hi we are time astronauts or such but are interested in working some if, then maybe time to associate surface information with a sewing pattern how about we find somebody else's data so it's possible that some die cutters can only cut 12 inch wide stuff (have not learned more about them at this time but it seems reasonably possible) it's also clear the one we tried only cuts 24 inch wide stuff, which means only spheroids of 3.8 inch radius in the current approach it's juts a little bit under the radius of a human head, 3.8 inches, a head is maybe 3.9 . hats have radii of 3.25 to 4 inches. but it may be that we can often only cut 12 inch wide things so that idea was to change how the fabric goes over the object surface, so it can spiral. then you could still cover an arbitrarily large surface with only 12 inch width fabric. it means solving a different surface problem also it might be interesting to consider solving it generally, for bumpy surfaces. we thought a little bit about bumpy surfaces and the sewing pattern problem sounds harder than originally considered (although also in a pretty different state of mind). consider a sphere with a length of fabric wrapping it; the middle of the fabric travels the circumference of the sphere the upper and lower parts of the fabric don't travel at the same rate, they have much shorter circumferences, making bumps in the fabric to try to change the rate of travel of the edges compared to the center. when the sphere is removed, the fabric makes a cylinder rather than a sphere. so what's notable is that the problem is not summarised by the concept of concavity; rather, it has to do with the surface coordinates. the fabric has orthogonal surface coordinates. we have to cut it to fit these onto other kinds of surfaces than rectangular ones. also, consider a flat area with a bump in the middle. the flat area will cover with fabric, but for the bump it seems one would have to cut the fabric and add an extra portion to make the total surface area correct. it wasn't immediately obvious, for an arbitrarily bumped region, how many cuts would be needed. great considering. some of us thought it sounds interesting to consider the concept of surface area. how might one calculate the surface area here? well, fabric could be seen as basically covering surface area. it covers it in orthogonal dimensions, so needs to be cut when the surface area is not rectangularly dimensioned. let's try it with a sphere, the surface area is the integral of an arc over an angle, right? kind of .. it's like many calculations of the circumference of a circle, which is the integral of a single point over it. trying to figure out. integral from 0 to 2pi, of pause time so regarding the sphere challenge, we did of course already solve that exact problem and basically we cut fabric so as to fit the amount there is of surface area (have not tested solution) by measuring it but it is notable that we didn't consider the orthogonality of the surface axes. the solution may be wrong. one proposal is to start by considering a point on the surface, and track orthogonal surface vectors along the object and see what they do. interesting because with a sphere, they do not stay orthogonal. they immediately begin curving. it looks almost as if fabric could not cover a sphere. let's look at that closer, noting that there are many cut cloth objects that are round and look fine X vector travels circumference Y vector travels latitudinally the problem is that as we travel in one direction ... i can imagine traveling 1 unit Y and 1 unit X then the same and connecting but i think this shape might not be square we start off at X=1 Y=0 Z=0 then we rotate by n degrees around Y, and around Z now we are at and at if we then do these again to unite somewhere, then we'd get to ummm the one that rotated around Z is rotating around Y now, and vice versa, so the last two coordinates are <, sin(n), sin(n)> cognitive issues all over the first coordinate we have the starting 3D location. we have the axis of rotation (i think? am i wrong?) what axis are we rotating around the second time. say -- one idea was to rotate around Y, the dimension containing sin n another idea is to rotate arounhd a slightly shifted axis if we rotate around Y we don't travel the same amount of distance instead we want to rotate around something like which is perpendicular to the vector so that all its length is included in the rotation if you rotate around Y the length you are rotating is cos n, so you travel (r cos n) theta if instead you rotate around then the length you are rotating is 1 so you travel r theta (where theta is n of course) ok i can rederive the rotation equation using matrices X rotation 0 0 0 0 sin cos 0 cos -sin