A Triple of Projective Billiards

Projektivni biljar je poligon u realnoj projektivnoj ravnini koji ima upisanu i opisanu koniku. Poput klasičnih biljara u konikama, sjecišta produljenih stranica projektivnog biljara se nalaze na familiji konika koje tvore pridruženu Ponceletovu mrežu. Proširujemo projektivni biljar unutarnjim i vanjskim biljarom i otkrivamo mnoštvo veza između pridruženih mreža i dijagonala, posebice drugih trojki projektivnih biljara.A projective billiard is a polygon in the real projective plane with a circumconic and an inconic. Similar to the classical billiards in conics, the intersection points between the extended sides of a projective billiard are located on a family of conics which form the associated Poncelet grid. We extend the projective billiard by the inner and outer billiard and disclose various relations between the associated grids and the diagonals, in particular other triples of projective billiards

O tetraedrima u dodekaedru

The 60 edges of the ten tetrahedra inscribed in a regular pentagon-dodecahedron form the so-called Gr"unbaum framework. It is already known that this structure is flexible. There are one-parameter motions which preserve the symmetry with respect either to a face axis or to a vertex axis. The paper treats analytical representations of these motions. Furthermore it is proved that both motions can blend into two-parametric motions which do not preserve any symmetry.60 bridova od 10 tetraedara upisanih u pravilan dodekaedar čine tzv. Grünbaumovu mrežu. Poznato je da je ta struktura fleksibilna. Postoje jednoparametarska gibanja koja čuvaju simetriju s obzirom na os stranice ili na os koja prolazi vrhom. Rad se bavi analitičkim reprezentacijama takvih gibanja. Osim toga dokazano je da se oba gibanja mogu spojiti u dvoparametarska gibanja koja ne čuvaju simetriju

Reflection in quadratic surfaces

“The transparent cup” is the title of pictures which show an interesting phenomenon: The circular boundary c of the depicted plate appears as an ellipse which seems to coincide with the view of the reflection of c in the coffee-cup. Is this just by chance or is there a geometric theory behind? In one example the circle c is the focal circle of the reflecting one-sheet hyperboloid, and for this particular case the displayed phenomen is a con- sequence of focal properties of quadratic surfaces. The tangent cones drawn from a fixed point P to a family of confocal quadrics are confocal and have therefore coinciding axes. These axes are the surface normals to the particular quadrics passing through P. Also the cones connecting P with the focal conics are included in the considered set of confocal cones. Therefore, all focal conics share the property: In each perspective, the images of these curves and their reflections belong to the same conic. The goal of the paper is to highlight the geometric background, i.e., to focus on confocal conics and their spatial counterparts

C2 popunjavanje praznina pomoću konveksne kombinacije ploha pod rubnim ograničenjima

Two surface generation methods are presented, one for connecting two surfaces with C2 continuity while matching also two prescribed border lines on the free sides of the gap, and one for G1 filling a three-sided hole in a special case. The surfaces are generated as convex combination of surface and curve constituents with an appropriate correction function, and are represented in parametric form.Dane su dvije metode za izvođenje ploha. Jedna za povezivanje dviju ploha sa C2 neprekinutošću koja odgovara i dvjema graničnim linijama, a druga za G1 popunjavanje posebnog slučaja trostrane rupe. Plohe se izvode kao konveksna kombinacija plošnih i krivuljnih sastavnih dijelova sa odgovarajućom korektivnom funkcijom, a dane su u parametarskom obliku

C2 popunjavanje praznina pomoću konveksne kombinacije ploha pod rubnim ograničenjima

Two surface generation methods are presented, one for connecting two surfaces with C2 continuity while matching also two prescribed border lines on the free sides of the gap, and one for G1 filling a three-sided hole in a special case. The surfaces are generated as convex combination of surface and curve constituents with an appropriate correction function, and are represented in parametric form.Dane su dvije metode za izvođenje ploha. Jedna za povezivanje dviju ploha sa C2 neprekinutošću koja odgovara i dvjema graničnim linijama, a druga za G1 popunjavanje posebnog slučaja trostrane rupe. Plohe se izvode kao konveksna kombinacija plošnih i krivuljnih sastavnih dijelova sa odgovarajućom korektivnom funkcijom, a dane su u parametarskom obliku

Steinerova krivulja: deltoide konstantne površine pridružene elipsi

The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.Negativno nožišna krivulja elipse s obzirom na neku njezinu točku M je zatvorena krivulja s tri šiljka koja je afina slika Steinerove deltoide. Za sve točke M na elipsi krivulje dobivene familije imaju istu površinu i niz zanimljivih svojstava

