knotR-package: Knot Diagrams using Bezier Curves

In knotR: Knot Diagrams using Bezier Curves

Knot Diagrams using Bezier Curves

Description

Makes visually pleasing diagrams of knot projections using optimized Bezier curves.

Details

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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)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <[e-mail]>

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)

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