BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV
DYNAMIC SYSTEM
TO GENERATE THE COORDINATES
OF AN INTRINSIC DEFINED CURVE
S. S. RADNEF
Abstract. The paper deals with the differential geometry problem of finding
the Cartesian coordinates of a curve defined by its own normal and torsion
curvatures. Analysing the Serret-Frenet frame kinematics, there are derived
the differential equations of its orientation angles. Then, by common
geometry tools, there are derived the differential equations of the spherical
coordinates, which are used to determine the cartesian ones. The result thus
obtained is used for some special cases of intrinsic defined curves. The
differential system that generates the coordinates of an intrinsic defined curve
is used when we intend to describe in an intrinsic manner the general motion
of a rigid body (namely an aircraft or space vehicle).
1. Introduction
A curve is defined in intrinsic manner when its curvature and torsion are given by two
functions of one parameter. For a plane curve thus defined, there is the Euler explicit
integral solution that gives the cartesian coordinates of the curve in terms of its curvature.
For the three-space the problem of finding such explicit solution (of integral or
differential type) is a very difficult task. Perhaps of no significant meaning for differential
geometry, this problem becomes an intrinsic one for the equations of motion of a free
rigid body (an aircraft for example) which must not be dependent on the coordinate
system of the reference frame. Because the equations of motion are of differential type,
we will derive an explicit differential system which provides, as its Cauchy solution, the
cartesian coordinates of a curve defined by its own intrinsic equations, that is
kn  Kn()  R  , kt  Kt ( )  R .
The main goal of this paper is attained by kinematics considerations, regarding the
movement of a geometrical point along its own trajectory and by this way emphasising
the kinematics of the Serret-Frenet axes system, denoted (TdF). We will use the spherical
coordinates and the intrinsic definition of the curve referred to this coordinate system,
that is a curve defined relative to a spherical surface (as an example we may consider the
curve that represent the flight path of an aircraft relative to the Earth surface). The figure
1 represents these coordinate axes systems. We denote as O  T and ff f fT




the angles defined in the following figures and as O  p O  I  q O  J  rO  K




f  p f  t  q f n  rf b the angular velocities of local spherical axes system and,

Dept. of Flight Dynamics, National Institute of AeroSpace Research, Bucharest
316
Dynamic system to generate the coordinates of an intrinsic defined curve
respectively, that of the Serret-Frenet axes system. The velocity of the geometrical point
along the trajectory is:

 


dR
 K 
vC 
R
(1)
O R
dt ( P )
Fig. 1
The angular coordinates of the Frenet frame are as follows:
Fig. 2
2. Equations of kinematics
 (TdF) kinematics
The (TdF) rotation relative to the spherical axes system, denoted (TdO), is expressed
as a result of succesive rotation represented by f angles:
Bulletin of the Transilvania University of Brasov Vol 13(48) - 2006
317
 
 



 f +  f +  f
f = pf  t + q f  n + rf  b = 
having:





 f  -
 f K = -
 f (sin  f t + cos f  cos f n - sin  f cos f b)






 f   f (cos f I - sin  f J ) =  f (sin  f n  cos  f b )





 f   f (cos  f sin  f I  cos  f cos  f J  sin  f K )   f  t
By comparing similar terms, in (Cxfyfzf) axes system, we derive the following differential
system:
 f cosf = -qf  cosf  rf  sin f


= q  sin  + r  cos
f
f
f
f
f
 f cosf = p f  cosf + q f  sin f  cosf + rf  sin f sin  f
expressed as :
 f   f f
(2)
using the matrix:
0  cos 1 f  cos f

 f  0
sin f
1
tan f  cos f

cos 1 f  sin f 

cos f 
tan f  sin f 
Defining the trajectory relative to a spherical surface of the spherical coordinates, with
the main curvatures denoted kn , kt, we may write the angular velocity of (TdF):



fN  v C k t  t  v C k n b
Regarding the linear velocity, we take into account that it may be expressed too as:





v C  v C  t (v C sin f cosf ) I (v C cosf cosf )J (v C sin f )K
(3)
 (TdO) kinematics
The angular velocity of (TdO), o, is expressed as a result of successive rotation
represented by O angles:
  
