utilities: Various utilities for knots

In knotR: Knot Diagrams using Bezier Curves

Various utilities for knots

Description

Various utilities for knots including reading files and creating objects

Usage

controlpoints(x)
inkscape(x)
minobj(x)
knotvec(x)
make_controlpoints_from_ink(a)
make_minobj_from_ink(a)
make_minobj_from_vector(vec)
make_ink_from_minobj(x)
make_inkscape_from_controlpoints(b)
make_minobj_from_knot(k)
make_knotvec_from_minobj(x)

Arguments

x	Suitable object for coercion; see details
a	An inkscape object: a two column matrix with rows representing the positions of nodes and control points
b	A controlpoints object
k	An object of class knot
vec	A vector of reals

Details

Functions inkscape(), minobj(), and knotvec() are low-level functions; these are the only places that objects have their classes assigned directly. These functions are not user-friendly and require very specific types of object; they perform some checks but are not really intended for the user. Functions as.foo() are much more user-friendly, and are documented at as.Rd.

Functions make_foo_from_bar() coerce bar objects into foo objects. Functions that involve symmetry are documented at symmetry.Rd.

Objects of class inkscape are in the form of a two-column matrix, with rows corresponding to 2D positions. The rows correspond to the (x,y) coordinates of points as held in the inkscape file.

There is quite a lot of redundancy in an inkscape object:

• The first row of an inkscape object is equal to the last row (this follows from the fact that the path is closed).

• If n=0 modulo 3, then a[n+2,]-a[n+1,]==a[n+1,]-a[n], corresponding to the fact that the handles are symmetric in inkscape. This is visualised best by knotplot2(k4_1,ink=TRUE,seg=TRUE)

Look at functions make_inkscape_from_minobj() and make_minobj_from_ink() to see this from a symbolic perspective. The vignette also gives some details.

The minobj class is a ‘MINimal OBJect’. Objects of class minobj are a list of two elements: $node and $handle_A. Each element has rows corresponding to 2D positions, the same as inkscape objects. Element $node shows the positions of the nodes, and element $handle_A shows the positions of (one of) the handles; the other handle is symmetrically positioned with respect to its node. Use knotplot2(k4_1,node=TRUE,seg=TRUE) to see the meaning of the entries; the nodes are indicated by a square and the handles by circles.

NB: objects of class minobj have no redundancy in the sense that changing any entry of either $node or $handle_A results in modifying the corresponding inkscape diagram. However, doing this to a knot which has imposed symmetry conditions (ie a nontrivial symobj) may introduce asymmetry into the inkscape diagram. For example, one might take a minobj object whose knot diagram is left-right mirror symmetric (eg k6_2) and alter one of the handle positions. Then the resulting inkscape object will be asymmetric. There is an example using k6_2 below.

An object of class controlpoints is a list of matrices of size 4-by-2. For each matrix, the four rows correspond to the points in 2D Cartesian space needed to specify a Bezier curve; further details and examples are given in bezier.Rd. There is lots of redundancy in a controlpoints object because the inkscape nodes are symmetric nodes with diametrically opposed handles.

The knotvec class is a named vector of independent reals suitable for use with optimization routines.

None of the functions here deal with symmetry relations. This is documented at symmetry.Rd.

Author(s)

Robin K. S. Hankin

See Also

as,symmetrize

Examples

 a <- as.minobj(k6_3)
 plot(a$node,asp=1,pch=16)
 segments(x0=a[[1]][,1],y0=a[[1]][,2],x1=a[[2]][,1],y1=a[[2]][,2],
 main="handle direction follows the string path")
 points(getstringpoints(a),type='l',col='gray',lwd=0.4)

 

knotR documentation built on June 22, 2024, 6:56 p.m.