{"id":219838,"date":"2026-01-15T15:05:28","date_gmt":"2026-01-15T13:05:28","guid":{"rendered":"https:\/\/azbuki.bg\/?p=219838"},"modified":"2026-01-16T09:19:00","modified_gmt":"2026-01-16T07:19:00","slug":"izsledvane-na-proekczii-na-rabesti-tela-v-sistemata-strunima","status":"publish","type":"post","link":"https:\/\/mathinfo.azbuki.bg\/en\/uncategorized\/izsledvane-na-proekczii-na-rabesti-tela-v-sistemata-strunima\/","title":{"rendered":"Investigating Projections of 3-Dimensional Shapes with the StruniMa System"},"content":{"rendered":"

Mladen Valkov<\/strong><\/p>\n

Institute of Mathematics and Informatics<\/em><\/p>\n

Bulgarian Academy of Sciences (Bulgaria)<\/em> (<\/em>Bulgaria<\/em>)<\/em><\/p>\n

https:\/\/doi.org\/10.53656\/voc25-3-4-13<\/a><\/p>\n

Abstract.<\/strong> In the StruniMa system there is a functionality for visualizing square grid in 3D \u2013 using it one can find the orthogonal projections of shapes that lie inside the grid or consisting of unit cubes with vertices that coincide with the vertices of the grid. Also there is a part of the system which consists of competitive levels which require finding the third projection when only the first and the second are given.<\/p>\n

Keywords: <\/em>StruniMa; technical drawing; projections; vertices; edges; faces<\/p>\n

 <\/p>\n

    \n
  1. Introduction<\/strong><\/li>\n<\/ol>\n

    1<\/strong>.1. <\/strong>\u0418\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u0442\u0435\u043b\u0430 \u0432 \u0443\u0447\u0438\u043b\u0438\u0449\u0435<\/strong><\/p>\n

    \u0418\u0437\u043e\u0431\u0440\u0430\u0437\u044f\u0432\u0430\u043d\u0435\u0442\u043e \u0438 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u0442\u0435\u043b\u0430 \u0441\u044a\u0441 \u0441\u043e\u0444\u0442\u0443\u0435\u0440 \u0435 \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u043d\u043e \u0432 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0438\u043d\u0434\u0443\u0441\u0442\u0440\u0438\u0438 \u2013 \u0441\u0442\u0440\u043e\u0438\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0438 \u0430\u0440\u0445\u0438\u0442\u0435\u043a\u0442\u0443\u0440\u0430, \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0438 \u0438\u0433\u0440\u0438, \u0432\u0438\u0440\u0442\u0443\u0430\u043b\u043d\u0430 \u0440\u0435\u0430\u043b\u043d\u043e\u0441\u0442, \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0444\u0438\u0437\u0438\u0447\u043d\u0438 \u043f\u0440\u043e\u0446\u0435\u0441\u0438 \u0438 \u0434\u0440. \u0421 \u043d\u0430\u043b\u0438\u0447\u0438\u0435\u0442\u043e \u043d\u0430 \u0434\u043e\u0441\u0442\u0430\u0442\u044a\u0447\u0435\u043d \u0431\u0440\u043e\u0439 \u0441\u043e\u0444\u0442\u0443\u0435\u0440\u0438 \u0437\u0430 \u0432\u0438\u0437\u0443\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u0442\u0435\u043b\u0430 \u0441\u0435 \u043f\u043e\u044f\u0432\u044f\u0432\u0430 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0442\u0430 \u0437\u0430 \u043f\u043e-\u0448\u0438\u0440\u043e\u043a\u043e \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430\u0442\u0430 \u0438\u043c \u0432 \u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u043e\u0442\u043e \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430, \u043a\u0430\u043a\u0442\u043e \u0438 \u043c\u0443\u043b\u0442\u0438\u0434\u0438\u0441\u0446\u0438\u043f\u043b\u0438\u043d\u0430\u0440\u043d\u0438 \u0434\u0435\u0439\u043d\u043e\u0441\u0442\u0438 \u0441 \u043f\u0440\u0435\u0434\u043c\u0435\u0442\u0438 \u043a\u0430\u0442\u043e \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u043e \u0447\u0435\u0440\u0442\u0430\u043d\u0435. \u0412 \u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f \u043f\u0440\u0438\u043c\u0435\u0440\u0438\u0442\u0435 \u0437\u0430 \u0442\u0430\u043a\u0438\u0432\u0430 \u0441\u043e\u0444\u0442\u0443\u0435\u0440\u0438 (AutoCAD, GeoGebra \u0438 \u0434\u0440.) \u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435\u0442\u043e \u0438\u043c \u0432 \u043e\u0431\u0443\u0447\u0438\u0442\u0435\u043b\u0435\u043d \u043f\u0440\u043e\u0446\u0435\u0441, \u0432\u043a\u043b\u044e\u0447\u0432\u0430\u0449 \u0434\u0435\u0441\u043a\u0440\u0438\u043f\u0442\u0438\u0432\u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f \u0437\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438, \u0441\u0430 \u043d\u0435\u043c\u0430\u043b\u043a\u043e (Zareva, Chehlarova, 2016; Chehlarova, Stanchev, 2001), \u043a\u0430\u043a\u0442\u043e \u0438\u043c\u0430 \u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u0438 \u0437\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 (Lebamovski, Nikolova, 2019). \u0412\u044a\u043f\u0440\u0435\u043a\u0438 \u0442\u043e\u0432\u0430 \u0438 \u0438\u0437\u0433\u0440\u0430\u0434\u0435\u043d\u0438\u0442\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e STEM \u0446\u0435\u043d\u0442\u0440\u043e\u0432\u0435 \u043f\u043e\u0434\u043e\u0431\u043d\u0438 \u0447\u0430\u0441\u043e\u0432\u0435 \u0438 \u0434\u0435\u0439\u043d\u043e\u0441\u0442\u0438 \u0441\u0430 \u0440\u044f\u0434\u043a\u043e\u0441\u0442, \u043a\u043e\u0435\u0442\u043e \u0438\u043c\u0430 \u0441\u0432\u043e\u0438\u0442\u0435 \u043f\u043e\u0441\u043b\u0435\u0434\u0441\u0442\u0432\u0438\u044f \u0432\u044a\u0440\u0445\u0443 \u0438\u0437\u0433\u0440\u0430\u0436\u0434\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435\u043d\u043e \u043c\u0438\u0441\u043b\u0435\u043d\u0435 \u0443 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435.<\/p>\n

