3 research outputs found
Analysis of johnson polyhedra using projective geometry techniques ΠΠ ΠΠΠΠ’ΠΠΠΠΠ ΠΠ€ΠΠ§ΠΠ‘ΠΠΠ ΠΠΠΠΠΠ ΠΠΠΠΠΠΠ ΠΠΠΠΠΠΠ ΠΠΠΠΠ‘ΠΠΠ
The authors analyze the capabilities of projective geometry techniques based on the method of tracing for diagrams, as applied to problems of Johnson polyhedra and formation of convex polyhedral structures. Johnson polyhedra, known as Johnson solids, demonstrate a specific type of symmetry. Each polyhedron can serve as the core for varied shapes capable of preserving their properties. The authors believe that the research into clusters of Johnson solids have a stronger potential than any research into a single Johnson polyhedron. The paper shows how the change of parameters (rotation angles, axis of symmetry, and number of facets) can be preserved for a variety of shapes; this is a very lucrative property in terms of architecture and design. Specialized computer software is used for the practical implementation of the method.<br>ΠΡΠΎΠ²Π΅Π΄Π΅Π½ Π°Π½Π°Π»ΠΈΠ· Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΠ΅ΠΉ ΠΏΡΠΎΠ΅ΠΊΡΠΈΠ²ΠΎΠ³ΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ Π°ΠΏΠΏΠ°ΡΠ°ΡΠ°, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΠΎΠ³ΠΎ Π½Π° ΠΌΠ΅ΡΠΎΠ΄Π΅ ΡΠ»Π΅Π΄ΠΎΠ²ΡΡ
ΡΠΏΡΡΠΎΠ², Π² ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠΈ ΠΊ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠ°ΠΌ ΠΠΆΠΎΠ½ΡΠΎΠ½Π° Π² Π·Π°Π΄Π°ΡΠ°Ρ
ΡΠΎΡΠΌΠΎΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°Π½ΠΈΡ Π²ΡΠΏΡΠΊΠ»ΡΡ
ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΡΡ
ΡΡΡΡΠΊΡΡΡ. Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½Ρ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠΈ ΠΠΆΠΎΠ½ΡΠΎΠ½Π°, Π΄Π»Ρ ΠΊΠΎΡΠΎΡΡΡ
ΡΠΊΠ°Π·Π°Π½Ρ ΡΠΈΠΏΡ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΠΈ. ΠΠ°ΠΆΠ΄ΡΠΉ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊ ΠΌΠΎΠΆΠ΅Ρ ΡΠ²Π»ΡΡΡΡΡ ΡΠ΄ΡΠΎΠΌ ΠΎΠ΄Π½ΠΎΡΠΏΡΡΠ½ΠΎΠΉ ΠΈΠ»ΠΈ ΠΌΠ½ΠΎΠ³ΠΎΡΠΏΡΡΠ½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ ΠΊΠ°ΠΊ ΠΏΡΠΈ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠΈ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² (ΡΠ³Π»ΠΎΠ² ΠΏΠΎΠ²ΠΎΡΠΎΡΠ°, ΠΎΡΠ΅ΠΉ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΠΈ, ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΡΠ΄Π΅Ρ) ΠΌΠΎΠΆΠ½ΠΎ ΠΏΠΎΠ»ΡΡΠ°ΡΡ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΠ΅ ΡΠΎΡΠΌ, ΠΈΠ½ΡΠ΅ΡΠ΅ΡΠ½ΡΡ
Ρ ΡΠΎΡΠΊΠΈ Π·ΡΠ΅Π½ΠΈΡ Π°ΡΡ
ΠΈΡΠ΅ΠΊΡΡΡΡ ΠΈ Π΄ΠΈΠ·Π°ΠΉΠ½Π°. ΠΠ»Ρ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅Π°Π»ΠΈΠ·Π°ΡΠΈΠΈ ΠΌΠ΅ΡΠΎΠ΄Π° ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½Π° ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΠΎ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠ°Π½Π½Π°Ρ ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½Π°Ρ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠ°
On the use of polar coordinate system in the projective graphic drawings
Projective graphics is a polyhedra simulation method, which is based on the use of trace diagrams of initial polyhedron. Previously developed computer software allows using Cartesian coordinates. In some cases it is advisable to use polar coordinate system for description of projective graphics drawings. Using the example of icosahedron the authors analyzed the advantages of using projective graphics drawings in the polar coordinate system. The transition to the polar coordinate system is a tool that allows using certain patterns of projective graphics drawings in the process of calculation. When using polar coordinate system the search of Polar correspondence for the directs is simplified. In order to analyze the two lines in the polar coordinate system it is enough to compare the corresponding coefficients of the equations of these lines. The authors consider a diagram of the icosahedron in polar coordinates, and a corresponding fragment of calculation program in the Mathematica system. Some examples of forming based on icosahedrons are offered. Optimization of computer programs using polar coordinate system will simplifies the calculations of projective graphics drawings, accelerates the process of constructing three-dimensional models, which expand the possibilities of selecting original solutions. Finally, the authors conclude that it is appropriate to use the polar coordinate system only in the construction of projective graphics diagrams of the planes system having rich symmetry. All Platonic and Archimedean solids, Catalan solid possess this property
Automatic receipt of projective geometry drawings of Johnson bodies
The article analyzes the possibilities of polyhedral structuresβ formation basing on the automated construction of orthographic drawings (trace diagrams) derived from Johnson bodies. Projective Graphical method makes it possible to simulate the new multi-faceted forms with the help of the trace diagrams selected as a basis of a polyhedron. The computer program developed for this aim allows receiving both trace diagrams and diverse visual images of newly created polygonal shapes. Due to the large number of possible solutions it is proposed to use trace diagrams themselves (based on their degree of symmetry) as a tool to assess the feasibility of using this or that Johnson body as a basis for further shaping