The clothing sizes that I am used to are in inches

UK and US peeps both can understand inches (Desolée, Es tut mir leid!)

I’m old-fashioned and hopelessly lazy

The units of measurement are incidental anyway – one day, I think I’ll publish a pattern in light years, just for the laugh. Maybe my next Dr. Who one (last one).

So if I give measurements elsewhere in the blog without giving the units, please assume inches.

Unless I’m being derogatory about a male enemy. Then it’s cm mm.

I previously explained why I think the Sleeve Cap can be described by an oval. Basically, a sleeve can be thought of as a cone which intersects the “plane” of the sweater/cardigan.

This produces an ellipse.

The perimeter of the ellipse is at least the same as the perimeter of the matching armscye on the body of the garment. If the perimeter of the ellipse is smaller than the armscye, it won’t “fit” – used in cap sleeves only. If the perimeter of the ellipse is larger than the armscye, you can ease the excess all around, or into the top to produce a gather. In any event, the armscye perimeter can be calculated from the garment body measurements.

You also will have a measurement for top arm circumference. The top arm circumference for a woman of average height is around

Bust

32

36

40

44

48

Top Arm Circumf.

10.5

11.5

13

14

15.5

Don’t forget to add ease! This is an unclothed measurement.

You can see from the diagram below, that the top arm circumference, i.e. the widest part of the sleeve, is equivalent to the width of two ellipses (0.5 + 1 + 0.5 ellipse widths). Hence, the horizontal radius of the sleeve cap ellipse is a quarter of the top arm circumference.

The sleeve cap height is twice the vertical radius of the ellipse, i.e. the same as the height of one ellipse.

If you plug the values for perimeter and horizontal radius into Ramanujan’s Approximation for the perimeter of an ellipse, you can get a good-enough value for the sleeve cap height.

The sleeve cap height is twice the value for c, the vertical radius of the ellipse, approximated by

Where p is the armscye perimeter / pi

And w is the top arm circumference / 4

The approximation above is essentially Ramanujan’s Approximation expressed in terms of one of the radii.

I find myself needing to knit a disc. I want to join two cylinders with different radii.

What I need to knit is the perpendicular plane where they join. I actually need to make a ring rather than a circle, but the same maths should apply.

The equation to get the number of stitches to decrease each row would seem to be

where n is the number of sts to increase or decrease per row, g_sts is the st gauge and g_rows is the row gauge. What’s really neat about this is it’s independent of start and end radius altogether. Sts will need to be distributed evenly across the row and staggered throughout all rows so as not to distort locally.

Still working on armholes and armscyes and caps and ellipse perimeters. Was very happy to discover Ramanujan’s approximation for the perimeter of an ellipse:

And this great page from My Measuring Tape on drafting different styles of sleeves is easily adaptable to knitted wares.

I have been slowly working my way down through Knitty’s Knitted Sleeve editorial from ’05, and I’m now around about the point where I realise I really have to get the armscye sorted before going ahead with designing the sleeve cap :)

I have the bodice armscye mostly sorted now, but I’ve been looking for some kind of indication as to what slope to use for the curve of the cast off (working up) at the armhole. If it’s too slow to decrease, the armscye will be triangular, and if it’s too quick, it’ll be square. What’s the happy, curvy, medium? I found a physical armscye curve online, having read this article on pattern drafting on Your Wardrobe Unlocked.

So, if the physical object exists, there must be a software equivalent, right? (my design’s totally digital) Or even just an equation… but can I find one? Nay, I say unto thee, my google searching skillz extends not to such a hit.

Still workin’ on it.

Boy is design hard.

But I absolutely looooove it!!!!!

Ooh, update! I’ve discovered from this UTube video by munnikusum, that a good place for the curve to end (just before the vertical rise to the shoulder) is halfway up that shoulder line.

Last night I popped up a formula for calculating the stitches on a particular row in the cap of a set-in sleeve. And I kind of just left it there, with no explanation. My excuse is my eyelids were sticking to my eyeballs from lack of sleep.

So, now let’s not talk about knitting or maths at all and go on to glass production.

Way back when, if you wanted a pane of glass, you got a glass blower to spin a disc of molten glass. The disc would be cut to shape when it cooled and every pane would have a pontil mark from the glassblower’s pipe. (circa mid 1700’s)

In 1953, the Pilkington brothers developed the float-glass method for making flat glass.

