The Geometry of Ships

8 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
Fig. 2
Local and global frames.
The simplest description of a local frame is to give
the coordinates X O (X O , Y O , Z O ) of its origin in the
global frame, plus a triple of mutually orthogonal unit
vectors {ê x , ê y , ê z } along the x, y, z directions of the
frame.
Non-cartesian coordinate systems are sometimes
useful, especially when they allow some geometric symmetry
of an object to be exploited. Cylindrical polar coordinates
(r, , z) are especially useful in problems that
have rotational symmetry about an axis. The relationship
to cartesian coordinates is:
x r cos, y r sin, z z (3)
or, conversely,
r [x 2 y 2 ] 1/2 , arctan (y / x), z z (4)
For example, if the problem of axial flow past a body
of revolution is transformed to cylindrical polar coordinates
with the z axis along the axis of symmetry, flow
quantities such as velocity and pressure are independent
of ; thus, the coordinate transformation reduces the
number of independent variables in the problem from
three to two.
Spherical polar coordinates (R, , ) are related to
cartesian coordinates as follows:
x R cos cos, y R cos sin, z R sin (5)
or conversely,
R [x 2 y 2 z 2 ] 1/2 ,
arctan(z/[x 2 y 2 ] 1/2 ), (6)
arctan (y / x)
2.2 Homogeneous Coordinates. Homogeneous coordinates
are an abstract representation of geometry,
which utilize a space of one higher dimension than
the design space. When the design space is 3-D, the
corresponding homogeneous space is four-dimensional
(4-D). Homogeneous coordinates are widely used for
the underlying geometric representations in CAD and
computer graphics systems, but in general the user
of such systems has no need to be aware of the fourth
dimension. (Note that the fourth dimension in the
context of homogeneous coordinates is entirely different
from the concept of time as a fourth dimension
in relativity.) The homogeneous representation
of a 3-D point [x y z] is a 4-D vector [wx wy wz w],
where w is any nonzero scalar. Conversely, the homogeneous
point [a b c d], d 0, corresponds to the
unique 3-D point [a / d b / d c / d]. Thus, there is an
infinite number of 4-D vectors corresponding to a given
3-D point.
One advantage of homogeneous coordinates is that
points at infinity can be represented exactly without exceeding
the range of floating-point numbers; thus, [a b c
0] represents the point at infinity in the direction from
the origin through the 3-D point [a b c]. Another primary
advantage is that in terms of homogenous coordinates,
many useful coordinate transformations, including
translation, rotation, affine stretching, and perspective
projection, can be performed by multiplication by a suitably
composed 4 4 matrix.
2.3 Coordinate Transformations. Coordinate transformations
are rules or formulas for obtaining the coordinates
of a point in one coordinate system from its
coordinates in another system. The rules given above relating
cylindrical and spherical polar coordinates to
cartesian coordinates are examples of coordinate transformations.
Transformations between cartesian coordinate systems
or frames are an important subset. Many useful coordinate
transformations can be expressed as vector and
matrix sums and products.
Suppose x (x, y, z) is a point expressed in frame coordinates
as a column vector; then the same point in
global coordinates is
X (X, Y, Z) X O Mx (7)
where X O is the global position of the frame origin, and
M is the 3 3 orthogonal matrix whose rows are the unit
vectors ê x , ê y , ê z . The inverse transformation (from
global coordinates to frame coordinates) is:
x M 1 (X X O ) M T (X X O ) (8)
(Since M is orthogonal, its inverse is equal to its transpose.)
A uniform scaling by the factor (for example, a
change of units) occurs on multiplying by the scaled
identity matrix:
S
⎧ 0 0 ⎫
⎪
⎪
I⎪
0 0 ⎪
⎪
⎪
⎩ 0 0 ⎭
(9)