Clean up comments

note/YarnCurve.tex

@@ -48,8 +48,6 @@
 
 \section{Overview}
 
-% TODO Motivation: plain knit is the most basic ``texture'' used to emulate appearance of knit garments, hence useful for visual applications.
-
 A \emph{plain stitch}, also known as a \emph{stockinette stitch}, \emph{jersey stitch}, \emph{flat stitch}, or \emph{felt stitch}, is a basic knitting pattern formed by rows of identical yarn loops.  Though a beautiful array of decorative knit stitches appear in the design of textiles, the plain stitch is the baseline pattern used for common garments, upholstery, \etc{}\  Hence, visual simulation of everyday objects can benefit from a cheap, simple, and controllable model of the kind presented here.  Parameterization of twisted fibers makes our model especially suitable for \emph{marled} yarns, where several fibers or \emph{rovings} of different color are twisted together to form a single yarn.
 
 The geometry of the plain stitch pattern arises from an interplay between elastic forces (bending and twisting of a rod) and collision forces (frictional contact between neighboring yarns), and has in the past been modeled via numerical optimization~\citep{yuksel2012stitch}, physical simulation~\citep{kaldor2010efficient}, and via a helicoid model~\citep{wadekar2020geometric}.  Parametric models have long been used in the textile industry to understand and predict physical properties of knitted garments (dimensions, weight, material usage, \etc{}).  For instance, \citet{Chamberlain:1926:HYF} gives a 2D description that is useful for reasoning about material properties, but not sufficient for capturing 3D appearance.  To our knowledge, the first parameterization of a three-dimensional plain knit stitch was given by \citet[Part IV]{Peirce:1947:GPA}, as a 2D piecewise function with straight and circular parts, composed with a mapping to a 3D cylinder.  The lack of curvature continuity, and the piecewise nature of this function, makes it difficult to develop a parameterization of the secondary twisted fibers.  Moreover, Peirce's curve does not provide many parameters to adjust the shape or appearance of the knit pattern (only the yarn diameter and wale/course spacing).  On the other hand, geometrically simple components make the curve and its parameters easy to analyze, whereas quantities like arc length and maximal thickness must be determined numerically for our model (Section \ref{sec:ArcLengthAndThickness}).  \citet{Leaf:1955:GPK} and \citet{Munden:1959:GDP} refine Peirce's model and connect it to empirical data; Munden in particular observes that yarn shape is largely geometric and variational in nature, \ie, arising from energy minimization that does not depend strongly on yarn properties or stitch length.  Twisted fibers can be seen in many papers from the computer graphics literature (\eg{}, \citet{kaldor2010efficient}), but no explicit equations are given.  In general, we were unable to find any parameterized version of the twisted fiber curves in the literature, though given the cultural importance of knitting it is very likely that such a parameterization has been written down somewhere.