In between, if you wanted flat glass without a pontil mark, your blower would blow up a cylinder, knock off the ends and cut the still malleable glass up one of the sides. He then opened and flattened the cylinder into a rectangle. It never went completely flat, which is why some old glass looks “wavy”.

Sleeves at their most basic are cylinders. If you knit one flat, you make a rectangle just like the glass cylinders above. Now, scale it up just a teensy bit, and a sleeve can probably be better approximated by a truncated cone intersecting a plane. Unless you’re making a cap sleeve, the armhole is also a closed curve. Hence, it’s an ellipse. Just needs a bit of chopping up to get it flat.

A set-in sleeve on a pattern schematic looks like the below. (Actually, usually they’re a bit wider, proportionally speaking).

Now, the equation I came up with last night is pretty much the same as the one I gave for knitting a circle. If you use it in the same way, you will knit an ellipse. Here’s that ellipse one again:

This reads “the number of stitches on row n is a (measured in stitches) by the root of (1 minus (the row number by the gauge over b (measured in stitches)) squared)”

a is half the width of the ellipse and b is half the height.

gauge is sts/10cm over rows/10cm

This will give you the stitches for a quarter of the ellipse (i.e. the curve in the positive x & y quadrant on the cartesian plane), you need to double it to get the stitch count for an entire row and mirror it to get the corresponding bottom half of the ellipse (where y<0).

However!

We want to do set-in sleeves. We’re not actually knitting an ellipse, we’re knitting the cylinder from which an ellipse has been cut, i.e. the negative space of the ellipse.

So the sleeve circumference at its widest point (usually, unless you’re doing a bishop or bell, more on that to come!!!) is usually at the bicep (check out this knitty winter ’04 feature). Call this circumference cAssuming a sleeve which is the same width as the armhole at the intersection (i.e. straight, not a puffy sleeve with extra material), then a is c/4. b is half the depth of the armhole. Very important note: these are the measurements of the fabric, not the model.

Assuming bottom-up construction, the sleeve width just as you are about to start decreasing for the sleeve cap is c. Set n = 1 and increase by 1 for each row. The rowcount decreases at a rate of to the midpoint of the set-in curve, when . Then, the rowcount continues to decrease, and the simplest way to calculate it thereafter is to just reset n=1 and use

So that’s my convoluted THEORY. Need to knit it up and see if it works! I’ll update again when I do.

This, I believe, is a formula for calculating the number of stitches on a row for the sleeve cap of a set-in sleeve, given a = half the width of the armhole, b = half the height of the armhole.

I shall write more tomorrow, I haven’t sufficiently explained how to use this to get a complete set of rowcounts for the sleeve cap. When it’s done though, I’d be interested to see if knitters out there concur.

So here’s a recipe for knitting up a circle for a given gauge and radius. It’s hardly a good way to go about putting less effort into knitting up a tension swatch as there’s a fair bit of maths to do first, but it does give a perfectly jaggy curve :)

Apologies for the eyeball-searing colour. I think I’ve been half-blinded from the real thing. I’m using Katia Monaco which is 100% mercerised cotton in a DK weight. The ball band claims a tension of 22 sts x 27 rows over 10cm on 3.5mm needles.

So this is the equation to get the stitch count for a chosen row (n) given the radius and the gauge. This will give you a quarter circle. To get a semicircle, double the value.

Which reads

“The number of stitches on row n is the square root of the radius (measured in stitches) squared, minus (the radius (measured in rows) less this row number) squared, times the gauge squared.”

A little cumbersome, perhaps, but all you have to do is plug in the numbers. With this yarn, for example, I wanted to get a 7″ Φ circle (17.7cm) and or 0.81.

So the radius is 8.8cm which means

And

So to calculate the appropriate number of stitches for each row, I made a table. I rounded the values to the nearest integers. I doubt even Elizabeth Zimmerma’am could knit 14.36 stitches!

The cast-on value is different to the number of calculated stitches because I adjusted for only casting on extra stitches at the start of each row. Note this is stockingette, so every even row is purled. Once you get to the end, work back down the rows, casting off instead of casting on stitches till you get to the last row and cast off the remaining 16.