    1<\/strong>.<\/strong>2<\/strong>. <\/strong>\u0418\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u0442\u0435\u043b\u0430 \u0432 \u201e\u0421\u0442\u0440\u0443\u043d\u0438\u041c\u0430\u201c<\/strong><\/p>\n

    \u0412 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u0421\u0442\u0440\u0443\u043d\u0438\u041c\u0430\u201c (https:\/\/strunima.valkovbg.com\/play-in-browser\/<\/a>) \u0438\u043c\u0430 \u043a\u043b\u0430\u0441 \u043e\u0442 \u043f\u043e\u0434\u0442\u0435\u043c\u0438, \u043a\u043e\u0438\u0442\u043e \u0441\u0435 \u0431\u0430\u0437\u0438\u0440\u0430\u0442 \u043d\u0430 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u0442\u0435\u043b\u0430 \u0438 \u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0442\u043e\u0440\u043d\u043e-\u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0432\u044a\u0440\u0445\u0443 \u0442\u044f\u0445 \u2013 \u0441\u0438\u043c\u0435\u0442\u0440\u0438\u044f \u0432\u044a\u0440\u0445\u0443 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u0434\u044a\u0441\u043a\u0430 (Valkov, 2024), \u043f\u043e\u043a\u0440\u0438\u0442\u0438\u044f \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u0434\u044a\u0441\u043a\u0430 \u0441 \u043f\u043b\u043e\u0447\u043a\u0438 \u043f\u043e\u043b\u0438\u043c\u0438\u043d\u043e, \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u0432\u044a\u0440\u0445\u0443 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u0434\u044a\u0441\u043a\u0430, \u0432\u044a\u0437\u043b\u0438 \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e\u0442\u043e, \u0443\u0441\u0443\u043a\u0432\u0430\u043d\u0438\u044f \u043d\u0430 \u041a\u043e\u043d\u0443\u0435\u0439 \u0438 \u0434\u0440. \u0418\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043f\u044a\u0440\u0432\u0430, \u0432\u0442\u043e\u0440\u0430 \u0438 \u0442\u0440\u0435\u0442\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u044f \u043d\u0430 \u0442\u0435\u043b\u0430, \u0441\u044a\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0432 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u0430 \u043c\u0440\u0435\u0436\u0430, \u0437\u0430\u0435\u0434\u043d\u043e \u0441 \u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0442\u043e\u0440\u043d\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u0441\u0435 \u043e\u043a\u0430\u0437\u0432\u0430 \u0434\u043e\u0431\u0440\u043e \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 \u0437\u0430 \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435\u043d\u043e\u0442\u043e \u043c\u0438\u0441\u043b\u0435\u043d\u0435 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435.<\/p>\n