O   
with the kinematic relations:





   (cos   J  sin   K) ,      I
So, we may write the differential equation of the spherical angles:
 O  O  O
using the matrix:
 0 cos  sin  
O  
0
0 
 1
3. Coordinates of intrinsic defined trajectory
(4)
Dynamic system to generate the coordinates of an intrinsic defined curve
318
Considering the equations (1), (2), (3), and (4) we derive the differential system

R  vC  sin f
    
O
O
(5)
O
 f   f f
and:
vC
 cos  f cos f
R
vC
qO 
 sin  f cos f
R
v
rO  tan   C  sin  f cos f
R
p f  vC  k t
pO 

qf  0
rf  v C  k n
Using the spherical coordinates obtained by solving the previous system, one may
derive the cartesian coordinates::
X p  R cos  cos 
Yp  R cos  sin 
Z P  R  sin 
Simplifying the (5) system by the factor vC we derive the differential system which
produces by common integration procedures (numerical or analytical if possible) the
coordinates of an intrinsic defined curve.
dR
 sin  f
ds
d O
ˆO
  O 
ds
d f
ˆf
  f 
ds
considering:
(6)
1
1
O   cos f cos f
vC
R
1
t
ˆf 

 f  k t 0 k n
vC
ˆO 

sin f cos f
tan   sin f cos f
t
4. Two sample intrinsic defined curves
To validate the differential system (6) we choose curves defined in cartesian
coordinates by three functions of one parameter and then derive their intrinsic equations.
Using these intrinsic equations we must obtain using the system (6) and the appropriate
starting point and direction the given cartesian coordinates. For example there were
chosen a spiral curve and an oscillating ellipse.
The spiral curve:
Bulletin of the Transilvania University of Brasov Vol 13(48) - 2006
319
x  a cos 
y  b sin 
z
h

2
has the curvature:
kn 
q ( )
2

 2 2
 a sin   b 2 cos 2    h  

 2  

3/ 2
and the torsion:
h
2
kt 
q( )
ab
2


 h  2
2
2
2
 a cos   b sin  . Giving some values to
 2spiral

curve
with the function q()  a 2 b2  
the parameters a, b, , h we have obtained, using the system (6), the same cartesian
coordinates, and thus the same curve , as in the following figure:
 X g  Y g  Z g Fig. 1 Spiral curve
The oscillating ellipse, defined by the cartezian coordinates:
x  a cos 
y  b sin 
z  c sin( 4)
has the curvature:
kn 
a
2
sin 2   b2 cos 2   16c 2 cos 2 (4)
and the torsion:
kt 
q( )
 60abc cos( 4)
q( )

3/ 2
Dynamic system to generate the coordinates of an intrinsic defined curve
320
2
traiectoria prescrisa
2 2
with the function q( )  16b c 4 cos  sin( 4 )  sin  cos( 4)  
16c 2 a 2 cos  cos( 4 )  4 sin  sin( 4 )   a 2 b 2
2
Giving values to the parameters a, b, c we have obtained, using the system (6), the same
cartezian coordinates, and thus the same curve, as in the following figure:
Fig.2 Oscillating ellipse
5. Conclusions
The differential
(6) tY
which
( x system
y  z)  ( tX
 tZ) gives the spherical coordinates of the curve defined
in intrinsic manner fulfilled the requirements to be integrated in the equations of motion
of a rigid body (an aircraft as a sample one), as it was used in the reference 2 for finding
the control laws to pursuit a given trajectory in a manner that in not dependent on
coordinate system.
Besides the two sample curves, the (6) system was used to provide the cartesian
coordinates for many other curves, defined by coordinate values given along a given set
of a free parameter values. The results where in a very good agreement with the defining
cartesian coordinates. In such cases the curvatures must be determined using numerical
tools with a great accuracy, which is a necessary requirement to obtain good results.
6. References
1.
2.
Iacob Caius, Mecanica teoretica, Didactic and Pedagogic Publishing House,
Bucharest, 1980
Sorin St. Radnef, Flight dynamics and control studies for pursuit problems, Ph d
thesis, Polytechnic University of Bucharest, 2002