Visualization (Assignment 7)

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University of Stuttgart

	Institute for Visualization and Interactive Systems
	Visualization (Assignment 7)

Betreuer: Daniel Weiskopf

Visualization (Assignment 7)

Assignment 7.1 [6 points] 2D Vector Field Topology

In the lecture, topology-guided analysis and classification of vector fields has been presented. The idea of this technique is to investigate critical points of a vector field by considering the eigenvalues of the Jacobian matrix J at these critical points. A critical point is defined as a point where the vector field vanishes. The following cases can be identified, based on the eigenvalues $\lambda_1$ and $\lambda_2$ :

Attracting node:: The eigenvalues are real and positive, corresponding to a sink.
Repelling node:: The eigenvalues are real and negative, corresponding to a source.
Saddle point:: The eigenvalues are real and have different sign.
Repelling focus:: Both eigenvalues are complex conjugates of each other and the real part is positive, i.e., a repelling spiral.
Attracting focus:: Both eigenvalues are complex conjugates of each other and the real part is negative, i.e., an attracting spiral.

(a) Classify these cases based on the determinant and trace of the matrix J. (Hint: For a 2D matrix we have trace $(J) = \lambda_1 + \lambda_2$ , det $(J) = \lambda_1\cdot\lambda_2$ .)

(b) Consider the vector field

$\displaystyle \vec{v}(\vec{x})=\left( \begin{array}{c} \mu x + \rho y \\ - \sigma x + \mu y \end{array} \right)$

Determine and classify the critical points for the following parameters:

Parameter set	$\mu$	$\rho$	$\sigma$
1	0	-1	1
2	1	0	0
3	-1	0	0
4	-2	-1	-1

Assignment 6.2 [4 points] Pathlines and Streaklines in Time-Varying Vector Fields

The aim of this assignment is to get accustomed to the difference between pathlines, streaklines, and streamlines in time-varying vector fields. A pathline describes the trajectory of a particle moving through a velocity field $\vec{v}(\vec{x},t)$ and can be determined according to the ordinary differential equation

$\displaystyle \frac{d \vec{x}(t)}{dt} = \vec{v}(\vec{x}(t), t) \quad ,$

for some given initial position $\vec{x}(0) = \vec{x}_0$ . Time is denoted t. Physically, such a pathline corresponds to a long time exposure of an illuminated fluid particle.

A streamline corresponds to a solution of

$\displaystyle \frac{d \vec{x}(\lambda)}{d\lambda} = \vec{v}(\vec{x}(\lambda), t) \quad ,$

where the time t is considered to be constant and $\lambda$ parameterizes the resulting curve.

A streakline corresponds to the curve traced out by particles injected at a fixed position $\vec{x}_1$ --analogously, to dye injection in classical fluid dynamic experiments. A streakline consists of all particles which pass by the injection point $\vec{x}_1$ during the time $t_1 \in [0,t]$ . Note that pathlines, streaklines, and streamlines are identical for a stationary flow.

During the lab hour we will develop a pseudo-code for a particle tracer for time-varying flows. We will consider pathlines, streaklines, and streamlines. As an example, we will investigate the vector field

$\displaystyle \vec{v}(\vec{x}, t) = (\cos t, \sin t) \quad ,$

and identify differences between the three visualization techniques.

Please review Assignment 4.2 and your notes on particle tracing. Please hand in a brief proposal (approx. 10 lines of text) which might deal with

differences in the required data structures between particle tracing in stationary and time-varying vector fields, and/or
differences in the actual particle tracing module, and/or
an educated guess of how pathlines, streaklines, and streamlines might differ in the above vector field.

The text need not contain the complete pseudo code or a sophisticated analysis of the three visualization techniques; it could rather be considered as a description of ``work in progress''!

Points will be given to those who hand in a short proposal and will participate in the lab hour.

Submission: Wednesday, 20. June 2001, before the lecture

Daniel Weiskopf
2001-06-13