5.8.1.3 Characteristic points

Starting points for tracing algebraic curves are identified by looking for characteristic points defined below:

1. Border points: The intersections of with all four boundary edges of the parameter space , e.g. .
2. Turning points: The -turning points are the points where the tangent of is parallel to the axis, which satisfies the simultaneous equations (with ). On the other hand the -turning points are the points where the tangent of is parallel to the axis, which satisfies the simultaneous equations (with ). Both types of turning points are shown in Fig. 5.18. If has a degree of in , then the degrees of and will be and , respectively. It can be shown that the total number of roots of two simultaneous polynomial equations in two variables whose degrees are and , respectively, is [95]. Therefore the number of -turning points and -turning points can be at most and , respectively over the entire complex plane. However within the square parameter space , the number of turning points is typically much reduced in practice and therefore methods that use the square as the search space of the roots such as the IPP algorithm in Sect. 4.9 or interval Newton methods [159] would typically outperform other methods.

3. Singular points: The points on the curve which satisfy the following three simultaneous equations are called singular points. Noting that , and , we deduce
   
   
   Similarly we obtain , and hence at singular points . This means that or that the normals of two surfaces are parallel and since at these points the two surfaces intersect tangentially. If has a degree of in , the degrees of and will be and , respectively, thus the number of singular points can be at most [95] over the entire complex plane and typically much less in number in . If the systems and are already solved, a small extra evaluation can identify their common roots which are the singular points. Alternatively the IPP algorithm for the overconstrained system can be used to find the singular points.

From the above discussions we can get upper bounds for the maximum number of -turning, -turning and singular points as listed in Table 5.4. These bounds refer to the maximum possible number of solutions in the entire complex plane. Biquadratic and bicubic surfaces in the first column of Table 5.4 are degree 8 and 18 implicit algebraic surfaces. It turns out that the number of such points in the real square is much smaller, but can still be quite large. Consequently, methods which focus only on the real solutions in are advantageous, such as IPP algorithm described in Chap. 4 or interval Newton's method [159].

Table 5.4: Maximum number of turning and singular points in various cases

		algebraic	max	max	max
		curve	number	number	number
			-turning	-turning	singular
		degree	pts	pts	points
plane	biquadratic	2, 2	6	6	5
plane	bicubic	3, 3	15	15	13
quadric	biquadratic	4, 4	28	28	25
quadric	bicubic	6, 6	66	66	61
torus	biquadratic	8, 8	120	120	113
torus	bicubic	12, 12	276	276	265
biquadratic	biquadratic	16, 16	496	496	481
bicubic	biquadratic	36, 36	2556	2556	2521
bicubic	bicubic	54, 54	5778	5778	5725