    \u0412 \u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u041f\u043e\u043a\u0440\u0438\u0442\u0438\u044f \u043d\u0430 \u0434\u044a\u0441\u043a\u0430\u201c \u0435 \u0438\u0437\u0440\u0430\u0431\u043e\u0442\u0435\u043d\u0430 \u0434\u044a\u0441\u043a\u0430, \u0440\u0430\u0437\u0434\u0435\u043b\u0435\u043d\u0430 \u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0447\u043d\u0438 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0447\u0435\u0442\u0430 (\u043a\u0443\u0431\u0447\u0435\u0442\u0430), \u043a\u043e\u0438\u0442\u043e \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0431\u044a\u0434\u0430\u0442 \u043e\u0446\u0432\u0435\u0442\u044f\u0432\u0430\u043d\u0438 \u0438\u043b\u0438 \u043f\u0440\u0435\u043c\u0430\u0445\u0432\u0430\u043d\u0438 \u043e\u0442 \u0434\u044a\u0441\u043a\u0430\u0442\u0430. \u0414\u044a\u0441\u043a\u0430\u0442\u0430 \u043c\u043e\u0436\u0435 \u0434\u0430 \u0441\u0435 \u0440\u043e\u0442\u0438\u0440\u0430 \u0438 \u0434\u0430 \u0438\u043c\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u043d\u0438 \u0440\u0430\u0437\u043c\u0435\u0440\u0438, \u043f\u043e-\u043c\u0430\u043b\u043a\u0438 \u043e\u0442 100. \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0435\u043d\u0438 \u0441\u0430 \u0434\u0432\u0435 \u043f\u043e\u0434\u0442\u0435\u043c\u0438. \u041f\u044a\u0440\u0432\u0430\u0442\u0430 \u0435 \u201e\u041f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u0434\u044a\u0441\u043a\u0430\u201c (Valkov, 2024) (\u0444\u0438\u0433. 1):<\/p>\n

    \"\"<\/p>\n

    \u0424\u0438\u0433\u0443\u0440\u0430 1.<\/strong> \u041f\u044a\u0440\u0432\u0430, \u0432\u0442\u043e\u0440\u0430 \u0438 \u0442\u0440\u0435\u0442\u0430 \u0443\u0441\u043f\u043e\u0440\u0435\u0434\u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u044f \u043d\u0430 \u0444\u0438\u0433\u0443\u0440\u0430\u0442\u0430, \u0441\u044a\u0441\u0442\u0430\u0432\u0435\u043d\u0430 \u043e\u0442 \u043f\u043e\u043a\u0430\u0437\u0430\u043d\u0438\u0442\u0435 \u043a\u0443\u0431\u0447\u0435\u0442\u0430 \u2013 https:\/\/strunima.free.bg\/CoveringProjections.html<\/a><\/p>\n

     <\/p>\n

    \u0412\u043f\u043e\u0441\u043b\u0435\u0434\u0441\u0442\u0432\u0438\u0435 \u0441\u0435 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e\u0441\u0442, \u043a\u043e\u044f\u0442\u043e \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438\u0442\u0435 \u043d\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u043d\u043e \u0440\u044a\u0431\u0435\u0441\u0442\u043e \u0442\u044f\u043b\u043e \u0432\u044a\u0440\u0445\u0443 \u0442\u0440\u0438 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u043e\u043d\u043d\u0438 \u0440\u0430\u0432\u043d\u0438\u043d\u0438. \u041c\u043e\u0436\u0435 \u0434\u0430 \u0441\u0435 \u0434\u043e\u0441\u0442\u044a\u043f\u0438 \u043f\u0440\u0435\u0437 \u0442\u0435\u043c\u0430\u0442\u0430 \u0437\u0430 \u043f\u043e\u043a\u0440\u0438\u0442\u0438\u044f \u0432\u044a\u0440\u0445\u0443 \u0434\u044a\u0441\u043a\u0430 \u0438 \u043f\u043e\u043a\u0440\u0438\u0432\u0430\u0449\u0438 \u043c\u043d\u043e\u0433\u043e\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438 (\u0444\u0438\u0433. 2) \u0438 \u0441\u043b\u0435\u0434 \u0442\u043e\u0432\u0430 \u0441 \u043f\u0440\u043e\u043c\u044f\u043d\u0430 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u0442\u0435 \u043f\u043b\u044a\u0437\u0433\u0430\u0447\u0430 \u0434\u0430 \u0441\u0435 \u043d\u0430\u043f\u0440\u0430\u0432\u0438 \u043f\u0440\u0430\u0437\u043d\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u0430 \u043c\u0440\u0435\u0436\u0430.<\/p>\n

    \"\"<\/p>\n

    \u0424\u0438\u0433\u0443\u0440\u0430 2.<\/strong> \u0414\u043e\u0441\u0442\u044a\u043f \u0438 \u0441\u0442\u043e\u0439\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u043f\u043b\u044a\u0437\u0433\u0430\u0447\u0438\u0442\u0435 \u0437\u0430 \u043d\u0430\u043f\u0440\u0430\u0432\u0430\u0442\u0430 \u043d\u0430 \u043c\u0440\u0435\u0436\u0430,
    \n\u0432 \u043a\u043e\u044f\u0442\u043e \u0432\u0441\u0438\u0447\u043a\u0438 \u043a\u0443\u0431\u0447\u0435\u0442\u0430 \u0441\u0430 \u0438\u0437\u043a\u043b\u044e\u0447\u0435\u043d\u0438<\/p>\n

