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A New Approach to Visualizing General Trees
Using Thickness-Adjustable Quadratic Curves
M. Ali Rostami 1 , Azin Azadi 2 ,andH.MartinBucker 1
1 Friedrich Schiller University Jena, Germany
{ a.rostami,martin.buecker } @uni-jena.de
2 Jovoto Company, Berlin, Germany
aazadi@gmail.com
In this abstract, we present a new algorithm for visualization of trees. This
algorithm illustrates the hierarchical data by using curve segments for each edge
and how edges evolve on a path from the root to the leaves. More clearly, we
visualize a tree based on the shape of a botanical tree. The software tool Graph-
Tea [3,4] implements this algorithm where the adjustable thicknesses of the curve
segments reflect the expansion [2] of botanical trees. Here, we consider a simple
expansion definition for each vertex as the sum over the number of children and
the children of its children up to the some specific level.
Given a tree and a layout of this tree, the algorithm computes a set of curve
segments as follows. A quadratic parameter curve segment is drawn for each
node starting from the tree root moving toward all leaves. Each curve segment
is associated to three consecutive nodes on a path. Suppose that these nodes
are located at the positions P 0 ,P 1 and P 2 in the layout. Then, the quadratic
parameter curve segment
t ) 2 P 0 +2(1
t ) tP 1 + t 2 P 2 ,
B ( t )=(1
t
[0 , 1] ,
is computed. After generating these curve segments for each node, a general
shape is generated from them by interpolating the segments. This approach
generates a tree like Fig. 1 (a) with a fixed thickness.
Although this visualization with a fixed thickness gives an overall image of
the tree structure, we can visualize more information using thickness. To control
the thickness, we visualize each curve segment by two so-called boundary curves.
As a result, we can generate different thicknesses by changing the start and end
positions of these boundary curves. More clearly, suppose we are given points
P 0 ,P 1 ,P 2 of the curve segment and the angles ʸ 0 , ʸ 1 , ʸ 2 , between their position
vectors. Also, let w 0 , w 1 ,and w 2 denote the starting, the middle, and the end
thicknesses of the curve segment, i.e., the distance between the boundary curves
in their different parts. First, we compute three values: the start width S ,the
middle width M , and the end width E ,
S = w 0 + w 1
2
, M = w 1 , and E = w 1 + w 2
2
.
These widths specify the distances between the two boundary curves at three
positions.
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