ISOGONAL AND ISOTOMIC CONJUGATES OF QUADRATIC RATIONAL Bézier CURVES

The Pure and Applied Mathematics.
2015.
Feb,
22(1):
25-34

- Received : August 01, 2014
- Accepted : December 13, 2014
- Published : February 28, 2015

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In this paper we characterize the isogonal and isotomic conjugates of conic. Every conic can be expressed by a quadratic rational Bézier curve having control polygon
b
_{0}
b
_{1}
b
_{2}
with weight
w
> 0. We show that the isotomic conjugate of parabola and hyperbola with respect to Δ
b
_{0}
b
_{1}
b
_{2}
is ellipse, and that the isotomic conjugate of ellipse with the weight
is identical. We also find all cases of the isogonal conjugate of conic with respect to Δ
b
_{0}
b
_{1}
b
_{2}
. Our characterizations are derived easily due to the expression of conic by the quadratic rational Bézier curve in standard form.
b
_{0}
b
_{1}
b
_{2}
is tangent to the lines
b
_{0}
b
_{1}
and
b
_{1}
b
_{2}
at the points
b
_{0}
and
b
_{2}
, respectively.
In triangle geometry there are two well-known conjugates. One is the isogonal conjugate and the other is the isotomic conjugate
[2
,
9]
. For given any point in the triangle, the three lines obtained by reflecting the lines passing through the given point and vertices with respect to the angle bisectors at each vertices meet at one point, which is the isogonal conjugate of the given point with respect to the triangle, as shown in
Figure 1
. For given any point
x
in the triangle
b
_{0}
b
_{1}
b
_{2}
, and for the point
obtained by reflecting the intersection point of lines
xb
_{i}
and
b
_{j}
b
_{k}
with respect to the midpoint on the side
b
_{j}
b
_{k}
, for mutually distinct
i, j, k
, the three lines
,
,
meet at one point, which is the isotomic conjugate of the given point with respect to the triangle, as shown in
Figure 2
. The definition of the isogonal and isotomic conjugates shall be more detailedly described in the next section. The conjugates map any point inside the triangle into a point inside the triangle.
The isogonal conjugate of x is x * with respect to the triangle Δb _{0}b _{1}b _{2}.
The isogonal conjugate of x is x ° with respect to the triangle Δb _{0}b _{1}b _{2}.
We consider the conjugate points of all points on the quadratic rational Bézier curve having control polygon
b
_{0}
b
_{1}
b
_{2}
with respect to the triangle
b
_{0}
b
_{1}
b
_{2}
. The motivation of our study is the question: what are the isogonal and isotomic conjugates of the quadratic rational Bézier curve with respect to Δ
b
_{0}
b
_{1}
b
_{2}
, respectively? The researches on the isogonal and isotomic conjugates of conic have been developed
[4
,
5
,
6
,
7
,
8]
, but we cannot find the previous results about the isogonal and isotomic conjugates of conic having the control polygon with respect to Δ
b
_{0}
b
_{1}
b
_{2}
. In this paper, using the barycentric coordinates and weight of the quadratic rational Bézier curve we clearly characterize the isogonal and isotomic conjugates for all cases of quadratic rational Bézier curve for all cases.
Our manuscript is organized as follows. In Section 2, the definitions for barycentric coordinates, isogonal and isotomic conjugates, and quadratic rational Bézier curve are introduced. In Section 3, we characterize the isogonal and isotomic conjugates of all quadratic rational Bézier curves which are ellipse, parabola or hyperbola, and summarize our work in Section 4.
b
_{0}
b
_{1}
b
_{2}
, every point
x
can be uniquely expressed by
x
=
u
b
_{0}
+
v
b
_{1}
+
w
b
_{2}
,
u + v + w
= 1. The triple coordinates (
u, v, w
) are called the
barycentric coordinates
of
x
, as shown in
Figure 3
, and the ratio (
ku , kv , kw
) is called the
homogeneous barycentric coordinates
of
x
for some positive
k
. Note that the ratio of areas of triangles Δ
xb
_{1}
b
_{2}
: Δ
xb
_{2}
b
_{0}
: Δ
xb
_{0}
b
_{1}
is equal to
u , v , w
. The centroid and incenter have the homogeneous barycentric coordinates (1 : 1 : 1) and (
a : b : c
), respectively
[9]
, where
a, b, c
are lengths of side lines
in order.
Barycentric coordinates with respect to Δb _{0}b _{1}b _{2}.
2.1. Isogonal conjugate and isotomic conjugate
For any point
x
inside the triangle
b
_{0}
b
_{1}
b
_{2}
, the lines
xb
_{i}
,
i
= 0, 1, 2, are called by cevians of
x
[9]
. Let the line
ℓ_{i}
,
i
= 0, 1, 2, be the reflection of the cevian
xb
_{i}
with respect to the angle bisector at vertex
b
_{i}
, respectively. The three lines
ℓ
_{0}
,
ℓ
_{1}
and
ℓ
_{2}
are concurrent at one point, which is called the
isogonal conjugate of
x
with respect to the triangle
b
_{0}
b
_{1}
b
_{2}
and it denoted by
x
*, as shown in
Figure 1
. For each
i
= 0, 1, 2, two lines
xb
_{i}
and
x
*
b
_{i}
are symmetric with respect to the bisector of ∠
b
_{i}
. The incenter is the uniquely fixed point under the isogonal conjugate transformation, and the centroid and the orthocenter are the isogonal conjugates of the symmedian point and the circumcenter, respectively, and vice versa. If
x
has the homogeneous barycentric coordinates (
u
:
v
:
w
), then its isogonal conjugate
x
* has
[9]
.
Let the point
x
_{i}
,
i
= 0, 1, 2, be the intersection point of the cevian
xb
_{i}
and the opposite side
b
_{j}
b
_{k}
, where
i
,
j
,
k
∈ {0, 1, 2} are mutually distinct. Let
be the reflection point of
x
_{i}
with respect to the midpoint of the line segment
b
_{j}
b
_{k}
. The three lines
,
and
are concurrent at one point, which is called the
isotomic conjugate of
x
with respect to the triangle
