Dimensionality of shape space for triangles: Sumary of answers regarding my previous question

[morphmet]

-------- Original Message --------
Subject: Dimensionality of shape space for triangles: Sumary of answers 
 regarding my previous question
Date: Mon, 30 Nov 2009 16:34:44 GMT
From: Oliver Betz <oliver.b...@uni-tuebingen.de>
To: morphmet@morphometrics.org

Dear morphometricians:

Last week I posted the following question on the
server. Please find below the summary of the
replies I received from several morphometricians.
They all say that the shape space of triangles
refers to the surface of a sphere, so that it is
2D and not 3D. Thanks to all who replied!
Oliver



Question:
If the shape space for triangles has the dimensionality
2k-4, i.e. 2x3-4 = 2, why is the shape space for triangles usually
drawn in the form of  a globe or sphere (e.g. in TPSTri), which is a
3D space. In my understanding, the triangles should be part of a 2D-
space  such as the tangent projection of the shape space.


Summary of replies:

The Kendall shape space for triangles is the
*surface* of the sphere, which is two-dimensional.
But you are right, this 2D space is embedded in three dimensions.
It is a bit like the surface of the earth (a 2D
surface around a 3D planet). The tangent
projection is a representation of a part of the
shape space, as a map is a flat representation of
a part of the Earth's surface.


The shape space for triangles is the _surface_ of a (hemi)sphere, not
the interior of the sphere.
Hence it is two-dimensional, yet not "flat". The tangent space to this
shape space is a two-dimensional Euclidean vector space.

As I understood it, the morphospace for triangles
actually corresponds to the plane that forms the
outer layer of the sphere (so a 2D space), so not
a full 3D sphere but just the skin.

...having triangles constrained to be on the
surface of a sphere with a constant radius makes
their df 2. The triangles cannot be anywhere in
the sphere: they can only be on its surface and that
means that even if the sphere is 3D, triangles have only 2 df.








Univ.-Prof. Dr. Oliver Betz
Institut für Evolution und Ökologie
Abt. Evolutionsbiologie der Invertebraten
Auf der Morgenstelle 28E
D-72076 Tübingen
Germany

phone: 0049-(0)7071-2972995
E-mail: oliver.b...@uni-tuebingen.de
http:/www.uni-tuebingen.de/agbetz

Mitglied des Netzwerks für Elektronenmikroskopie Tübingen (NET)
Member of the Network for Electron microscopy Tuebingen
http://www.uni-tuebingen.de/zet







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