| knotR-package {knotR} | R Documentation |
Knot Diagrams using Bezier Curves
Description
Makes visually pleasing diagrams of knot projections using optimized Bezier curves.
Details
The DESCRIPTION file:
| Package: | knotR |
| Type: | Package |
| Title: | Knot Diagrams using Bezier Curves |
| Version: | 1.0-4 |
| Authors@R: | person(given=c("Robin", "K. S."), family="Hankin", role = c("aut","cre"), email="hankin.robin@gmail.com", comment = c(ORCID = "0000-0001-5982-0415")) |
| Depends: | R (>= 2.10) |
| Maintainer: | Robin K. S. Hankin <hankin.robin@gmail.com> |
| LazyData: | TRUE |
| Description: | Makes visually pleasing diagrams of knot projections using optimized Bezier curves. |
| License: | GPL-2 |
| Author: | Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>) |
Index of help topics:
as Conversions between various forms of a knot
badness Badness of knots
bezier Various functionality for Bezier curves
bezier_angle Intersection of two Bezier curves
bezier_find_length Solve for arclength
bezier_integrals Arcwise integrals over Bezier curves
crossing Crossing Metrics for knots
getstringpoints Returns the coordinates of a knot's path
head.inkscape Head and tail methods for inkscape objects
knotoptim Optimization of knot appearance
knotplot Plotting of knots
knotR-package Knot Diagrams using Bezier Curves
knots Optimized knots
overunder Functionality for specifying overstrands and
understrands
reader Reading and writing svg files
symmetrize Symmetry and knots
utilities Various utilities for knots
The package contains a large number of knots, optimized for visual appearance in the sense that the knot path is nice and smooth, and strands cross at close to right angles. These can be displayed by typing
knotplot(k9_23)
at the R prompt. The package includes all prime knots up to and including 9 crossings, and a number of other interesting and attractive knots.
The package facilitates the creation and optimization of new knots. The
basic workflow is to create an .svg file in inkscape comprising a
single closed path (that is, the first and last node are the same
point). Control nodes should all be symmetrical. Many examples of
correctly formatted .svg files are given in the inst/
directory.
The best way to reproduce a knot from an image of its projection is to fire up inkscape, then import the image into inkscape, resize and rotate as desired, then follow the string with the ‘Bezier curves and straight lines’ tool (also called the ‘pen tool’ by Kirsanov; the keyboard shortcut is shift-F6). Use 1-2 nodes per segment, or 3 nodes for longer or more visually prominent segments.
Keep the lines straight at first. Close the path by making a final click on the initial node; now you have a closed polygon, which will self-intersect at the path crossing points. To smoothen the path, select the ‘edit paths by node’ tool (shift-F2), then convert the corner nodes of the path to symmetric Bezier nodes (‘make selected nodes symmetric’). You can then tweak the path by moving the control nodes about with the mouse. Be aware that adding or deleting nodes changes the adjacent nodes to asymmetrical Bezier control points; make them symmetric by selecting all nodes (‘Ctrl-A’), then hit the ‘make selected nodes symmetric’ button. Do this frequently to avoid confusion.
An .svg file may be imported into R using the
reader() function, which creates an inkscape object. This
represents the path of the knot: it does not include over and under
information. The package assumes that inkscape uses absolute
coordinates (as opposed to relative coordinates); see reader.Rd
for more information.
The package provides four classes of objects that specify the path of a knot:
inkscape, minobj, controlpoints, and
knotvec. These four classes have different uses, and objects may
be converted from one form to another by using functions such as
as.minobj(), documented at as.Rd and utilities.Rd.
A knot requires information on which strands pass over or under which
other strands; full documentation at ?overunder.
Knots sometimes have symmetry constraints such as horizontal or vertical
symmetry, or rotational symmetry. Symmetry is imposed by using the
symmetrize() function: this takes a knot path (coereced to
minobj form) and a symmetry object. Symmetry objects are created
with function symmetry_object(), which takes a knot path and a
series of matrices and vectors that specify the symmetry of the knot.
Author(s)
NA
Maintainer: Robin K. S. Hankin <hankin.robin@gmail.com>
Examples
a <- reader(system.file("7_6.svg",package="knotR"))
knotplot2(a) # shows curvature
# Now use text=TRUE to display strand numbers so you can figure out the
# overunder relations:
knotplot2(a,text=TRUE,lwd=1)
ou76 <- matrix(c(
12,01,
02,11,
07,03,
04,15,
16,06,
14,08,
10,13
),byrow=TRUE,ncol=2)
# Now we can do a proper knot plot:
knotplot(a,ou76)
# To symmetrize a knot we use the symmetry functionality of the knot:
a <- reader(system.file("3_1_not_symmetric.svg",package="knotR"))
knotplot2(a,seg=TRUE,text=TRUE,lwd=1,node=TRUE)
# First specify the vertical symmetry:
Mver <- matrix(c(
08,10,
07,11,
02,04,
01,05,
12,06
),ncol=2,byrow=TRUE)
# Then the rotational symmetry:
Mrot <- matrix(c(
09,05,01,
10,06,02,
08,04,12
),byrow=TRUE,ncol=3)
# Now the overunder information:
ou31 <- matrix(c(
03,08,
11,04,
07,12
),byrow=TRUE,ncol=2)
# create a symmetry object:
sym31 <- symmetry_object(a, Mver=Mver,xver=c(9,3),Mrot=Mrot)
knotplot(symmetrize(a,sym31),ou31)
# Symmetric-- but ugly as a burglar's bulldog.
# to beautify, either use the knotoptim() function, or do it by hand:
objective <- function(m) {badness(make_minobj_from_minsymvec(m, sym31))}
startval <- make_minsymvec_from_minobj(as.minobj(a),sym31)
## Not run:
# Following examples take a long time to run.
# nlm() is the best optimization method, I think. Limit to 1 iteration:
o <- nlm(f=objective, p=startval, iterlim=1)
# extract the evaluate:
oo <- make_minobj_from_minsymvec(o$estimate, sym31)
# create a knot:
k31_marginally_better <-
knot(x = oo, overunderobj = ou31, symobj = sym31)
# then plot it:
knotplot(k31_marginally_better)
## End(Not run)