[HN Gopher] Constructing a Four-Point Egg
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Constructing a Four-Point Egg
Author : fanf2
Score : 104 points
Date : 2024-01-31 08:15 UTC (14 hours ago)
(HTM) web link (dotat.at)
(TXT) w3m dump (dotat.at)
| heckelson wrote:
| Great egg. Thanks for posting
| dr_kiszonka wrote:
| One free tool for making such diagrams is GeoGebra.
|
| Online: https://www.geogebra.org/geometry
|
| Offline: https://www.geogebra.org/download
|
| (I have no affiliation with it. It was used in an article OP's
| submission links to.)
| danbruc wrote:
| You can also try paper, pencil, ruler and compass.
| hughes wrote:
| You can try a bit of burnt wood and a cave wall.
| lapetitejort wrote:
| You can use a stick and some dirt, as long as no Macedonian
| kings block your sun
| blt wrote:
| wow, apparently this comment touched a nerve
| dylan604 wrote:
| Do schools even teach paper/pencil/ruler/compass any more? I
| get not teaching slide rule, but using a straight edge and
| compass is even older than slide rule. I actually enjoyed
| drawing things like this, but I know that's not something
| _all_ do. I can easily see where this is something lost with
| CAD and other graphing apps.
| lupire wrote:
| Yes, but they use a "non-collapsing" compass that can
| transmit distances without constructing them, not Euclid's
| compass (which can be used to transmit distances, with more
| work).
|
| http://euclidea.xyz
| dylan604 wrote:
| I like how you responded to a question about not using
| apps with an app.
| shashanoid wrote:
| Thanks having fun on geogebra
| rantallion wrote:
| What makes this a four-point egg if only the first three points
| are required to derive the rest of the shape?
| burkaman wrote:
| See Figure 15 on page 8:
| https://web.archive.org/web/20200618202007/https://www.dynam...
|
| The egg is constructed of four distinct arcs based on four
| different points, plus two of them mirrored.
| rantallion wrote:
| Thanks. This makes more sense. I was curious why the _inner_
| point mattered, when the fourth point is actually the one on
| the side.
| eduhetxub wrote:
| I don't know exactly what you mean, but the name of this
| shape is consistent with the naming of traditional
| geometric constructions like the "five-centered arch" and
| the "three-centered arch" (see many diagrams online for
| details of these constructions).
|
| So "four-centered egg" might make more sense to you, though
| it could also be called the "six-centered egg", because it
| includes both sides, unlike the classical arch
| constructions.
| a_t48 wrote:
| There are four points inside the egg.
| kazinator wrote:
| Also, why do we say "four legged table" in reference to common,
| rectangular tables? The position of the fourth leg is
| determined from the other three, so it shouldn't count!
| rantallion wrote:
| I'd love to see a three-legged table stand up with the fourth
| corner under load.
| kazinator wrote:
| It has a leg there; it's just _called_ a three legged
| table, because one of the legs is geometrically implied by
| the other three.
| SamBam wrote:
| Surely you only need two, as you could use two diagonal
| legs.
|
| Let's see if the name "two-legged table" takes off.
| lupire wrote:
| You only need 1 center leg.
| dylan604 wrote:
| my grandfather was a custom cabinet builder, and for
| whatever reason 4 legged tables were his absolute
| nemesis. i have the last remaining 3 legged table of his.
| it's a round table, so i thought the 3 legs were a design
| choice. that notion was corrected when my dad started
| laughing at remembering his dad's sheer frustration when
| he realized 3 of 4 table legs were the same length...on
| multiple occasions.
| kazinator wrote:
| Even if a four legged table's legs are all perfectly
| equal so their tips lie in the same plane, that perfectly
| will be thwarted by an uneven floor.
| addaon wrote:
| > Even if a four legged table's legs are all perfectly
| equal so their tips lie in the same plane, that perfectly
| will be thwarted by an uneven floor.
|
| Unless you're able to rotate it:
| https://www.scientificamerican.com/article/strange-but-
| true-...