     <\/p>\n

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    \"\"<\/p>\n

    \u0424\u0438\u0433\u0443\u0440\u0430 3.<\/strong> \u041f\u043e\u0441\u0442\u0440\u043e\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0442\u0435\u0442\u0440\u0430\u0435\u0434\u044a\u0440 \u0432\u044a\u0440\u0445\u0443 \u043c\u0440\u0435\u0436\u0430\u0442\u0430<\/p>\n

     <\/p>\n

    \u041f\u0440\u0438 \u0438\u0437\u0431\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u201e\u0422\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f\u201c \u043e\u0442 \u043f\u0430\u0434\u0430\u0449\u043e\u0442\u043e \u043c\u0435\u043d\u044e \u0434\u043e \u0431\u0443\u0442\u043e\u043d\u0430 \u0437\u0430 \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438\u0442\u0435 \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0441\u0435 \u043d\u0430\u043f\u0440\u0430\u0432\u044f\u0442 \u043e\u0440\u0442\u043e\u0433\u043e\u043d\u0430\u043b\u043d\u0438\u0442\u0435 (Elias, 2012) \u0442\u0430\u043a\u0438\u0432\u0430, \u043a\u043e\u0438\u0442\u043e \u0441\u0435 \u043d\u0430\u0431\u043b\u044e\u0434\u0430\u0432\u0430\u0442 \u043e\u0442\u043b\u044f\u0432\u043e, \u043e\u0442\u0433\u043e\u0440\u0435 \u0438 \u043e\u0442\u043f\u0440\u0435\u0434. \u0412\u0438\u0434\u0438\u043c\u0438\u0442\u0435 \u0438 \u043d\u0435\u0432\u0438\u0434\u0438\u043c\u0438\u0442\u0435 \u0440\u044a\u0431\u043e\u0432\u0435 \u0441\u0435 \u043f\u0440\u043e\u0435\u043a\u0442\u0438\u0440\u0430\u0442 \u0441\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u043e \u0432 \u043e\u0440\u0430\u043d\u0436\u0435\u0432 \u0438 \u0441\u0438\u043d \u0446\u0432\u044f\u0442, \u043a\u0430\u0442\u043e \u0430\u043a\u043e \u043d\u0435\u0432\u0438\u0434\u0438\u043c \u0440\u044a\u0431 \u043b\u0435\u0436\u0438 \u0432\u044a\u0440\u0445\u0443 \u0432\u0438\u0434\u0438\u043c, \u0441\u0435 \u0432\u0438\u0436\u0434\u0430 \u0441\u0430\u043c\u043e \u0432\u0438\u0434\u0438\u043c\u0438\u044f\u0442. \u0412\u0441\u0435\u043a\u0438 \u043e\u0442 \u0432\u044a\u0440\u0445\u043e\u0432\u0435\u0442\u0435 \u043d\u0430 \u0440\u044a\u0431\u0435\u0441\u0442\u043e\u0442\u043e \u0442\u044f\u043b\u043e \u0441\u0435 \u043f\u0440\u043e\u0435\u043a\u0442\u0438\u0440\u0430 \u0432\u044a\u0440\u0445\u0443 \u0441\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u0430\u0442\u0430 \u0440\u0430\u0432\u043d\u0438\u043d\u0430.<\/p>\n

    \"\"<\/p>\n

    \u0424\u0438\u0433\u0443\u0440\u0430 4.<\/strong> \u041f\u043e\u0441\u0442\u0440\u043e\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u0440\u044a\u0431\u0435\u0441\u0442\u043e \u0442\u044f\u043b\u043e<\/p>\n

     <\/p>\n

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    \"\"<\/p>\n

    \u0424\u0438\u0433\u0443\u0440\u0430 5.<\/strong> \u041d\u0430\u0447\u0430\u043b\u043d\u043e \u0441\u044a\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043d\u0430 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u0442\u0435\u043b\u043d\u043e\u0442\u043e \u043d\u0438\u0432\u043e \u0438 \u043e\u0447\u0430\u043a\u0432\u0430\u043d\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u0435,<\/p>\n

    \u043f\u0440\u0435\u0434\u0438 \u0434\u0430 \u0441\u0435 \u043d\u0430\u0442\u0438\u0441\u043d\u0435 \u0431\u0443\u0442\u043e\u043d\u044a\u0442 \u0437\u0430 \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u0430<\/p>\n