The man who invented descriptive geometry

Gaspar Monž je poznat kao otac moderne nacrtne i diferencijalne geometrije . Godine 1764. angažovan je da izradi detaljan nacrt utvrđenja u svom rodnom gradu i njegov rad je primetio jedan oficir iz vojne škole École Royale du Génie de Mézières. Budući da je nacrt bio jako dobar, metode koje je Monž koristio čuvane su kao vojna tajna dugi niz godina. Godine 1780. Monž je postao član Akademije nauka i učestvovao je u radu Komiteta za tegove i mere, koji je imperijalni merni sistem prevodio u metrički. Gaspar je pomogao u osnivanju škole École Centrale des Travaux Publics (kasnije École Polytechnique) gde je i predavao nacrtnu geometriju. Godine 1798. Napoleon je krenuo u pohod na Egipat i zamolio je čuvenog hemičara Kloda Bertolea da regrutuje istaknute naučnike koji bi mu se pridružili u pohodu. Među njima su bili Furije, Monž, Dolomju i Malu. Napoleon je osnovao Egipatski institut i Monž je bio njegov prvi direktor. Gaspar Monž je preminuo 28. jula 1818. godine u Parizu. Njegovo ime je urezano u temelj Ajfelovog tornja i to na mestu tačno preko puta Vojne akademije. Pored nacrtne geometrije, Monž se bavio hemijom i fizikom.Gaspard Monge is known as the father of modern descriptive and differential geometry. In 1764, he was engaged to draw a detailed plan of a fortification in his hometown, which was seen by an officer at the École Royale du Génie de Mézières. This plan was a success and his techniques were marked as a military secret for a long period of time. In 1780, he was elected to the Academy of Science and participated in the work of the Commission for Weights and Measures, that was in charge of moving the system from imperial to metric. In 1794, Monge helped setting up the École Centrale des Travaux Publics (later École Polytechnique) where he was lecturing Descriptive Geometry. In 1798, Napoleon undertook a campaign in Egypt. The famous chemist Claude Louis Berthollet was asked to recruit prominent scientists. Among them were Fourier, Monge, Dolomieu and Malus. Institut d'Egypte was established by Napoleon and Monge was named as its first president. Monge passed away on July 28, 1818. His name is inscribed on the base of the Eiffel Tower and it is located on the third façade opposite the Military Academy. Besides descriptive geometry, he carried on many different researches in chemistry and physics

The man who invented descriptive geometry

Gaspar Monž je poznat kao otac moderne nacrtne i diferencijalne geometrije . Godine 1764. angažovan je da izradi detaljan nacrt utvrđenja u svom rodnom gradu i njegov rad je primetio jedan oficir iz vojne škole École Royale du Génie de Mézières. Budući da je nacrt bio jako dobar, metode koje je Monž koristio čuvane su kao vojna tajna dugi niz godina. Godine 1780. Monž je postao član Akademije nauka i učestvovao je u radu Komiteta za tegove i mere, koji je imperijalni merni sistem prevodio u metrički. Gaspar je pomogao u osnivanju škole École Centrale des Travaux Publics (kasnije École Polytechnique) gde je i predavao nacrtnu geometriju. Godine 1798. Napoleon je krenuo u pohod na Egipat i zamolio je čuvenog hemičara Kloda Bertolea da regrutuje istaknute naučnike koji bi mu se pridružili u pohodu. Među njima su bili Furije, Monž, Dolomju i Malu. Napoleon je osnovao Egipatski institut i Monž je bio njegov prvi direktor. Gaspar Monž je preminuo 28. jula 1818. godine u Parizu. Njegovo ime je urezano u temelj Ajfelovog tornja i to na mestu tačno preko puta Vojne akademije. Pored nacrtne geometrije, Monž se bavio hemijom i fizikom.Gaspard Monge is known as the father of modern descriptive and differential geometry. In 1764, he was engaged to draw a detailed plan of a fortification in his hometown, which was seen by an officer at the École Royale du Génie de Mézières. This plan was a success and his techniques were marked as a military secret for a long period of time. In 1780, he was elected to the Academy of Science and participated in the work of the Commission for Weights and Measures, that was in charge of moving the system from imperial to metric. In 1794, Monge helped setting up the École Centrale des Travaux Publics (later École Polytechnique) where he was lecturing Descriptive Geometry. In 1798, Napoleon undertook a campaign in Egypt. The famous chemist Claude Louis Berthollet was asked to recruit prominent scientists. Among them were Fourier, Monge, Dolomieu and Malus. Institut d'Egypte was established by Napoleon and Monge was named as its first president. Monge passed away on July 28, 1818. His name is inscribed on the base of the Eiffel Tower and it is located on the third façade opposite the Military Academy. Besides descriptive geometry, he carried on many different researches in chemistry and physics