A least upper bound for the number of singular points on normal arcs and curves of cyclic order four

Differentiable curves of cyclic order four

“…The order of a point p is the minimum of the orders of all neighborhoods of p on j^I A point of (minimal) order three is called an ordinary point, a point of order greater than three a singular point, and a point of support of s/ with respect to C(p 3 ) a vertex. With regard to an arc or curve of cyclic order four: (i ) It contains at most four singular points [9].…”

“…Using the notion of the characteristic, this paper classifies all possible simple differentiable curves of cyclic order four in the conformal plane in regard to the number and type of singular points. It is well known that such curves contain at most four singular points [9].…”

Normal arcs and curves of K-order k+1

“…(B) Let E be the set of all circles in the (conformal) plane with k = 3. Then a normal arc or curve of K-order 4 contains at most four K-singular points [3].…”

“…Proof The following slight modification of the proof of 4.2 is used (cf. 3 • 6 of[3] for the conformal case).Let L t3 {z2} t_J M be a small two-sided neighbourhood of z2 on Ak+l. Since Z 2 is /3~ or 72 k then K~-(z2) does not meet [a, z2).…”

A characterization of curves of cyclic order four

“…The only supporting general osculating circle ofd at z is the point circle z[5].2.3. For (3, 1)-singular points that are not (1,3)-singular clearly C-(z) supports d at z while C ÷ (z) intersects.…”