Morphometric Data - An Invariant Approach to Statistical Analysis of Shapes - page 51

Figure 2.7 Pictorial representation of the operations of translation and rotation. The
triangle drawn with an unbroken line is the original configuration. The triangle drawn
with a broken line represent a change in the triangle due to rotation on an axis with
no translation. The triangle drawn with a dotted line represents a change in the trian-
gle due to translation across the plane with no rotation. The triangle drawn with dots
and dashes represents the result of the combination of rotation and translation.

Notice that rotation of a three-dimensional object requires
specification of two angles of rotation α

and o / .

d) Translation of an object : Let the landmark coordinate matrix

for a given two dimensional object be denoted by M . Suppose
we translate this object so that all the X-coordinates are
moved by t 1 units and the Y-coordinates are moved by t 2 units.
The landmark coordinate matrix of the translated object is

given by M + 1 t where 1= , a K 1 column vector of 1's and

t is a 1 by 2 row vector. We call “ t ” the translation vector or a
translation parameter.