Distance product cubics
The locus of points that determine a constant product of their distances to the sides of a triangle is a cubic curve in the projectively closed Euclidean triangle plane. In this paper, algebraic and geometric properties of these distance product cubics shall be studied. These cubics span a pencil of...
| Permalink: | http://skupni.nsk.hr/Record/nsk.NSK01001128692/Details |
|---|---|
| Matična publikacija: |
Kog (Online) (2020), 24 ; str. 29-40 |
| Glavni autor: | Odehnal, Boris (Author) |
| Vrsta građe: | e-članak |
| Jezik: | eng |
| Predmet: | |
| Online pristup: |
https://doi.org/10.31896/k.24.3 Hrčak |
| LEADER | 02970naa a22003494i 4500 | ||
|---|---|---|---|
| 001 | NSK01001128692 | ||
| 003 | HR-ZaNSK | ||
| 005 | 20220224153833.0 | ||
| 006 | m d | ||
| 007 | cr|||||||||||| | ||
| 008 | 220221s2020 ci a |o |0|| ||eng | ||
| 024 | 7 | |2 doi |a 10.31896/k.24.3 | |
| 035 | |a (HR-ZaNSK)001128692 | ||
| 040 | |a HR-ZaNSK |b hrv |c HR-ZaNSK |e ppiak | ||
| 041 | 0 | |a eng |b hrv | |
| 042 | |a croatica | ||
| 044 | |a ci |c hr | ||
| 080 | 1 | |a 51 |2 2011 | |
| 100 | 1 | |a Odehnal, Boris |4 aut | |
| 245 | 1 | 0 | |a Distance product cubics |h [Elektronička građa] / |c Boris Odehnal. |
| 300 | |b Ilustr. | ||
| 504 | |a Bibliografija: 10 jed. | ||
| 504 | |a Abstract ; Sažetak. | ||
| 520 | |a The locus of points that determine a constant product of their distances to the sides of a triangle is a cubic curve in the projectively closed Euclidean triangle plane. In this paper, algebraic and geometric properties of these distance product cubics shall be studied. These cubics span a pencil of cubics that contains only one rational and non-degenerate cubic curve which is known as the Bataille acnodal cubic determined by the product of the actual trilinear coordinates of the centroid of the base triangle. Each triangle center defines a distance product cubic. It turns out that only a small number of triangle centers share their distance product cubic with other centers. All distance product cubics share the real points of inflection which lie on the line at infinity. The cubics' dual curves, their Hessians, and especially those distance product cubics that are defined by particular triangle centers shall be studied. | ||
| 520 | |a U projektivno zatvorenoj Euklidskoj ravnini trokuta geometrijsko mjesto točaka trokuta kojima je umnožak udaljenosti od stranica trokuta konstantan je jedna kubika. Proučavat će se algebarska i geometrijska svojstva tih kubika konstantnog umnoška udaljenosti. Takve kubike čine pramen kubika koje sadrže samo jednu racionalnu nedegeneriranu kubiku poznatu kao Batailleova kubika s izoliranom točkom, a koja je određena umnoškom pravih trilinearnih koordinata težišta temeljnog trokuta. Svaka točka trokuta određuje jednu kubiku konstantnog umnoška udaljenosti. Ispostavlja se da mali broj točaka trokuta međusobno dijele kubiku konstantnog umnoška udaljenosti. Sve kubike konstantnog umnoška udaljenosti dijele realne točke infleksije koje leže na pravcu u beskonačnosti. Proučavat će se dualne krivulje kubike, njihove Hessianove matrice i posebno one kubike konstantnog umnoška udaljenosti koje su odre\5ene poznatim točkama trokuta. | ||
| 653 | 0 | |a Trokut |a Točke trokuta |a Kubika |a Trilinearna udaljenost |a Konstantni umnožak |a Steinerova upisana elipsa | |
| 773 | 0 | |t Kog (Online) |x 1846-4068 |g (2020), 24 ; str. 29-40 |w nsk.(HR-ZaNSK)000628952 | |
| 981 | |b Be2020 |b B04/20 | ||
| 998 | |b tino2202 | ||
| 856 | 4 | 0 | |u https://doi.org/10.31896/k.24.3 |
| 856 | 4 | 0 | |u https://hrcak.srce.hr/248412 |y Hrčak |
| 856 | 4 | 1 | |y Digitalna.nsk.hr |