     <\/p>\n

      \n
    1. \u041f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438 \u0443\u0447\u0435\u0431\u043d\u0438 \u0447\u0430\u0441\u043e\u0432\u0435 \u0441 \u0438\u0437\u0433\u0440\u0430\u0434\u0435\u043d\u0438\u0442\u0435 \u0434\u0438\u0433\u0438\u0442\u0430\u043b\u043d\u0438 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u0438 \u0432 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u0421\u0442\u0440\u0443\u043d\u0438\u041c\u0430\u201c<\/strong><\/li>\n<\/ol>\n

      \u0412 STEM \u043a\u0430\u0431\u0438\u043d\u0435\u0442\u0430 \u043d\u0430 \u041f\u041f\u041c\u0413 \u201e\u041d\u0430\u043d\u0447\u043e \u041f\u043e\u043f\u043e\u0432\u0438\u0447\u201c \u0441\u0435 \u043f\u0440\u043e\u0432\u0435\u0434\u043e\u0445\u0430 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u0443\u0447\u0435\u0431\u043d\u0438 \u0447\u0430\u0441\u0430 \u0441 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043e\u0442 V \u0438 VII \u043a\u043b\u0430\u0441, \u0431\u0430\u0437\u0438\u0440\u0430\u043d\u0438 \u043d\u0430 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u0442\u044f\u043b\u043e, \u0441\u044a\u0441\u0442\u0430\u0432\u0435\u043d\u043e \u043e\u0442 \u043a\u0443\u0431\u0447\u0435\u0442\u0430. \u0417\u0430\u0434\u0430\u0447\u0430\u0442\u0430 \u0441\u0435 \u0441\u044a\u0441\u0442\u043e\u0438 \u0432 \u0442\u043e\u0432\u0430 \u043f\u0440\u0438 \u0434\u0430\u0434\u0435\u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u0434\u044a\u0441\u043a\u0430 3\u04453\u04453 (\u0444\u0438\u0433. 6.) \u0434\u0430 \u0441\u0435 \u043f\u0440\u0435\u043c\u0430\u0445\u043d\u0435 \u043d\u0430\u0439-\u0433\u043e\u043b\u0435\u043c\u0438\u044f\u0442 \u0431\u0440\u043e\u0439 \u043a\u0443\u0431\u0447\u0435\u0442\u0430, \u0442\u0430\u043a\u0430 \u0447\u0435 \u0434\u044a\u0441\u043a\u0430\u0442\u0430 \u0434\u0430 \u0437\u0430\u043f\u0430\u0437\u0438 \u0442\u0440\u0438\u0442\u0435 \u0441\u0438 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438.<\/p>\n

       <\/p>\n

      \"\"<\/strong><\/p>\n

      \u0424\u0438\u0433\u0443\u0440\u0430 6.<\/strong> \u041e\u0440\u0442\u043e\u0433\u043e\u043d\u0430\u043b\u043d\u0438 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u043a\u0443\u0431 3\u04453\u04453<\/p>\n

       <\/p>\n

      \u041c\u0438\u043d\u0438\u043c\u0430\u043b\u043d\u0438\u044f\u0442 \u0431\u0440\u043e\u0439 \u043e\u0441\u0442\u0430\u0432\u0430\u0449\u0438 \u043a\u043b\u0435\u0442\u043a\u0438 \u0435 9, \u0442\u044a\u0439 \u043a\u0430\u0442\u043e \u0442\u0440\u044f\u0431\u0432\u0430 \u0434\u0430 \u0438\u043c\u0430\u043c\u0435 \u043f\u043e\u043d\u0435 \u0435\u0434\u043d\u0430 \u043a\u043b\u0435\u0442\u043a\u0430 \u0432\u044a\u0432 \u0432\u0441\u0435\u043a\u0438 \u0431\u043b\u043e\u043a 1\u04453. \u041f\u044a\u0440\u0432\u043e\u043d\u0430\u0447\u0430\u043b\u043d\u0430\u0442\u0430 \u0440\u0435\u0430\u043a\u0446\u0438\u044f \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435 \u0431\u0435\u0448\u0435, \u0447\u0435 \u0442\u043e\u0432\u0430 \u043d\u0435 \u0435 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e, \u0434\u043e\u043a\u0430\u0442\u043e \u043d\u0435 \u0441\u0435 \u0434\u0430\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u0440 \u0441 \u043f\u044a\u0440\u0432\u0438\u044f \u0441\u043b\u043e\u0439 3\u04453, \u043e\u0442 \u043a\u043e\u0439\u0442\u043e \u043c\u043e\u0436\u0435 \u0434\u0430 \u0441\u0435 \u043e\u0441\u0442\u0430\u0432\u044f\u0442 \u0441\u0430\u043c\u043e \u0442\u0440\u0438\u0442\u0435 \u043a\u043b\u0435\u0442\u043a\u0438 \u043f\u043e \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0430.<\/p>\n

