16 research outputs found
Shape Derivatives
Shape Theory, together with Shape-and-Scale Theory, comprise Relational
Theory. This consists of -point models on a manifold , for which some
geometrical automorphism group is regarded as meaningless and is thus
quotiented out from the -point model's product space .
Each such model has an associated function space of preserved quantities,
solving the PDE system for zero brackets with the sums over of each of
's generators. These are smooth functions of the -point geometrical
invariants. Each pair has moreover a `minimal nontrivially relational
unit' value of ; we now show that relationally-invariant derivatives can be
defined on these, yielding the titular notions of shape(-and-scale)
derivatives. We obtain each by Taylor-expanding a functional version of the
underlying geometrical invariant, and isolating a shape-independent derivative
factor in the nontrivial leading-order term. We do this for translational,
dilational, dilatational and projective geometries in 1-, the last of which
gives a shape-theoretic rederivation of the Schwarzian derivative. We next
phrase and solve the ODEs for zero and constant values of each derivative. We
then consider translational, dilational, rotational, rotational-and-dilational,
Euclidean and equi-top-form (alias unimodular affine) cases in -. We
finally pose the PDEs for zero and constant values of each of our -
derivatives, and solve a subset of these geometrically-motivated PDEs. This
work is significant for Relational Motion and Background Independence in
Theoretical Physics, and foundational for both Flat and Differential Geometry.Comment: 15 pages including 2 figure
Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles
We provide a natural answer to Lewis Carroll's pillow problem of what is the
probability that a triangle is obtuse, Prob(Obtuse). This arises by
straightforward combination of a) Kendall's Theorem - that the space of all
triangles is a sphere - and b) the natural map sending triangles in space to
points in this shape sphere. The answer is 3/4. Our method moreover readily
generalizes to a wider class of problems, since a) and b) both have many
applications and admit large generalizations: Shape Theory. An elementary and
thus widely accessible prototype for Shape Theory is thereby desirable, and
extending Kendall's already-notable prototype a) by demonstrating that b)
readily solves Lewis Carroll's well-known pillow problem indeed provides a
memorable and considerably stronger prototype. This is a prototype of, namely,
mapping flat geometry problems directly realized in a space to shape space,
where differential-geometric tools are readily available to solve the problem
and then finally re-interpret it in the original `shape-in-space' terms. We
illustrate this program's versatility by posing and solving a number of
variants of the pillow problem. We first find Prob(Isosceles is Obtuse). We
subsequently define tall and flat triangles, as bounded by regular triangles
whose base-to-median ratio is that of the equilateral triangle. These
definitions have Jacobian and Hopfian motivation as well as entering Kendall's
own considerations of `splinters': almost-collinear triangles. We find that
Prob(Tall) = 1/2 = Prob(Flat) is immediately apparent from regularity's
symmetric realization in shape space. However, Prob(Obtuse and Flat),
Prob(Obtuse is Flat) and all other nontrivial expressions concerning having any
two of the properties mentioned above, or having one of these conditioned on
another, constitute nontrivial variants of the pillow problem, and we solve
them all.Comment: 13 pages, 6 figure
Maximal Angle Flow on the Shape Sphere of Triangles
There has been recent work using Shape Theory to answer the longstanding and
conceptually interesting problem of what is the probability that a triangle is
obtuse. This is resolved by three kissing cap-circles of rightness being
realized on the shape sphere; integrating up the interiors of these caps
readily yields the answer to be 3/4. We now generalize this approach by viewing
rightness as a particular value of maximal angle, and then covering the shape
sphere with the maximal angle flow. Therein, we discover that the kissing
cap-circles of rightness constitute a separatrix. The two qualitatively
different regimes of behaviour thus separated both moreover carry distinct
analytic pathologies: cusps versus excluded limit points. The equilateral
triangles are centres in this flow, whereas the kissing points themselves --
binary collision shapes -- are more interesting and elaborate critical points.
