On the Construction of Curves of Constant Width

Abstract

1. DefimtIOn and general propertIes In a paper by E. Barbier,l a CUlve of constant wIdth IS defined as a closed CUlve wIth a constant dIstance between parallel tangents A method of constructIOn, accredIted by BarbIer to Puiseux, was glVen which ,YIll produce examples of such curves other than circles. By takmg half of an ellIpse, where the hne of dIvIsIOn IS the maJor aXIS, and markmg off on each normal a dIstance equal to the major aXIS, such a curve can sometImes be obtamedThe followmg propertIes of a curve of constant wIdth were stated Property 1 The curve is composed of two arcs with a common evolute Property 2 The common evolute of the two arcs must be a curve wIth one and only one tangent m each dIrectIOn 2 Property 3 The envelope of the famIly of hnes each of which hes mIdway between parallel tangents to the curve wIll be a curve wIth one and only one tangent m each direction. PlOperty 4 All curves of constant wIdth wIth the same dIstance between parallel tangents wIll have the same length The reasonmg used by BarbIer m the proof of Property 4 was based upon probabIlIty consideratIOns Another derIvation can be based upon the idea of average radIUs of curvature, that is, the ratio of arc length to dIfference of mclination between the end pomts of an arc Suppose P and pI are opposite points of a curve of constant wIdth. That is, P and pI lIe on the same normal. Let the line ppl be of length 2a, the wIdth of the curve. If R IS the radius of curvature at P and R' is the radIUs of curvature at pI,for there is a common point on the evolute for opposite points of the curve. Pairmg opposite points, the average radIUs of curvature for the complete closed curve IS seen to be a. The total difference of inclinatIOn is 211". The length of a curve of constant wIdth 2a is therefore 211"a.The problem conSIdered in this paper is to find m general the type of configuration whIch may be used to construct a curve of constant width. Exact conditions upon the parts of such a configuratIOn are presented and a general method of construction is given.2. Evolute star. An evolute star IS a configuration of n pomts or vertices { V, I and n curvilmear sides {&,

Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$

“…Multiplying ( 16) with N and B 2 , we get λ ′ = 0, µ ′ = 0, respectively, i.e., λ, µ are real non-zero constants. So, we have λ 2 + µ 2 = 1, which is the second equality in (6). Moreover, from ( 13) and ( 14), we have…”

“…It follows that p and r are non-zero constants. Then, from (10), we get p 2 + r 2 = 1 and we have first equality in (6). Now, (11…”

“…In the same study, equation of enveloping curve of the family of normal planes for space curve has been also defined. Gere and Zupnik studied involute-evolute curves by considering a curve composed of two arcs with common evolute [6]. Fukunaga and Takahashi defined evolutes and involutes of fronts in the plane and introduced some properties of these curves [4,5].…”

On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space

“…For any s ∈ I, if the (0,2)-tangent plane of α at α(s) coincides with the (1,3)-normal plane at α * (s) of α * , then α * is called the (0,2)-involute curve of α in E 4 1 and α is called the (1,3)-evolute curve of α * in E 4 1 . An arbitrary curve, α(s) in E 4 1 , can locally be space-like, time-like, or null (light-like) if all of its velocity vectors, α (s), are respectively space-like, time-like, or null [18]. A null curve, α, is parametrized by the pseudo-arc s if g(α (s), α (s)) = 1 [19].…”

“…Suleyman and Seyda [3] determined the concept of parallel curves, which means that if the evolute exists, then the evolute of the parallel arc will also exist and the involute will coincide with the evolute. Brewster and David [4] stated that a curve is composed of two arcs with a common evolute, and the common evolute of two arcs must be a curve with only one tangent in each direction. In general, the evolute of a regular curve has singularities, and these points correspond to vertices.…”