      \"\"<\/p>\n

      \u0424\u0438\u0433\u0443\u0440\u0430 7.<\/strong> \u041e\u0440\u0442\u043e\u0433\u043e\u043d\u0430\u043b\u043d\u0438 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u043a\u0443\u0431 3\u04453\u04453<\/p>\n

       <\/p>\n

      \u0421\u043b\u0435\u0434 \u043d\u0430\u043c\u0438\u0440\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u0435\u0442\u043e \u0437\u0430 \u043a\u0443\u0431 3\u04453\u04453 \u043d\u0435\u043c\u0430\u043b\u043a\u043e \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0443\u0441\u043f\u044f\u0445\u0430 \u0434\u0430 \u0441\u0435 \u0441\u043f\u0440\u0430\u0432\u044f\u0442 \u0441 \u043c\u0438\u043d\u0438\u043c\u0430\u043b\u043d\u0430 \u043f\u043e\u043c\u043e\u0449 \u0441\u044a\u0441 \u0441\u044a\u0449\u0430\u0442\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u0437\u0430 \u043a\u0443\u0431 4\u04454\u04454. \u041f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0442\u0435 \u0447\u0430\u0441\u043e\u0432\u0435 \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0441\u0435 \u0441\u0447\u0438\u0442\u0430\u0442 \u0437\u0430 \u043c\u0443\u043b\u0442\u0438\u0434\u0438\u0441\u0446\u0438\u043f\u043b\u0438\u043d\u0430\u0440\u043d\u0430 \u0434\u0435\u0439\u043d\u043e\u0441\u0442, \u0432\u043a\u043b\u044e\u0447\u0432\u0430\u0449\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 (\u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0442\u043e\u0440\u0438\u043a\u0430), \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u043e \u0447\u0435\u0440\u0442\u0430\u043d\u0435 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438, \u043a\u0430\u0442\u043e \u0441\u043e\u0444\u0442\u0443\u0435\u0440\u044a\u0442 \u0434\u0430\u0432\u0430 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442 \u0434\u0430 \u0432\u0438\u0436\u0434\u0430\u0442 \u043d\u0430 \u0432\u0441\u044f\u043a\u0430 \u0441\u0442\u044a\u043f\u043a\u0430 \u043f\u0440\u043e\u043c\u0435\u043d\u0438\u0442\u0435 \u043f\u043e \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438\u0442\u0435, \u043a\u0430\u043a\u0442\u043e \u0438 \u0434\u0430 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u0442 \u0441\u0430\u043c\u043e\u0442\u043e \u0442\u044f\u043b\u043e, \u0441\u044a\u0441\u0442\u0430\u0432\u0435\u043d\u043e \u043e\u0442 \u043a\u0443\u0431\u0447\u0435\u0442\u0430\u0442\u0430 \u0432 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u0440\u043e\u0442\u0430\u0446\u0438\u0438. \u0418\u043c\u0430 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0435\u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e\u0441\u0442, \u043a\u043e\u044f\u0442\u043e \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0432\u0430 \u0438\u0437\u043e\u0431\u0440\u0430\u0437\u044f\u0432\u0430\u043d\u0435\u0442\u043e \u0438 \u043d\u0430 \u0440\u044a\u0431\u043e\u0432\u0435\u0442\u0435 \u043d\u0430 \u0442\u044f\u043b\u043e\u0442\u043e, \u043a\u043e\u0435\u0442\u043e \u043f\u0440\u0438 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u0442\u0435\u043b\u043d\u0438\u0442\u0435 \u043d\u0438\u0432\u0430 (\u043d\u0430\u043c\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0442\u044f\u043b\u043e, \u0441\u044a\u0441\u0442\u0430\u0432\u0435\u043d\u043e \u043e\u0442 \u043a\u0443\u0431\u0447\u0435\u0442\u0430 \u043f\u043e \u0434\u0430\u0434\u0435\u043d\u0438 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438) \u0432\u043d\u0430\u0441\u044f \u0434\u043e\u043f\u044a\u043b\u043d\u0438\u0442\u0435\u043b\u043d\u0430 \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442.<\/p>\n

      \"\"<\/p>\n

      \u0424\u0438\u0433\u0443\u0440\u0430 8.<\/strong> \u041f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u0441 \u0440\u044a\u0431\u043e\u0432\u0435 \u043d\u0430 \u0442\u044f\u043b\u043e\u0442\u043e, \u0441\u044a\u0441\u0442\u0430\u0432\u0435\u043d\u043e \u043e\u0442 \u043a\u0443\u0431\u0447\u0435\u0442\u0430<\/p>\n

