symmetrize: Symmetry and knots

In knotR: Knot Diagrams using Bezier Curves

Symmetry and knots

Description

Various functionality to impose different types of symmetry on knots

Usage

force_nodes_mirror_images_LR(x,symobj)
force_nodes_mirror_images_UD(x,symobj)
force_nodes_exactly_horizontal(x,symobj)
force_nodes_exactly_vertical(x,symobj)
force_nodes_on_V_axis(x,xver)
force_nodes_on_H_axis(x,xhor)
force_nodes_rotational(x,symobj)
symmetrize(x,symobj)
tag_notneeded(x, Mver, xver, Mhor, xhor, Mrot,exact_h,exact_v)
make_minsymvec_from_minobj(x,symobj)
minsymvec(vec)
make_minobj_from_minsymvec(minsymvec,symobj)
symmetry_object(x, Mver=NULL, xver=NULL, Mhor=NULL, xhor=NULL,
Mrot=NULL, exact_h=NULL, exact_v=NULL,
mcdonalds=FALSE, celtic=FALSE, reefknot=FALSE,center_crossing=FALSE)
knot(x, overunderobj, symobj, Mver=NULL, xver=NULL, Mhor=NULL,
xhor=NULL, Mrot=NULL, mcdonalds=FALSE, celtic=FALSE,
reefknot=FALSE,center_crossing=FALSE)

Arguments

x	Object coerced to class minobj
Mver, Mhor	Matrices specifying vertical (resp. horizontal) symmetry, with two columns. The rows specify pairs of symmetric nodes about a vertical (resp. horizontal) axis. Nodes specified by the first column should be on the left (resp. upper) side; these are fixed. Used by functions force_nodes_mirror_images_LR() and force_nodes_mirror_images_UD() which move the right (resp. lower) nodes and their associated handles to the positions required for exact vertical (resp. horizontal) symmetry
Mrot	A matrix specifying rotational symmetry. Each row corresponds to a set of nodes in a rotational relationship. The number of columns specifies the order of the rotational symmetry. The first column corresponds to nodes whose position is fixed. Used by force_nodes_rotational(), which also moves handles appropriately
xver, xhor	Vector specifying nodes to be on the vertical (resp. horizontal) axis of symmetry. The nodes are assumed to flow from left to right. Used by functions force_nodes_on_V_axis() and force_nodes_on_H_axis() respectively, which also move the handles
exact_h, exact_v	Vector specifying nodes to be exactly horizontal or exactly vertical. A node is exactly horizontal (resp. vertical) if the y (resp. x) coordinate of the node is the same as the y (resp. x) coordinate of the handle. Note that the position of an exactly horizontal or vertical node is not restricted, and may be anywhere. Used by functions force_nodes_exactly_horizontal() and force_nodes_exactly_vertical()
symobj	An object representing the symmetry of the knot, usually created by function symmetry_object()
mcdonalds	For vertical symmetry, argument mcdonalds is Boolean, defaulting to FALSE, with TRUE meaning that the symmetric pairs of strands approach the vertical line of symmetry in the same sense (either both moving inward, or both moving outward). It is hard to explain (and named for the gesture one makes when tracing the top two strands a knot with this type of symmetry). The only common knot that needs this is 7_2
celtic	Like mcdonalds but for horizontal symmetry
reefknot	Like mcdonalds but for the reefknot
center_crossing	Implements a peculiar type of rotational symmetry in which the strands pass through the geometrical center of the knot projection. The only common knot needing this is 9_29
minsymvec	A “minimal symmetric vector”. This is a numeric vector containing just the independent degrees of freedom of a knot, after symmetry constraints have been imposed. The idea is that one may optimize a minsymvec object using nlm(), and then reconstruct a knot using make_minobj_from_minsymvec() together with a symmetry object
vec	A vector, given to function minsymvec()
overunderobj	A matrix specifying the overs and the unders; a two-column matrix with rows corresponding to pairs of strands intersecting. The first element of a row identifies the overstrand and the second element specifies the understrand

Details

Function symmetry_object() creates a symmetry object from Mver et seq, but if given a knot object, returns the embedded symmetry object.

There are seven types of symmetry that may be imposed on a knot. These are imposed by the following seven force_nodes_foo() functions:

• Functions force_nodes_mirror_images_LR() and force_nodes_mirror_images_UD() symmetrize a knot about a vertical (resp. horizontal) axis by taking ordered pairs of nodes, specified by matrix Mver (resp. Mhor) and forcing the second node to be symmetrically placed with respect to the first. It does the same thing to the handles too.

• Functions force_nodes_exactly_horizontal() and force_nodes_exactly_vertical() force nodes to be exactly horizontal (resp. vertical) by restricting the position of their handles. Nodes so forced do not need to be on an axis of symmetry; they can be anywhere

• Functions force_nodes_on_V_axis() and force_nodes_on_H_axis() force nodes specified by xver (resp. xhor) to be on the vertical (resp. horizontal) axis, and to have appropriately placed handles

• Function force_nodes_rotational() imposes the rotational symmetry specified by Mrot

Function symmetrize() imposes the seven kinds of symmetry by calling each of the force_nodes_foo() functions in turn.

Function tag_notneeded() is an internal function, not really intended for the end-user. It takes a minobj object and marks a maximal set of dependent entries with a ‘not needed’ value. The values of the entries so marked may be determined by a combination of the imposed symmetry relations and the unmarked values. The unmarked entries constitute a minsymvec object (see above). These are the real degrees of freedom in the symmetrical knot. Only these unmarked values are modified by the optimization routines in knotoptim()

Note

You can achieve up-down symmetry (that is, a horizontal line of symmetry) by making a left-right symmetric knot and rotating by 90 degrees. D'oh.

Author(s)

Robin K.S. Hankin

Examples

# each row of M = a pair of symmetrical nodes; each element of v is a
# node on the vertical axis

M <- matrix(c(6,4,13,11,7,3,2,8,9,1,14,10),byrow=TRUE,ncol=2) 
v <-  c(5,12) # on vertical axis

sym_7_3 <- symmetry_object(k7_3, M, v)

k <- symmetrize(as.minobj(k7_3), sym_7_3)

knotplot2(k)  #nice and symmetric!


## OK now convert to and from a mimimal vector for a symmetrical knot:

mii <- make_minsymvec_from_minobj(k, sym_7_3)
pii <- make_minobj_from_minsymvec(mii,sym_7_3)
knotplot2(pii)


##   So 'mii' is a minimal vector for a symmetrical knot, and 'pii' is
##   the corresponding minobj object.  Note that you can mess about with
##   mii, but whatever you do the resulting knot is still symmetric:

mii[2] <- 1000
knotplot2(make_minobj_from_minsymvec(mii,sym_7_3))   # still symmetric.

## and, in particular, you can optimize the badness, using nlm():

## Not run: 
fun <- function(m){badness(make_minobj_from_minsymvec(m,sym_7_3))}
o <- nlm(fun,mii,iterlim=4,print.level=2)

knotplot2(make_minobj_from_minsymvec(o$estimate,sym_7_3))

## End(Not run)

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