Flow-lines are one of the most conspicuous means for visualizing two- and three-dimensional vector fields. The construction of flow-lines is most often accomplished by numerical integration of the defining differential equation
This defines a flow-line 
as a function of a scalar parameter
for the given field 
which passes through the
point 
.
In modern visualization systems, a broad spectrum of methods is used
which ranges from simple linear backward Euler to higher order
integration schemes.
Within the VISTA visualization concept
the restriction of the data to simplex sets enables us to use a
promising alternative. The value of the field 
is only known at
the vertices of the VVF data.
Whenever the value of 
within the simplex
is required, it must be interpolated from the
underlying grid points, independent of the construction method for the
flow-line. Within the vista visualization, all scalar quantities are
interpolated piecewise linear on the simplex sets. This suggests to
attempt a piecewise linear interpolation for the vector quantities as
well. The aim of this approach (which also exhibits some deficiencies)
is to end up with a (piecewise) analytical solution of the
flow-line differential equation which promises significant performance
and accuracy advantages compared to numerical schemes.
A linear interpolation of within a simplex can
always be expressed as
and a solution of Equation 5.7, 5.8 is obtained by choosing
After reformulating Equation 5.10 one ends up with the following system of linear ordinary (inhomogeneous, autonomous) differential equations.
The solution of the homogeneous system is
and a particular solution can be found by variation of
the constant a
so that 
becomes
The superposition of homogeneous and particular solution
must
satisfy the boundary condition
and we end up with the final solution for the flow-line
All statements so far are valid and applicable independent of the dimension, and the problem is solved from a theoretical point of view. However, two open problems remain towards the numerical implementation.
(and 
)
be found from the values of 
given at the
grid points 
(the corner points of a simplex)?
be computed?
