3-D Shadows of 4-D Algebraic Hypersurfaces in a 4-D Perspective

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11410%2F25%3A10509352" target="_blank" >RIV/00216208:11410/25:10509352 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21450/25:00389578

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=dILJ.s6x0Q" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=dILJ.s6x0Q</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

3-D Shadows of 4-D Algebraic Hypersurfaces in a 4-D Perspective

Popis výsledku v původním jazyce

Visualizing a scene of objects in 4-D space faces several challenges. Mere projections into 3- or less-dimensional spaces usually contain overlapping parts, making them difficult to comprehend or study. Illuminating the scene can enhance intuition about its &quot;dimensionality.&quot; Our contribution describes a geometric approach to creating visualizations of 4-D hypersurfaces represented by implicit algebraic equations without their parametrization. By geometric, we mean methods using constructions of geometric objects without their approximation, for example, by polyhedral meshes. Therefore, instead of sets of many points and operating with meshes, we work with implicitly represented hypersurfaces, their projections, contours, intersections, etc. We provide a general algorithm to find shadow boundaries in an arbitrary dimension and apply it in a 4-D space. Furthermore, we design a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective. The results of our algorithm are components of the 3-D model of a scene image represented by polynomial equations and inequalities prepared for plotting by standard computer algebra systems with visualization tools. The method is presented on three 4-D scenes with gradual many properties of the visualized shapes, they are suitable for precise mathematical or scientific visualization. On the other hand, processing higher-degree time.

Název v anglickém jazyce

3-D Shadows of 4-D Algebraic Hypersurfaces in a 4-D Perspective

Popis výsledku anglicky

Visualizing a scene of objects in 4-D space faces several challenges. Mere projections into 3- or less-dimensional spaces usually contain overlapping parts, making them difficult to comprehend or study. Illuminating the scene can enhance intuition about its &quot;dimensionality.&quot; Our contribution describes a geometric approach to creating visualizations of 4-D hypersurfaces represented by implicit algebraic equations without their parametrization. By geometric, we mean methods using constructions of geometric objects without their approximation, for example, by polyhedral meshes. Therefore, instead of sets of many points and operating with meshes, we work with implicitly represented hypersurfaces, their projections, contours, intersections, etc. We provide a general algorithm to find shadow boundaries in an arbitrary dimension and apply it in a 4-D space. Furthermore, we design a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective. The results of our algorithm are components of the 3-D model of a scene image represented by polynomial equations and inequalities prepared for plotting by standard computer algebra systems with visualization tools. The method is presented on three 4-D scenes with gradual many properties of the visualized shapes, they are suitable for precise mathematical or scientific visualization. On the other hand, processing higher-degree time.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10102 - Applied mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal for Geometry and Graphics

ISSN

1433-8157

e-ISSN

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Svazek periodika

29

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

23

Strana od-do

199-221

Kód UT WoS článku

001664223400007

EID výsledku v databázi Scopus

2-s2.0-105028979464