Evolutes of fronts in the Euclidean plane

2015

Beitr Algebra Geom

The notions of involutes (also known as evolvents) and evolutes were studied by C. Huygens. For a regular plane curve, an involute of it is the trajectory described by the end of stretched string unwinding from a point of the curve. Even if a regular curve, the involute of the curve have singularities. By using a moving frame of the front and the curvature of the Legendre immersion in the unit tangent bundle, we define an involute of the front in the Euclidean plane and discuss properties of them. We also consider about relationship between evolutes and involutes of fronts without inflection points. As a result, we observe that the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral in classical calculus.

“…Here, we recall an alternative definition of the evolute of the front as follows, see Theorem 3.3 in [9].…”

Section: Example 29 Let N M and K Be Natural Numbers Withmentioning

confidence: 99%

Section: Example 22mentioning

confidence: 99%

Section: Example 29 Let N M and K Be Natural Numbers Withmentioning

confidence: 99%

“…The properties of evolutes discussed by using distance squared functions and the theories of Lagrangian, Legendrian singularity (cf. [2,3,4,9,17,18,20,22]). …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Involutes of fronts in the Euclidean plane

Fukunaga

2015

Beitr Algebra Geom

Dualities and envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐spaces

Chen

Pei

Mathematische Nachrichten

2020

Self Cite

We consider envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one-parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one-parameter families of frontals in hyperbolic and de Sitter 2-space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes. K E Y W O R D Sde Sitter 2-space, envelope, frontal, hyperbolic 2-space, Legendrian duality M S C ( 2 0 1 0 ) 53A35, 53B30, 58K05

One‐parameter families of Legendre curves and plane line congruences

Kabata

Mathematische Nachrichten

2022

Families of curves in the Euclidean plane naturally contain singular curves, where the frame of classical differential geometry does not work well. We introduce the notions of one-parameter family of Legendre curves in the Euclidean plane, congruent equivalence and curvature. Especially, a one-parameter family of Legendre curves can contain singular curves, and is determined by the curvature up to congruence. We also give properties of one-parameter families of Legendre curves. As applications, we give a relation between one-parameter families of Legendre curves and Legendre surfaces. Moreover, we study plane line congruences (one-parameter families of lines in plane) in terms of the curvatures as one-parameter families of Legendre curves.