| adrianmonk wrote:
| There are ways:
|
| (1) Diagonal legs that give the table a wide stance. Draw a
| triangle that the tabletop fits entirely inside of. The
| legs touch the ground at the vertices of the triangle.
|
| (2) Make the table itself arbitrarily heavy. A downward
| force on the corner of the tabletop will create torque
| around an axis, and this torque wants to tip it over. But
| the table's center of mass is on the other side of this
| axis, and the table's weight creates torque too. If the
| table's mass is high enough, this torque is greater, and it
| won't tip.
|
| (3) Screw the table to the floor. Is this a table? I think
| it's still a table. You often see tables attached to the
| floor.
|
| (4) Make one leg really wide so that it stretches from one
| corner to the other. The other two legs can be traditional.
| kazinator wrote:
| [delayed]
| dylan604 wrote:
| why do we say a rectangle has four sides, when it's just 2
| reflected
| gowld wrote:
| The Wikipedia article about the 2-point (Moss) egg (3 points
| derived from 2 freely chosen points) cites Freyja's article
| https://web.archive.org/web/20200618202007/https://www.dynam...
|
| which has a 2-point (Moss) egg, a 4-point egg, and a 5-point
| egg. The OP article has only the 4-point egg.
| fanf2 wrote:
| Moss's egg is a 3 point egg: it has 4 arcs, two of which are
| a mirrored pair.
| SamBam wrote:
| Only the very first circle is needed to derive the rest of the
| shape. So that's a point and a radius, or two points.
| nick238 wrote:
| Two points (x1, y1, x2, y2) or 4 total parameters (x, y,
| rotation, scale) assuming you're using the exact shape
| provided, but from the interactive egg you can free the
| location/angles that they used for construction so you can
| wind up with less egg-like 4-point eggs.
|
| In the interactive link provided, you control the positions
| of 4 circle centers, each restricted along one of the axes.
| So you're giving 4 parameters (which also gives scale), and
| after including x-y offsets and rotation, it's 7 total
| numbers to make an arbitrary '4-point' egg anywhere in the
| plane
| lupire wrote:
| That's for a specific case of 4-point egg. The points can be
| moved to make different egg shapes. The OP links to an
| interactive app.
| samstave wrote:
| It looks like the four-point egg is defined by 8 circles, and
| connecting the intersections, of which their are four pair.
|
| Im just surprised the author never once used the proper term
| for the "west" "north" points; Quadrant. Its even the name for
| the snap function in all snap menus.
| fanf2 wrote:
| A quadrant is a quarter of a planar shape, not a point.
| etwigg wrote:
| I didn't know an egg could be so beautiful <3
| 1970-01-01 wrote:
| Eggcellent :)
| ks2048 wrote:
| Nice. I wonder how closely it could be approximated with a small
| number of Bezier curves. (for example, a quadratic with six
| points. 4 control points and two side points).
| ryandrake wrote:
| I wonder if you could make it even more "beautiful" by gradually
| easing between the various curvatures rather than having the
| discontinuities at the intersections of the curves. Kind of like
| the difference between a square with rounded corners and a
| "squircle"[1].
|
| 1: https://webflow.com/blog/squircle-vs-rounded-squares
| alexchamberlain wrote:
| Do you have discontinuities though? The tangents at the point
| of intersection match.
| fanf2 wrote:
| There is a discontinuity in curvature where each arc meets.
| It's possible to make joins between curves smoother using
| Euler spirals, but I don't know how to use them well enough
| to apply them to eggs.
| https://en.m.wikipedia.org/wiki/Euler_spiral
|
| I recently tried to draw f holes using Euler spirals, but
| they turned out too short and fat, and I gave up on them. But
| they are quite fun to draw on a computer by simple numerical
| integration, ignoring all the more complicated mathematics in
| the wikipedia article. Perhaps I could have made my f holes
| more elegant by understanding the parts I ignored...
| https://en.m.wikipedia.org/wiki/Sound_hole
| pimlottc wrote:
| The issue is more the smoothness in the change in curvy-ness:
|
| https://www.figma.com/blog/desperately-seeking-squircles/
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