      \u00a0<\/strong><\/p>\n

      \u041b\u0418\u0422\u0415\u0420\u0410\u0422\u0423\u0420\u0410<\/strong><\/p>\n

      \u0412\u044a\u043b\u043a\u043e\u0432, \u041c. (2024). \u0426\u0435\u043d\u0442\u0440\u0430\u043b\u043d\u0430 \u0441\u0438\u043c\u0435\u0442\u0440\u0438\u044f \u0432\u044a\u0440\u0445\u0443 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 \u0434\u044a\u0441\u043a\u0430, Central symmetry on a 3-dimensional board, 10 \u0433\u043e\u0434\u0438\u043d\u0438 \u0412\u0438\u0440\u0442\u0443\u0430\u043b\u0435\u043d \u0443\u0447\u0438\u043b\u0438\u0449\u0435\u043d \u043a\u0430\u0431\u0438\u043d\u0435\u0442 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430<\/em>, ISBN 978-954-561-636-5.<\/p>\n

      \u0412\u044a\u043b\u043a\u043e\u0432\u0430, \u0414., \u0412\u044a\u043b\u043a\u043e\u0432, \u041c. (2024). \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u0421\u0442\u0440\u0443\u043d\u0438\u041c\u0430 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u043e \u0447\u0435\u0440\u0442\u0430\u043d\u0435, \u0410\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u0437\u0430\u0446\u0438\u044f \u043d\u0430 \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u043e\u0442\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u0441\u0442\u0432\u043e<\/em>, 2, ISSN 2682-9584, mf.tu-sofia.bg\/adpmagazine\/includes\/archive\/2024_2.pdf<\/a>.<\/p>\n

      \u0417\u0430\u0440\u0435\u0432\u0430, \u0426. (2016). \u041c\u043e\u0434\u0435\u043b \u0437\u0430 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0434\u0435\u043a\u0441\u0440\u0438\u043f\u0442\u0438\u0432\u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f \u0441 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u043d\u0438 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u0438 \u0437\u0430 \u0441\u0442\u0440\u0443\u0434\u0435\u043d\u0442\u0438 \u043f\u043e \u0430\u0440\u0445\u0438\u0442\u0435\u043a\u0442\u0443\u0440\u0430 \u0438 \u0441\u0442\u0440\u043e\u0438\u0442\u0435\u043b\u0441\u0442\u0432\u043e<\/em>, \u0411\u0410\u041d, \u0410\u0432\u0442\u043e\u0440\u0435\u0444\u0435\u0440\u0430\u0442.<\/p>\n

      \u0417\u0430\u0440\u0435\u0432\u0430, \u0426., \u0427\u0435\u0445\u043b\u0430\u0440\u043e\u0432\u0430, \u0422. (2016). \u0413\u0440\u0435\u0448\u043a\u0438 \u043f\u0440\u0438 \u043d\u0430\u043c\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0440\u0430\u0432\u043d\u0438\u043d\u043d\u043e \u0441\u0435\u0447\u0435\u043d\u0438\u0435 \u043d\u0430 \u0442\u044f\u043b\u043e \u0432 \u0430\u043a\u0441\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u044f \u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430 \u0437\u0430 \u043f\u0440\u0435\u043e\u0434\u043e\u043b\u044f\u0432\u0430\u043d\u0435\u0442\u043e \u0438\u043c. \u041f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438 \u0444\u043e\u0440\u0443\u043c<\/em>, 3.<\/p>\n

      \u041b\u0435\u0431\u0430\u043c\u043e\u0432\u0441\u043a\u0438, \u041f., \u041d\u0438\u043a\u043e\u043b\u043e\u0432\u0430, \u041c. (2019)., \u0418\u043d\u043e\u0432\u0430\u0442\u0438\u0432\u0435\u043d \u043f\u043e\u0434\u0445\u043e\u0434 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0441\u0442\u0435\u0440\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f \u0432 \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u0441\u0442\u0435\u0440\u0435\u043e\u0441\u0438\u0441\u0442\u0435\u043c\u0430 StereoMV<\/em>, DOI: 10.55630\/STEM.2019.0101<\/a>.<\/p>\n

      \u0427\u0435\u0445\u043b\u0430\u0440\u043e\u0432\u0430, \u0422., \u0421\u0442\u0430\u043d\u0447\u0435\u0432, \u0413. (2001). \u0424\u043e\u0440\u043c\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0443\u043c\u0435\u043d\u0438\u044f \u0443 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438\u0442\u0435 \u0437\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 \u0441 \u0433\u0440\u0435\u0448\u043a\u0438 \u043d\u0430 \u0437\u0430\u043d\u044f\u0442\u0438\u044f\u0442\u0430 \u043f\u043e \u0410\u0412\u0418\u0422\u041e. \u0412: \u041d\u0430\u0443\u0447\u043d\u0438 \u0442\u0440\u0443\u0434\u043e\u0432\u0435 \u043d\u0430 \u041f\u0423 \u201e\u041f. \u0425\u0438\u043b\u0435\u043d\u0434\u0430\u0440\u0441\u043a\u0438\u201c.<\/em> 38<\/em>(2), \u041c\u0435\u0442\u043e\u0434\u0438\u043a\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e, 107 \u2013 112.<\/p>\n

