Page 21 - 2024F

14 UEC Int’l Mini-Conference No.53

Figure 1: LineSets visualization of restaurant categories on a map (left) and communities on a social
network (right) [2].

rameters such as maximum curvature and cross-
ing angles of the curves based on Bézier splines,
aiming to find the optimal solution through a
multi-objective optimization process.

2 Methodology

This section describes the methodology used in Figure 2: An example of cubic Bézier curve.
this study, including mathematical foundations,
analysis of geometric factors in LineSets, algo-
rithm design, and system implementation.
A cubic Bézier spline is a curve constructed
2.1 Cubic Bézier splines by joining multiple cubic Bézier curves at end-
points. A set of n points can be represented by
The foundation of this work lies in the use of a cubic Bézier spline consisting of n − 1 cubic
cubic Bézier splines to represent the curves in Bézier curve segments. The given points are the
LineSets. The cubic Bézier spline is particularly endpoints of the Bézier curves that make up the
suited for LineSets and its optimization due to spline, ensuring that the spline passes through
its mathematical properties, which allow for pre- all points in the given set.
cise control over the shape of the curve while
maintaining continuity. An important property of cubic Bézier splines
A cubic Bézier curve is defined by four control is their ability to ensure smooth transitions be-
points, denoted as P 0 , P 1 , P 2 , P 3 . The curve is tween consecutive curve segments. For each cu-
defined as: bic Bézier curve, the direction of the tangent at
P 0 is aligned with P 0 P 1 , while the tangent at P 3
1
3 2 is aligned with P 2 P 3 . Therefore, C continuity
B(t) = (1 − t) P 0 + 3(1 − t) tP 1
(1) can be achieved by ensuring that the tangents
3
2
+ 3(1 − t)t P 2 + t P 3 ,
on either side of a connecting point are aligned.
where P 0 and P 3 are the endpoints of the curve, For two curve segments defined by P 0 , P 1 , P 2 , P 3
and P 1 , P 2 can move freely to control the shape and P 3 , P 4 , P 5 , P 6 , connected at P 3 , the require-
1
of the curve. ment of C continuity at P 3 is: