ISOGONAL AND ISOTOMIC CONJUGATES OF QUADRATIC RATIONAL Bézier CURVES

Abstract

In this paper we characterize the isogonal and isotomic conjugates of conic. Every conic can be expressed by a quadratic rational B$\acute{e}$zier curve having control polygon $b_0b_1b_2$ with weight w > 0. We show that the isotomic conjugate of parabola and hyperbola with respect to ${\Delta}b_0b_1b_2$ is ellipse, and that the isotomic conjugate of ellipse with the weight $w={\frac{1}{2}}$ is identical. We also find all cases of the isogonal conjugate of conic with respect to ${\Delta}b_0b_1b_2$. Our characterizations are derived easily due to the expression of conic by the quadratic rational B$\acute{e}$ezier curve in standard form.

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 b0b1b2 is tangent to the lines b0b1 and b1b2 at the points b0 and b2, 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 b0b1b2, and for the point obtained by reflecting the intersection point of lines xbi and bjbk with respect to the midpoint on the side bjbk, 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.

Figure 1.The isogonal conjugate of x is x* with respect to the triangle Δb0b1b2.

Figure 2.The isogonal conjugate of x is x° with respect to the triangle Δb0b1b2.

We consider the conjugate points of all points on the quadratic rational Bézier curve having control polygon b0b1b2 with respect to the triangle b0b1b2. The motivation of our study is the question: what are the isogonal and isotomic conjugates of the quadratic rational Bézier curve with respect to Δb0b1b2, 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 Δb0b1b2. 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.

 

2. DEFINITIONS

In a given triangle b0b1b2, every point x can be uniquely expressed by x = ub0 + vb1 + wb2, 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 Δxb1b2 : Δxb2b0 : Δxb0b1 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.

Figure 3.Barycentric coordinates with respect to Δb0b1b2.

2.1. Isogonal conjugate and isotomic conjugate For any point x inside the triangle b0b1b2, the lines xbi, i = 0, 1, 2, are called by cevians of x [9]. Let the line ℓi, i = 0, 1, 2, be the reflection of the cevian xbi with respect to the angle bisector at vertex bi, respectively. The three lines ℓ0, ℓ1 and ℓ2 are concurrent at one point, which is called the isogonal conjugate of x with respect to the triangle b0b1b2 and it denoted by x*, as shown in Figure 1. For each i = 0, 1, 2, two lines xbi and x*bi are symmetric with respect to the bisector of ∠bi. 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 xi, i = 0, 1, 2, be the intersection point of the cevian xbi and the opposite side bjbk, where i, j, k ∈ {0, 1, 2} are mutually distinct. Let be the reflection point of xi with respect to the midpoint of the line segment bjbk. The three lines , and are concurrent at one point, which is called the isotomic conjugate of x with respect to the triangle b0b1b2 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 bi, i = 0, 1, 2 and weights wi > 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 b0b1b2 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.

Figure 4.Quadratic rational Bézier curve having control polygon b0b1b2 with weight w > 0 is ellipse(green) if w < 1, parabola(magenta) if w = 1, and hyperbola(blue) if w > 1.

 

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 Δb0b1b2 using the expression of conic by the quadratic rational Bézier curve having the polygon b0b1b2. The isotomic and isogonal conjugates relative to the control polygon have a particularly nice form.

Theorem 3.1. With respect to Δb0b1b2 the isotomic and isogonal conjugates of the quadratic rational Bézier curve having the control polygon b0b1b2 are also quadratic rational Bézier curves with control polygon b2b1b0.

Proof. The quadratic rational Bézier curve r(t) having the control polygon b0b1b2 with weight w is

r(t) = τ0b0 + τ1b1 + τ2b2

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 b2,b1,b0 with weight .

Also we have

where

Thus this quadratic rational Bézier curve has control points b2,b1,b0 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 b0b1b2 into b2b1b0. 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.

Figure 5.Isotomic conjugate transformation maps parabola (magenta) into an ellipse (green, ). The ellipse with weight (blue) is fixed uniquely under this transformation.

Table 1.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 b0b1b2 into b2b1b0. The quadratic rational Bézier curve with weight is uniquely fixed under the transformation. Thus there are three cases: b2-4ac is negative, zero or positive. It looks like the discriminant of quadratic equation.

If b2−4ac < 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.

Figure 6.When b2−4ac < 0, the isogonal conjugate transformation maps parabola(magenta) into an ellipse(green, ), and the ellipse with weight (blue) is uniquely fixed under this transformation.

Table 2.Isogonal conjugate for b2−4ac < 0.

If b2−4ac = 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.

Figure 7.For b2−4ac = 0, the parabola(magenta) is uniquely fixed under the isogonal conjugate transformation.

Table 3.Isogonal conjugate for b2−4ac = 0.

If b2−4ac > 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.

Figure 8.When b2−4ac > 0, the isogonal conjugate transformation maps parabola(magenta) into a hyperbola(green, ), and the hyperbola with weight (blue) is uniquely fixed under this transformation.

Table 4.Isogonal conjugate for b2−4ac > 0.

Our results in this section are the improvement and reconstruction of thesis[10] of the first author in this paper.

 

4. CONCLUSION

In this paper we characterized all cases of the isogonal and isotomic conjugates of quadratic rational Bézier curve having control polygon b0b1b2 with respect to Δb0b1b2. 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.