The blue shape on the left shows the rotated sleeve cap that matches the blue rotated armscye on the right. The more symmetrical red shape on the left shows the un-rotated sleeve cap matching the un-rotated armscye on the right.
Our original model for obtaining these shapes was based on the arm hanging at rest at the side of the body with a slight forward and outward rotation. This model didn’t quite work. That orientation produced the following unrealistic sleeve/armhole intersection.
In order to get more realistic intersection shapes, it was necessary to hold the arm a bit outward from the body. The shapes above are both based on this revised model, with the red based on the arm outward at the side and the blue based on the arm outward and rotated forward.
The math is based on taking a cylinder and slicing it at an angle. The cylinder represents the sleeve, and the angle slice is the angle at which the cylinder sleeve intersects the side of the body (plane). The original strategy we came up with was based on using a cone to represent a sleeve. Our Stanford Research International friend suggested a method based on using a cylinder because it is a close enough approximation and is easier to calculate. A cylinder is a cone with its apex at infinity, not just past the wrist point, but the difference doesn’t seem to matter in terms of producing a correct shape and the math is far simpler.
Since I have to sew a 3rd V8543 muslin anyway I am going to try to sew the blue sleeve cap, to scale. I’m curious whether this sleeve cap will be ideal in terms of what you’d actually want to wear because although it closely resembles the anatomically correct sleeve cap depicted in Kathleen Fasanella’s post, it isn’t exactly the same. I am guessing the differences are due to modifications necessary to make the sleeve more comfortable to wear, but I don’t know. The sleeve I sewed on the 2nd bodice muslin was based on a curve very close to the blue curve, so the blue curve may actually be correct. Please note that there was a slight error in the 2nd bodice muslin: I didn’t get the shoulder top seam point aligned with the peak of the sleeve cap and I think I should have. The dot position got lost in the various pattern iterations.
Yesterday Kathleen posted a link to a very interesting Ph.D dissertation that seems, at first glance, to explain in depth what we’ve been discovering through experimentation, and to provide detailed drafting specifications. I wish I could stop life for the next three months and try to understand that dissertation!
Anyway, here’s the procedure Martin worked out based on our friend’s math that produced the above shapes:
Math for cylinder-plane-shoulder arm model:
1) Determine the (x,y,z) position of the shoulder joint and the center wrist. S(x,y,z) and W(x,y,z)
2) Determine a unit vector from the center wrist to the shoulder joint. (A unit vector has length 1 and describes the direction from the wrist to the shoulder joint.) This line is the axis of the cylinder and is the center of the arm when the elbow is not bent.
3) Determine two unit vectors perpendicular to each other and to the axis of the cylinder, u(xyz) and v(x,y,z). Then, parameterize a circle in terms of radius, R and angle, t and W(x,y,z) in terms of these two unit vectors: G = W + uRcos(t) + vRsin(t). G describes the points on the circle of the cylinder at the location of the wrist.
4) For each point on the circle determine the distance from the wrist to the plane containing the shoulder joint in the along the direction parallel to the axis of the cylinder. Call this distance S(R,t). A graph of S as a function of Rt is a plot of the sleeve laid flat.
5) A graph of the (x,y,z) coordinates in the plane containing the shoulder joint then describes the arm hole.
Aaron Heller, a computer scientist in the Artificial Intelligence Center at SRI in Menlo Park, CA.
Geometric Approaches to Nonplanar Quadric Surface Intersection Curves by James R. Miller, Control Data Corporation.
Martin Partlan, Physics Professor, Cañada College, Redwood City, CA.