Geometry

Frenet's Trihedron of the Second Order

Authors: Abel Cavași
Comments: 9 Pages.

Based on the remarkable property of the Darboux vector to be perpendicular to the normal, I define a new trihedron associated with curves in space and prove that this trihedron also satisfies Frenet's formulas. Unlike the previous paper, where I used the trigonometric form of Frenet's formulas for simplicity, in this paper I construct a proof based only on curvature and torsion, respectively, darbuzian and lancretian.
Category: Geometry

Triedrul Lui Frenet de Ordinul al Doilea (Frenet's Trihedron of the Second Order)

Authors: Abel Cavași
Comments: 9 Pages. In Romanian

Bazându-mă pe proprietatea remarcabilă a vectorului lui Darboux de a fi perpendicular pe normală, definesc un nou triedru asociat curbelor din spațiu și demonstrez că și acest triedru satisface formulele lui Frenet. Spre deosebire de lucrarea anterioară, unde am folosit pentru simplitate forma trigonometrică a formulelor lui Frenet, în această lucrare construiesc o demonstrație bazată doar pe curbură și torsiune, respectiv, pe darbuzian și lancretian.

Based on the remarkable property of the Darboux vector to be perpendicular to the normal, I define a new trihedron associated with curves in space and prove that this trihedron also satisfies Frenet's formulas. Unlike the previous paper, where I used the trigonometric form of Frenet's formulas for simplicity, in this paper I construct a proof based only on curvature and torsion, respectively, darbuzian and lancretian.
Category: Geometry

Quick Tiling

Authors: Volker W. Thürey
Comments: 4 Pages.

In the first part, we tile the plane with k-gons for natural numbers k which have the rest three if we devide it by four. The proof is by pictures. In a second part, we extend the result to all natural numbers larger than two. The foundation is the tiling of the plane by rectangles or hexagons. We use at most two different tiles for the covering.
Category: Geometry