Skewers

Tabachnikov, Serge

doi:10.1007/s40598-016-0037-7

Cited by 2 publications

(5 citation statements)

References 33 publications

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“…Let Q, P ∈ E and let q, p be the associated unit sliding vectors, then we have the useful formulas x ´ (y ´ z) Proof. As already observed by Skopenkov [28] and Tabachnikov [30], this is really a consequence of the Petersen-Morley theorem, that is, of the Jacobi identity. Indeed, y × z has screw axis perpendicular to the plane of the triangle passing through point A, and hence x × (y × z) has axis on the plane perpendicular to the side a.…”

Section: Euclidean Geometry Via D-module Geometrysupporting

confidence: 59%

The theory of screws derived from a module over the dual numbers

Minguzzi¹

2022

Preprint

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It is known that the real 6-dimensional space of screws can be endowed with an operator E, E 2 = 0, that converts it into a rank 3 module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 module over the dual numbers endowed with orientation and a suitable scalar product (D-module geometry), we show that it is possible to define, in a natural way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the same effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view. As a case study, it is shown that D-module geometry is effective in deriving results of Euclidean geometry. For instance, in a pretty much algorithmic way we prove classical results in triangle geometry, among which, Ceva's theorem, existence of the Euler line, and Napoleon's theorem.

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Section: Euclidean Geometry Via D-module Geometrysupporting

confidence: 59%

The theory of screws derived from a module over the dual numbers

Minguzzi¹

2022

Preprint

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show abstract

“…The second result is a different skewer version of the Pappus theorem. We proved this result, in the hyperbolic case, by a brute force calculation using the approach to hyperbolic geometry, developed in [14]; see [47] for details. It is not clear whether this theorem is a part of a general pattern.…”

Section: Skewersmentioning

confidence: 71%

“…This section is based upon the recent paper [47]. The main idea is that planar projective configuration theorems have space analogs where points and lines in the projective plane are replaced by lines in space, and the two operations, connecting two points by a line and intersecting two lines at a point, are replaced by taking the common perpendicular of two lines.…”

Section: Skewersmentioning

confidence: 99%

“…And so on... Next one would like to define line analogs of conics. A first step in this direction is made in [47], but much more work is needed. In particular, one would like to have skewer analogs of various configuration theorems involving conics, including the Pascal theorem and the whole hexagrammum mysticum, the Poncelet Porism, and the theorems described in Section 4.…”

Section: Identities In the Lie Algebras Of Motionsmentioning

confidence: 99%

“…This section is based upon the recent paper[47]. The main idea is that planar projective configuration theorems have space analogs where points…”

mentioning

confidence: 99%

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