I was sitting in Elder's Quorum [1] last week watching the fellow in front of me playing with his infant. The little baby boy was wearing a tiny pair of jeans. Just for fun, I began sketching in my notebook the pattern of yellow stitching. Since I was drawing in only two dimensions, I flattened everything. Rather than simulate three dimensions through the use of shading, I sought to capture the entire surface. As with rendering a two-dimensional map of a three-dimensional Earth, I was forced to introduce distortions in order to capture the details I wanted and present them in a particular way.[2] So, I present to you my map projection of blue jeans.
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Stitches that were perpendicular to each other I sought to maintain that way throughout their length. So, for example, in three dimensions you have the waistseam (red) [3] and the center seam (yellow) which are perpendicular to each other. The inseam (yellow) is perpendicular to the center seam but bends to follow the leg and so ends up perpendicular to the waistseam (left). But in two dimensions, using my projection, the belt stitch and the inseam are parallel (right). This also means that the hemline, which is normally parallel to the waistseam ends up being perpendicular in the final projection.
A second problem that I had to deal with was that blue jeans can be considered three tubes stuck together: the waist tube, the right leg tube, and the left leg tube. To flatten them out I have to 'cut' them so the entire surface can be seen (much the same way many map projections of the Earth make a 'cut' through the Pacific Ocean). I decided to make the 'cuts' in the seams that don't have yellow stitching—the side seam. (Actually, there are a few inches of stitching coming down from the waistseam, but most of the seam is unstitched.) This ends up cutting the waist tube twice and has the additional effect of placing the back pockets [4] upside down in relation to the zipper.
There is one more distortion which I added mostly for aesthetic reasons. I was primarily interested in making a projection of the stitching and associated phenomena (rivets, button, belt loops, etc.), not the entire surface of the fabric. In reality the projection should show the fabric more or less occupying a + shape (actually, more like a flattened German cross). But I didn't care for the empty corners, so I filled them in with denim.
And here it is! (Click to see a larger version.) When I drew this as a vector graphic in Inkscape [5], I actually made it life-sized (i.e. the inseam is actually 32 inches in both directions). I exported the .png at 90 dpi (which is apparently the resolution of 'real life'). But that image was 5641×2270 px, which is much too large to upload to Blogger. So the image above is a scaled-down version.
Here is an unscaled fragment so you can see the detail. I drew the denim by following a video posted over at the Inkscape tutorial blog.[6]
Notes:
[1] For those of you unfamiliar with the LDS Church (sometimes colloquially refered to as the 'Mormon' Church), we refer to a group of men who share the same office in the priesthood a quorum. Other offices include deacon, teacher, and priest. For many other Christian denominations the term priesthood simply refers to members of the clergy. But in the LDS Church it refers not only to the men who hold an office, but also to the power and authority of God granted to the men who hold that office. This power and authority includes (but is not limited to) performing saving ordinances, such as baptism. To learn more, see here and here.
[2] Since the blue jeans were constructed using all flat pieces of material I cannot say whether the distortion is unaviodable per Gauss's Theorema Egregium.
[3] If you're not sure what seams I'm talking about, see this diagram.
[4] I left the decorative stitching off the back pockets because this varies widely according to the brand. This doesn't really count as a distortion, though.
[5] If you're unsure what Inkscape or a vector graphic, see my post Raster Graphics and Vector Graphics.
[6] See http://inkscapetutorials.wordpress.com/2012/11/07/create-a-denim-texture-and-use-it-to-create-a-jeans-pocket/.
Stitches that were perpendicular to each other I sought to maintain that way throughout their length. So, for example, in three dimensions you have the waistseam (red) [3] and the center seam (yellow) which are perpendicular to each other. The inseam (yellow) is perpendicular to the center seam but bends to follow the leg and so ends up perpendicular to the waistseam (left). But in two dimensions, using my projection, the belt stitch and the inseam are parallel (right). This also means that the hemline, which is normally parallel to the waistseam ends up being perpendicular in the final projection.
A second problem that I had to deal with was that blue jeans can be considered three tubes stuck together: the waist tube, the right leg tube, and the left leg tube. To flatten them out I have to 'cut' them so the entire surface can be seen (much the same way many map projections of the Earth make a 'cut' through the Pacific Ocean). I decided to make the 'cuts' in the seams that don't have yellow stitching—the side seam. (Actually, there are a few inches of stitching coming down from the waistseam, but most of the seam is unstitched.) This ends up cutting the waist tube twice and has the additional effect of placing the back pockets [4] upside down in relation to the zipper.
There is one more distortion which I added mostly for aesthetic reasons. I was primarily interested in making a projection of the stitching and associated phenomena (rivets, button, belt loops, etc.), not the entire surface of the fabric. In reality the projection should show the fabric more or less occupying a + shape (actually, more like a flattened German cross). But I didn't care for the empty corners, so I filled them in with denim.
And here it is! (Click to see a larger version.) When I drew this as a vector graphic in Inkscape [5], I actually made it life-sized (i.e. the inseam is actually 32 inches in both directions). I exported the .png at 90 dpi (which is apparently the resolution of 'real life'). But that image was 5641×2270 px, which is much too large to upload to Blogger. So the image above is a scaled-down version.
Here is an unscaled fragment so you can see the detail. I drew the denim by following a video posted over at the Inkscape tutorial blog.[6]
Notes:
[1] For those of you unfamiliar with the LDS Church (sometimes colloquially refered to as the 'Mormon' Church), we refer to a group of men who share the same office in the priesthood a quorum. Other offices include deacon, teacher, and priest. For many other Christian denominations the term priesthood simply refers to members of the clergy. But in the LDS Church it refers not only to the men who hold an office, but also to the power and authority of God granted to the men who hold that office. This power and authority includes (but is not limited to) performing saving ordinances, such as baptism. To learn more, see here and here.
[2] Since the blue jeans were constructed using all flat pieces of material I cannot say whether the distortion is unaviodable per Gauss's Theorema Egregium.
[3] If you're not sure what seams I'm talking about, see this diagram.
[4] I left the decorative stitching off the back pockets because this varies widely according to the brand. This doesn't really count as a distortion, though.
[5] If you're unsure what Inkscape or a vector graphic, see my post Raster Graphics and Vector Graphics.
[6] See http://inkscapetutorials.wordpress.com/2012/11/07/create-a-denim-texture-and-use-it-to-create-a-jeans-pocket/.
Demasiado diría yo.
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