Implicitization using Moving Lines

where the equations a0x+ b0y + c0 = 0 and a1x+ b1y + c1 = 0 define any two distinct lines. It is known that any conic section can be “generated by the intersection of corresponding lines of two related pencils in a plane” [Som51], p.388. In other words, given two distinct pencils, (a00x+b00y+c00)(1−t)+(a10x+b10y+c10)t = 0 and (a01x+b01y+c01)(1−t)+(a11x+b11y+c11)t = 0, to each value of t corresponds exactly one line from each pencil, and those two lines intersect in a point. The locus of points thus created for −∞ ≤ t ≤ ∞ is a conic section, as illustrated in Figure 18.1. This is reviewed in section 18.2. This chapter examines the extension of that idea to higher degrees. A degree n family of lines intersects a degree m family of lines in a curve of degree m+ n, which is discussed in section 18.3. Section 18.4 shows that any rational curve can be described as the intersection of two families of lines, from which the multiple points and the implicit equation of the curve can be easily obtained. For example, any cubic rational curve can be described as the intersection of a pencil of lines and a quadratic family of lines. The pencil axis lies at the double point of the cubic curve. Section 18.5 discusses the family of lines which is tangent to a given rational curve. Such families of lines are useful for analyzing the singularities of the curve, such as double points, cusps, and inflection points, and also for calculating derivative directions.