Does a "real" spline carry any of the same guarantees as a mathematical spline or does it just create a curve that looks good? How does the curve differ depending on the material used for the spline?
@leoadberg - Based on the article linked on this slide—which tries to explain the geometrical underpinnings of splines from and for a layperson’s perspective (and is quite good; would recommend the read)—the answer to your question about the mathematical guarantees of physical splines is yes. This is because the supple wood of the spline acts as a model for the way other woods would react*: “An optimally smooth, attractive, and mechanically sound curvature is guaranteed by the uniform distribution of stress throughout the long elastic spline as it tries to regain its original straightness.” The fact that this method was relied-upon for many centuries (and even to this day, apparently, in translating drawings to full-scale parts for boats) seems to indicate that the mathematical basis was always there; our modern computational methods of calculating curves has been derived from our observation of the physical materials upon which we used to (and in some cases, still do) rely. So, in effect, the physical spline tool was a way for boatbuilders and architects of the past to utilize mathematics that hadn’t yet been articulated.
*even other materials, maybe? I’m not 100% sure about this, but it seems to me that the way the stress works out in the curve is best modeled by a material with high elasticity, such as a thin cut of supple wood, but that the stress will then be consistent for that curve, regardless of the material it ends up being made out of … so the material of the spline might just need to be something with sufficient elasticity to model a “fair” curve (borrowing some of the jargon from the boatbuilding section of the article) that will then be “fair” for any material that it’s applied to? I would welcome someone else’s input on this since I just reasoned it intuitively and without any knowledge of the physics/mechanics/material science involved.