started looking up. a rotation matrix is: cos -sin sin cos it's right-multiplied on wikipedia so cos -sin x sin cos * y so x = x cos - y sin y = x sin + y cos let's test by rotating <0,1> by 30. cos 30 = 0.866, sin 30 = 0.5 0 * 9.866 - 1 * 0.5 = -0.5 0 * 0.5 + 1 * 0.866 = 0.866 we rotated y=1 by 30, and ended up with <-0.5, 0.866> it rotated counterclockwise, and is higher than to the left. it seems ok. things rotate counterclockwise because they're considered starting at <1,0> and cos falls while sin rises so now how do we rotate around an axis i'll look up again it reads as kind of complex. maybe we can think of it in terms of arcs. we have 4 arcs of length r*n that make a square-like shape on the surface of the sphere. the surface can be subdivided into smaller arcs arbitrarily. what's the surface area? well, consider it as a portion of a sphere ... it's made of arcs of angle n. ... their angular portion is n / (2 * pi) meanwhile the whole circumference is r * 2 * pi so their angular length ... their part of the whole circumference ... is n / (2 * pi) * r * 2 * pi = n * r . which is the definition of radians ;p then we have that in the other direction, too n * r in one direction, n * r in the other the middle is not bigger than the edges here? no it might actually be square ...? because each point was rotated about the center of the sphere and traveled a similar distance? i'm not sure they did. how do we properly perform the n * r integral in the other direction. i guess it does depend on how the rotation is done let's do a normal spherical surface area to figure it out. start with circle. integrate a parametric definition of circle. x = cos(a), y = sin(a) when we integrate with regard to a, we are considering a an angle already integral(x) = sin(a) + C integral(y) = -cos(a) + C ok but we want to integral the distance travelled so let's first take the derivative x' = -sin(a), y' = cos(a) now calculate arclength s' = sqrt((-sin(a))**2 + cos(a)**2) now integrate s from 0 to 2pi to find circumference, is harder wait isn't sin(x)**2 + cos(x)**2 == 1 yeah wolfram alpha drew a chart of a line of y=1 travelling from 0 to 2pi the integral is 2pi * 1 = 2pi which is of course the correct answer ok let's try a sphere so with a sphere we probably rotate the arc in such a way that each point travels a different distance we could use a 1pi arc, 180 degrees, and rotate it about its poles. now each point on the arc covers a different area. we can calculate i think, slice radius = sin(theta) total circumference = 2 pi r = 2 pi sin(theta) the proposal is that each point on the arc produces a surface region with a circumference of 2 pi sin(theta) becase we aren't considering the surface axes here, it works, it's accumulating points. there are 2pi sin(theta) points for each point on theta what happens if we go integral(2 pi sin theta, theta = 0 to pi) = 2 pi integral(sin theta, theta, 0, pi) integral of sin is -cos = 2 pi [-cos pi - -cos 0] cos pi = -1, cos 0 = 1 = 2 pi [--1 - -1] = 2 pi * (2) = 4 pi this appears to be correct. we left out r. the formula is 4 pi r^2, and we have an r of 1. ok so we can approximate a cloth solution by assuming that small regions are disregardable. so if we have a small region and the edges are roughly parallel, we coudl assume a small bit of cloth fits it :s maybe i can think about it a different way. if we had an infinitely thin strip of cloth it would easily wrap a sphere. once it has width, it doesn't wrap it well. we could instead cut a triangle from an end and wrap it partway ... maybe there is something harsher than a sphere that would make this clearer like if we have a triangle, it is clear that a rectangle of cloth covers it only partway, and part folds oh i am finally feeling cold. mostly my outside though. still hot inside. will move one seat over this seat is cold too. we'll try the warmer room. head feels hot. torso feels cold. confusing. maybe i'll [poop] and see how i feel on return ok so you'd kind of consider one part of the sphere, and do the circumference around, the go up a smidge, note the circumference is smaller, and cut the cloth to be a little shorter to accommodate. - i guess this assumes the cloth can shear so as to align its endpoints up, and reducing that shear is probably why the alignment is made at the middle of the cloth, and why there are multiple panels. i'm thinking a shear would also change the height of the sheared area as the cosine of the angle of shear (and that this value would change throughout the length) - thinking one could model cloth as a mesh of threads to sort that out in precision if desired or needed huhh ok you'd have a circumference thread, then a thread a little above that is shorter, and on the sides