The other curves' formulae and associated area integrals are more complicated,
and yet remain evaluable. As a particular example, we evaluate the probability
that a triangle is Fermat-acute, meaning that its Fermat point is nontrivially
located; the critical maximal angle in this case is 120 degrees.Comment: 18 pages including 9 figure
Isotropy Groups and Kinematic Orbits for 1 and 2- -Body Problems
Mitchell and Littlejohn showed that isotropy groups and orbits for -body
problems attain a sense of genericity for . The author recently showed
that the arbitrary- generalization of this 3- result is that genericity
in this sense occurs for . The author also showed that a second
sense of genericity -- now order-theoretic rather than a matter of counting --
occurs for , excepting , for which it is not 7 but 8.
Applications of this work include 1) that some of the increase in complexity in
passing from 3 to 4 and 5 body problems in 3- is already present in the
more-well known setting of passing from intervals to triangles and then to
quadrilaterals in 2-. 2) That not but is a
natural theoretical successor of . 3) Such consideration isotropy
groups and orbits is moreover a model for a larger case of interest, namely
that of GR's reduced configuration spaces. The current Article presents the
lower- cases explicitly: 0, 1 and 2-, including also the topological and
geometrical form of the corresponding isotropy groups and orbits.Comment: 16 pages, including 4 figures. arXiv admin note: text overlap with
arXiv:1807.0839
Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
We show that the recent derivation that triangleland's topology and geometry
is from Heron's formula does not extend to quadrilaterals by considering
Brahmagupta, Bretschneider and Coolidge's area formulae. That -a-gonland is
more generally (with recovering the triangleland
sphere) follows from Kendall's extremization that is habitually used in Shape
Theory, or the generalized Hopf map. We further explain our observation of
non-extension in terms of total area not providing a shape quantity for
quadrilaterals. It is rather the square root of of sums of squares of subsystem
areas that provides a shape quantity; we clarify this further in
representation-theoretic terms. The triangleland moreover also
generalizes to -simplexlands being topologically by
Casson's observation. For the 3-simplex - alias tetrahaedron - while volume
provides a shape quantity and is specified by the della Francesca-Tartaglia
formula, the analogue of finding Heron eigenvectors is undefined. -volume
moreover provides a shape quantity for the -simplex, specified by the
Cayley-Menger formula generalization of the Heron and della Francesca-Tartaglia
formulae. While eigenvectors can be defined for the even- Cayley-Menger
formulae, the dimension count does not however work out for these to provide
on-sphere conditions. We finally point out the multiple dimensional
coincidences behind the derivation of the space of triangles from Heron's
formula. This article is a useful check on how far the least technically
involved derivation of the smallest nontrivial shape space can be taken. This
is significant since Shape Theory is a futuristic branch of mathematics, with
substantial applications in both Statistics (Shape Statistics) and Theoretical
Physics (Background Independence: of major relevance to Classical and Quantum
Gravitational Theory).Comment: 17 pages, including 7 figure
Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities
Sides and medians are both Jacobi coordinate magnitudes, moreover then
equably entering the spherical coordinates on Kendall's shape sphere and the
Hopf coordinates. This motivates treating medians on the same footing as sides
in triangle geometry and the resulting Shape Theory. In this paper, we
consequently reformulate inequalities for the medians in terms of shape
quantities, and proceed to find inequalities on the mass-weighted Jacobi
coordinates. This work moreover identifies the -- powers of which occur
frequently in the theory of medians -- as the ratio of Jacobi masses.
One of the Hopf coordinates is tetra-area. Another is anisoscelesness, which
parametrizes whether triangles are left-or-right leaning as bounded by
isoscelesness itself. The third is ellipticity, which parametrizes
tallness-or-flatness of triangles as bounded by regular triangles. Whereas
tetra-area is clearly cluster choice invariant, Jacobi coordinates,
anisoscelesness and ellipticity are cluster choice dependent but can be
`democratized' by averaging over all clusters. Democratized ellipticity
moreover trivializes, due to ellipticity being the difference of base-side and
median second moments, whose averages are equal to each other. Thus we
introduce a distinct `linear ellipticity' quantifier of tallness-or-flatness of
triangles whose democratization is nontrivial, and find inequalities bounding
this. Some of this paper's inequalities are shape-independent bounds, whereas
others' bounds depend on the isoperimetric ratio and arithmetic-to-geometric
side mean ratio shape variables.Comment: 22 pages including 8 figure
Polynomial Eulerian shape distributions
In this paper a new approach is derived in the context of shape theory. The
implemented methodology is motivated in an open problem proposed in
\citet{GM93} about the construction of certain shape density involving Euler
hypergeometric functions of matrix arguments.
