 from: *3/8 to: *5/8

 4. Draw the egg's lower right arc, centred on the unit circle's west
    point, from the right end of the bottom arc up to the positive X
    axis.

x: -1 y: 0 radius: 2+[?]2 from: +*2/8 to: +*3/8

 5. Draw the egg's lower left arc mirroring the lower right arc.

x: +1 y: 0 radius: 2+[?]2 from: -*3/8 to: -*2/8

 6. The lower right and lower left arcs meet the X axis at the egg's
    east and west points. Imagine a scaffolding circle centred at the
    origin joining these points.

 7. The egg's north centre is where this scaffolding circle meets the
    Y axis at +1+[?]2.

    Imagine two new diagonal scaffolding lines: one through the egg's
    west point and north centre; and another through its east point
    and north centre.

 8. Draw the egg's upper right arc, centred on the egg's west point,
    from the egg's east point up to a scaffolding line.

x: -1-[?]2 y: 0 radius: 2+2[?]2 from: +*1/8 to: +*2/8

 9. Draw the egg's upper left arc mirroring the upper right arc.

x: +1+[?]2 y: 0 radius: 2+2[?]2 from: -*2/8 to: -*1/8

10. Draw an arc on the egg's north centre joining the ends of the
    upper right and upper left arcs. This arc forms the little end of
    the egg.

x: 0 y: 1+[?]2 radius: [?]2 from: -*1/8 to: +*1/8

The six coloured points marked in the diagram above are the centres
of the six arcs that make the egg. These comprise the four points the
egg is named after, plus their mirror images.

Moss's egg is a three-point egg. It has the same top half as this
four-point egg, but its bottom half is a simple semi-circle.

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⇐ 2023-12-23 ⇐ Popcorn &
DoubleLeftArrow; ⇒ ☆ ⇒ ☆
⇒
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Tony Finch <dot@dotat.at>