Master's Thesis - Studierstube Augmented Reality Project - Graz ...

2.1 Technical Visualization
Arnold1998]. The order defines the number of pre-integration steps before they are
averaged added to the previous step x k . Equation 2.3 can be amplified to
n∑
x k+1 = x k + h b j c j , (2.4)
with procedure depending coefficients b j and the intermediate steps c j . Each c j is
calculated per step with a basic Euler integration step. Defining a concrete n, for
example n = 4, will lead to the popular Runge-Kutta method of fourth oder or denoted
by RK4. An approximated integration with four intermediate steps will then be defined
by
j=1
x k+1 = x k + h 6 (c 1 + 2c 2 + 2c 3 + c 4 ), where (2.5)
c 1 = v(x k , t k , τ),
c 2 = v(x k + h 2 c 1, t k + h 2 , τ),
c 3 = v(x k + h 2 c 2, t k + h , τ) and
2
c 4 = v(x k + hc 3 , t k + h, τ).
Compared to the former described explicit Euler method the benefits of this technique
are shown in figure 2.13.
Consequently, it can be denoted, that the Runge-Kutta
method will produce more accurate results with less sampled supporting points of the
desired trajectory. This technique will be favored later on.
Storing the supporting points as a result of the calculation of a certain amount of
particles will produce a n × m-array with all sampled trajectory points depending on
the step sizes ∆t and the time parametrization τ.
This array can be utilized for the further presented visualization techniques except
of a concrete particle system effect itself. Visualizing particles in a flow field directly will
only require the calculation of the next position of the particle, so previous trajectory
points need not to be stored. Line based visualizations, or similar approaches must not
discard these ”older” positions.
Based on simple particle calculations a vast number of trajectory based flow visualization
techniques have been investigated. Referring to [Weiskopf2007] the following
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