The associated distribution is obtained by establishing a connection between
the required shape invariants and a known result on canonical correlations
available since 1963; as usual in statistical shape theory and the addressed
result, the densities are expressed in terms of infinite series of zonal
polynomials which involves considerable difficulties in inference. Then the
work proceeds to solve analytically the problem of computation by using the
Eulerian matrix relation of two matrix argument for deriving the corresponding
polynomial distribution in certain parametric space which allows to perform
exact inference based on exact distributions characterized for polynomials of
very low degree. A methodology for comparing correlation shape structure is
proposed and applied in handwritten differentiation.Comment: 26 page
Absolute versus Relational Debate: a Modern Global Version
Suppose one seeks to free oneself from a symmetric absolute space by
quotienting out its symmetry group. This in general however fails to erase all
memory of this absolute space's symmetry properties. Stratification is one
major reason for this, which is present in both a) Kendall-type Shape Theory
and subsequent Relational Mechanics, and b) General Relativity configuration
spaces. We consider the alternative starting point with a generic absolute
space, meaning with no nontrivial generalized Killing vectors whatsoever. In
this approach, generically Shape-and-Scale Theory is but trivially realized,
there is no separate Shape Theory and indeed no stratification. While the GR
configuration space version of these considerations was already expounded in
1996 by Fischer and Moncrief, the Kendall-type shape theory version is new to
the current article. In each case, this amounts to admitting some small
deformation by which symmetry's hard consequences at the level of reduced
configuration spaces are warded off.We end by discussing the senses in which
each of the above two strategies retain absolutist features, each's main known
technical advantages and disadvantages, and the desirability of replacing
Kendall-type Shape Theory with a Local-and-Approximate Shape Theory. This
article is in honour of Prof. Niall \'{o} Murchadha, on the occasion of his
Festschrift.Comment: 22 page
Manifolds of Projective Shapes
The projective shape of a configuration of k points or "landmarks" in RP(d)
consists of the information that is invariant under projective transformations
and hence is reconstructable from uncalibrated camera views. Mathematically,
the space of projective shapes for these k landmarks can be described as the
quotient space of k copies of RP(d) modulo the action of the projective linear
group PGL(d). Using homogeneous coordinates, such configurations can be
described as real k-times-(d+1)-dimensional matrices given up to
left-multiplication of non-singular diagonal matrices, while the group PGL(d)
acts as GL(d+1) from the right. The main purpose of this paper is to give a
detailed examination of the topology of projective shape space, and, using
matrix notation, it is shown how to derive subsets that are in a certain sense
maximal, differentiable Hausdorff manifolds which can be provided with a
Riemannian metric. A special subclass of the projective shapes consists of the
Tyler regular shapes, for which geometrically motivated pre-shapes can be
defined, thus allowing for the construction of a natural Riemannian metric
Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
We extend finding geometrically-significant preserved quantities by solving
specific PDEs to the affine transformations and subgroups. This can be viewed
not only as a purely geometrical problem but also as a subcase of finding
physical observables, and furthermore as part of the comparative study of
Background Independence level-by-level in mathematical structure. While cross
and scalar-triple products (combined with differences and ratios) suffice to
formulate these preserved quantities in 2- and 3- respectively, the
arbitrary-dimensional generalization evokes the theory of forms. The affine
preserved quantities are ratios of -volume forms of differences, -volume
forms being the `top forms' supported by dimension , and referring moreover
to -volumes of relationally-defined subsystems.Comment: 15 pages, including 3 figures. Updated reference