b
_{0}
b
_{1}
b
_{2}
and we denote it by
x
°, as shown in
Figure 2
. The centroid is the uniquely fixed point under the isotomic conjugate transformation. If
x
has the homogeneous barycentric coordinates (
u
:
v
:
w
), then its isotomic conjugate
x
* has
.
2.2. Quadratic rational Bézier curve
The quadratic rational Bézier curve
r
(
t
) having the control points
b
_{i}
,
i
= 0, 1, 2 and weights
w_{i}
> 0 is defined by
where
is the Bernstein polynomial of degree two
[1
,
3]
. By changing the variable, it can be rewritten without change of the shape in the standard form
where
. For each
t
∈ [0, 1], the point
r
(
t
) has the barycentric coordinates
For nonlinear polygon
b
_{0}
b
_{1}
b
_{2}
the quadratic rational Bézier curve
r
(
t
) is a conic, which is ellipse iff
w
< 1, parabola iff
w
= 1, and hyperbola iff
w
> 1, as shown in
Figure 4
.
Quadratic rational Bézier curve having control polygon b _{0}b _{1}b _{2} with weight w > 0 is ellipse(green) if w < 1, parabola(magenta) if w = 1, and hyperbola(blue) if w > 1.
b
_{0}
b
_{1}
b
_{2}
using the expression of conic by the quadratic rational Bézier curve having the polygon
b
_{0}
b
_{1}
b
_{2}
. The isotomic and isogonal conjugates relative to the control polygon have a particularly nice form.
Theorem 3.1.
With respect to
Δ
b
_{0}
b
_{1}
b
_{2}
the isotomic and isogonal conjugates of the quadratic rational Bézier curve having the control polygon
b
_{0}
b
_{1}
b
_{2}
are also quadratic rational Bézier curves with control polygon
b
_{2}
b
_{1}
b
_{0}
.
Proof.
The quadratic rational Bézier curve
r
(
t
) having the control polygon
b
_{0}
b
_{1}
b
_{2}
with weight
w
is
where
which are the barycentric coordinates of
r
(
t
). Let
r
°(
t
) and
r
*(
t
) be the isotomic and isogonal conjugates of the quadratic rational Bézier curve
r
(
t
). We have
where
Thus
r
°(
t
) is the quadratic rational Bézier curve having control points
b
_{2}
,
b
_{1}
,
b
_{0}
with weight
.
Also we have
where
Thus this quadratic rational Bézier curve has control points
b
_{2}
,
b
_{1}
,
b
_{0}
with weight
in order, which can be expressed in standard form with the weight
.
Remark 3.2.
The isotomic conjugate of the quadratic rational Bézier curve can be written in standard form
Thus the isotomic conjugate transformation maps the weight
w
into
, control polygon
b
_{0}
b
_{1}
b
_{2}
into
b
_{2}
b
_{1}
b
_{0}
. The quadratic rational Bézier curve with weight
is fixed uniquely under the transformation. This transformation maps hyperbola and parabola into only ellipse. Also it maps ellipse with weight
w
less than, equal to, and greater than
into hyperbola, parabola, and ellipse, respectively, as shown in
Figure 5
and
Table 1
.
Isotomic conjugate transformation maps parabola (magenta) into an ellipse (green, ). The ellipse with weight (blue) is fixed uniquely under this transformation.
Isotomic conjugate r °(t ) of quadratic rational Bézier curve r (t ).
Remark 3.3.
The isogonal conjugate transformation maps the weight
w
into
, control polygon
b
_{0}
b
_{1}
b
_{2}
into
b
_{2}
b
_{1}
b
_{0}
. The quadratic rational Bézier curve with weight
is uniquely fixed under the transformation. Thus there are three cases:
b
^{2}
-4
ac
is negative, zero or positive. It looks like the discriminant of quadratic equation.
If
b
^{2}
−4
ac
< 0, the isogonal conjugate transformation maps hyperbola and parabola into ellipse. The ellipse with weight
and
are mapped into hyperbola, parabola and ellipse, respectively, as shown in
Figure 6
and
Table 2
.
When b ^{2}−4ac < 0, the isogonal conjugate transformation maps parabola(magenta) into an ellipse(green, ), and the ellipse with weight (blue) is uniquely fixed under this transformation.
Isogonal conjugate for b ^{2}−4ac < 0.
If
b
^{2}
−4
ac
= 0, the isogonal conjugate transformation maps ellipse into hyperbola and vice versa, and parabola is fixed uniquely, as shown in
Figure 7
and
Table 3
.
For b ^{2}−4ac = 0, the parabola(magenta) is uniquely fixed under the isogonal conjugate transformation.
Isogonal conjugate for b ^{2}−4ac = 0.
If
b
^{2}
−4
ac
> 0, the isogonal conjugate transformation maps ellipse and parabola into hyperbola. The hyperbola with weight
and
are mapped into ellipse, parabola and hyperbola, respectively, as shown in
Figure 8
and
Table 4
.
When b ^{2}−4ac > 0, the isogonal conjugate transformation maps parabola(magenta) into a hyperbola(green, ), and the hyperbola with weight (blue) is uniquely fixed under this transformation.
Isogonal conjugate for b ^{2}−4ac > 0.
Our results in this section are the improvement and reconstruction of thesis
[10]
of the first author in this paper.
b
_{0}
b
_{1}
b
_{2}
with respect to Δ
b
_{0}
b
_{1}
b
_{2}
. We showed that the isotomic conjugate transformation maps parabola, hyperbola and ellipse with the weight
w
> 1/4 into ellipse, maps ellipse with
into parabola, and maps ellipse with
into hyperbola. The isogonal conjugate transformation of conic is more complicated. We also identified all cases of the isogonal conjugates of quadratic rational Bézier curves which are ellipse, parabola or hyperbola. We could derive our characterizations very easily due to the expression of conic by the quadratic rational Bézier curve in standard form.
There are many natural and interesting problems along the modern geometry with the isogonal and isotomic conjugate transformations. In future work, we plan to exploit these problems concerned with the rational Bézier curves and surfaces of higher degree than two.