you'd want threads of the same length. but if you cut it flat, you'll end up stretching the threads at the edges and squishing the threads at the middle, or somesuch? so that must be what the seam allowance is for, for threads to pull in or out of before a final cut it seems like the threads in tension are kind of the final state. the bunched up threads aren't maybe as important. - as the cloth curves over the sphere, the perpendicular it curves to the side revolves as well; it's not a horizontal cross section like when modeling surface area or volume with perpendicular integrals. (of course you could use thread bunching to pretend it is) so for that second/final point, we might change the angle or plane over which the circumference is changed as things progress we might want a plane that is coplanar to the normal to the sphere where the point touches thinking again that it's the threads in tension that define the size and shape of the surface. so, the edge threads have a different tension. they could pull up. thinking you'd want the sizes to be such that regardless of which thread is in tension, the cloth can only cover a given region. arc length is r n arc area is ... integral(r n ... , theta, 0, n) confusing when we're not clear on how the rotation over the surface is performed. so the way the current approach tries to do it is by limiting the size of the horizontal threads. this is the function of cutting. it may not have the curvature correct. but when the horizontal threads are linked, it makes it form a sphere, possibly, in theory. it ideally would use the tension of the vertical threads to define where it is on the sphere, and then the tension of the horizontal threads to define the circumference at that point. some confusion on whether that's the horizontal circumference (which i think is what we calculated) vs a different circumference ... and i think the hope is that there are enough panels that they are the same. if it's not the horizontal circumference then the threads are imagined a little bunched at the edges whereas if it is, then i guess ... then i think that works better almost? the side threads can travel up the object equally with the middle thread. (oh except they have to shear to line up.) so what's the edge of a cloth like that's cut diagonally like that; the vertical threads keep stopping, but we could assume that the weaving of the cloth keeps them roughly attached to the horizontal threads. it shears a little bit. it's kind of like a surface that can shear or bend. what is it like if a triangle goes on a sphere, made of cloth? there is a common sphere-triangle where the base of the triangle is r pi / 2 and the two sides are also r pi / 2. because it's a spherical coordinate system, the triangle can have all 3 sides the same length, and perpendicular angle at each corner. what does cloth look like if we cut it to do that? you could put cloth at a point, and bend it so that it makes a cylinder around the sphere to put it at another point. you could then bend it down either way to touch the third point, and now two edges are aligned and the third edge is ... bent in the air or bunched up maybe...? there isn't a third edge, there are two edges, you'd cut the cloth at that point, and there'd be bunching it's confusing that it appears to be an impossible problem that has been repeatedly solved. i imagine if i had bunching, i might cut the bunch off, and connect the sides i cut, and the cloth would then fit the object. you could calculate that bunching to do that. (it might be very confusing to do so atm) considering same sphere-triangle thing if i connect both of the heights to the top point i end up having a bunch between them of the same length as the base. i could cut a line, from the middle of the base to each of the tips, making a shape like "|\/|" i imagine it would be approximate. we could plot the a quarter circumference going up the arc of the object. this might look like kind of like the upper half of "()" . the idea is each row of thread has the right length to fit onto the sphere. it might be wrong. but it might work. because the horizontal threads stay laying horizontal, it makes sense to calculate the horizontal circumference. but the vertical threads do curve, so it makes sense to use the arclength. well it's helpful to think about it a bit. i'm not sure it's quite accurate, but it's something. it sounds like there's play, and it partly depends on how you tension it when assembling it ! in the end, it's likely to be assembled laying flat, so you're defining the distances between all the given points as-is, i think :s i guess that means it's the short lengths that control the shape; the ones that are too long just bunch a smidge 2024-08-24 1705