# eryar

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Abstract. Although polynomials offer many advantages, there exist a number of important curve and surface types which cannot be represented precisely using polynomials, e.g., circles, ellipses, hyperbolas, cylinders, cones, spheres, etc. So we introduce the concepts of rational curves and homogeneous coordinates to solve the problem. To understand rational curves in a homogenous coordinate system is more straightforward. If you define irrational Bezier curves in 4D space and then project them back into 3D space, you obtain rational curves.

Key Words. OpenCASCADE, Rational Curves, Homogenous Coordinate, Geom_BezierCurve

1. Introduction

2. Homogeneous Coordinates

Figure 2.1 A representation of Euclidean points to homogeneous form

3. Bezier Curve Perspective Map

4. The Effects of Weighting Factors

4.1 可退化性

Figure 4.1 Degeneracy of Rational Bezier Curve

#
#    Copyright (c) 2014 eryar All Rights Reserved.
#
#        File    : degenerate.tcl
#        Author  : eryar@163.com
#        Date    : 2014-09-19 18:10
#        Version : 1.0v
#
#    Description : Demonstrate the degeneracy of Rational Bezier Curve.
#

# Bezier Curve with Weighted Poles:
# {P(1,0), w(1)}, {P(1,1), w(1)}, {P{0,1}, w(1)}

2dbeziercurve bc1 3 1 0 1 1 1 1 0 1 1

v2d
2dfit

# Bezier Curve without weights:
2dbeziercurve bc2 3 1 0 1 1 0 1
mkedge e1 bc2
0 1

vdisplay e1
vtop
vfit

4.2 当ω0≠0，ωn≠0时，对任意u∈[0,1]，

Figure 4.2 Rational Bezier Curve with different Weights

#
#    Copyright (c) 2014 eryar All Rights Reserved.
#
#        File    : degenerate.tcl
#        Author  : eryar@163.com
#        Date    : 2014-09-19 18:10
#        Version : 1.0v
#
#    Description : Demonstrate the Rational Bezier Curve with different weights.
#

set w1
0.1
set w2
0.5
set w3
1.0
set w4
2.0
set w5
5.0

set u
0.5

# 4 Bezier Curve with defferent Weighted Poles:
# {P(1,0), w(1)}, {P(1,1), w(\$w)}, {P{0,1}, w(1)}

2dbeziercurve bc1 3 1 0 1 1 1 \$w1 0 1 1
2dbeziercurve bc2
3 1 0 1 1 1 \$w2 0 1 1
2dbeziercurve bc3
3 1 0 1 1 1 \$w3 0 1 1
2dbeziercurve bc4
3 1 0 1 1 1 \$w4 0 1 1
2dbeziercurve bc5
3 1 0 1 1 1 \$w5 0 1 1

# mark weight factor.
2dcvalue bc1 \$u x1 y1
2dcvalue bc2
\$u x2 y2
2dcvalue bc3
\$u x3 y3
2dcvalue bc4
\$u x4 y4
2dcvalue bc5
\$u x5 y5

dtext x1 y1 w1
=\$w1
dtext x2 y2 w2
=\$w2
dtext x3 y3 w3
=\$w3
dtext x4 y4 w4
=\$w4
dtext x5 y5 w5
=\$w5

v2d
2dfit

5. Bezier Geometry Curve

6. Conclusion

7. References

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