isogonal conjugate
;
isotomic conjugate
;
conic
;
quadratic rational Bézier curve
;
ellipse
;
parabola
;
hyperbola

1. INTRODUCTION

Quadratic rational Bézier curve is one of the most widely used curves in the field of CAD/CAM and Computer Graphics. Any conic can be expressed by a quadratic rational Bézier curve, and vice versa if the control points of quadratic rational Bézier curve are not collinear. The quadratic rational Bézier curve having control polygon
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2. DEFINITIONS

In a given triangle
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3. ISOTOMIC AND ISOGONAL CONJUGATE OF QUADRATIC RATIONAL BÉZIER CURVES.

In this section we characterize the isotomic and isogonal conjugates of conic with respect to Δ
- r(t) =τ0b0+τ1b1+τ2b2

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Isotomic conjugater°(t) of quadratic rational Bézier curver(t).

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Isogonal conjugate forb2−4ac< 0.

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Isogonal conjugate forb2−4ac= 0.

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Isogonal conjugate forb2−4ac> 0.

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4. CONCLUSION

In this paper we characterized all cases of the isogonal and isotomic conjugates of quadratic rational Bézier curve having control polygon
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Acknowledgements

This study was supported by research funds from Chosun University, 2014. The authors are grateful to the editor and the anonymous referees for their valuable comments and constructive suggestions.

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Citing 'ISOGONAL AND ISOTOMIC CONJUGATES OF QUADRATIC RATIONAL Bézier CURVES
'

@article{ SHGHCX_2015_v22n1_25}
,title={ISOGONAL AND ISOTOMIC CONJUGATES OF QUADRATIC RATIONAL Bézier CURVES}
,volume={1}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.1.25}, DOI={10.7468/jksmeb.2015.22.1.25}
, number= {1}
, journal={The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={YOU, CHAN RAN
and
AHN, YOUNG JOON}
, year={2015}
, month={Feb}