      Elias, M. (2012). 3D Model reconstruction from vector perpendicular projections, Scientific Monographs in Automation and Computer Science, 1<\/em>.<\/p>\n

      Dimitrov, J. (2002). Automatic reconstruction from scanned image of mechanical drawing in axonometry,\u00a0 Annual of the University of Mining and Geology \u201cSt. Ivan Rilski\u201d, 44 \u2013 45<\/em>, part III, Mechanization, electrification and automation in mines, Sofia, 93 \u201397.<\/p>\n

      Fang, F. (2014). Algorithms for the Recovery of 3D Objects from 2D Line Drawings, School of Mechanical & Aerospace Engineering<\/em>, Nanyang Technological University.<\/p>\n

      Idesawa M. (1973). A system to generate a solid figure from three view, Bulletin of the JSME,<\/em> 16<\/em>(92).<\/p>\n

      Hoang, L. (2020). 3D Solid Reconstruction from 2d Orthographic views, Mixed Reality and Three-Dimensional Computer Graphics,<\/em> DOI: 10.5772\/intechopen.91977.<\/p>\n

      Park, H. S., Shiratori, T., Matthews, I., Sheikh, Y. (2010). 3D Reconstruction of a Moving Point from a Series of 2D Projections.<\/em><\/p>\n

      Wei, L., Yuanjun, H. (2006). 3D model retrieval based on orthogonal projecitons, Computer Aided Design and Computer Graphics,<\/em> 2005, 10.1109\/CAD-CG.2005.1<\/a>.<\/p>\n

       <\/p>\n

      REFERENCES<\/strong><\/p>\n

      Chehlarova, T., Stanchev, G. (2001). Formation of skills in students to work with errors in AVITO classes. In: Scientific works of PU \u201cP. Hilendarski\u201d,<\/em> 38<\/em>(2), Teaching methodology, 107 \u2013 112.<\/p>\n

      Dimitrov, J. (2002). Automatic reconstruction from scanned image of mechanical drawing in axonometry, Annual of the University of Mining and Geology \u201cSt. Ivan Rilski\u201d, 44 \u2013 45,<\/em> part III, Mechanization, electrification and automation in mines, Sofia, 93 \u2013 97.<\/p>\n

      Elias, M. (2012). 3D Model reconstruction from vector perpendicular projections, Scientific Monographs in Automation and Computer Science<\/em>, 1<\/em>.<\/p>\n

      Fang, F. (2014). Algorithms for the Recovery of 3D Objects from 2D Line Drawings, School of Mechanical & Aerospace Engineering,<\/em> Nanyang Technological University.<\/p>\n

      Hoang, L. (2020). 3D Solid Reconstruction from 2d Orthographic views, Mixed Reality and Three-Dimensional Computer Graphics,<\/em> DOI: 10.5772\/intechopen.91977.<\/p>\n

      Idesawa, M. (1973). A system to generate a solid figure from three view, Bulletin of the JSME<\/em>, 16<\/em>(92).<\/p>\n

      Lebamovski, P., Nikolova, M. (2019). An innovative approach to teaching stereometry at school using the StereoMV stereo system<\/em>, DOI: 10.55630\/STEM.2019.0101<\/p>\n

      Park, H. S., Shiratori T., Matthews I., Sheikh Y. (2010). 3D Reconstruction of a Moving Point from a Series of 2D Projections.<\/em><\/p>\n

      Valkov, M. (2024). Central symmetry on a 3-dimensional board, 10 years of Virtual School Mathematics Classroom,<\/em> ISBN 978-954-561-636-5.<\/p>\n

      Valkova, D., Valkov, M. (2024). Using the StruniMa system in technical drawing training, Automation of discrete production,<\/em> 2, ISSN 2682-9584, mf.tu-sofia.bg\/adpmagazine\/includes\/archive\/2024_2.pdf.<\/p>\n

      Wei, L., Yuanjun H. (2006). 3D model retrieval based on orthogonal projecitons, Computer Aided Design and Computer Graphics,<\/em> 2005, 10.1109\/CAD-CG.2005.1.<\/p>\n

      Zareva, T. (2016). A model for studying descriptive geometry with dynamic structures for students of architecture and construction,<\/em> Bulgarian Academy of Sciences, Abstract.<\/p>\n

      Zareva, T., Chehlarova, T. (2016). Errors when finding a plane section of a body in axonometry and means of overcoming them im. Pedagogical Forum<\/em>, 3.<\/p>\n

       <\/p>\n

      \u00a0<\/strong><\/p>\n

      INVESTIGATING PROJECTIONS OF 3-DIMENSIONAL SHAPES WITH THE STRUNIMA SYSTEM<\/strong><\/p>\n

       <\/p>\n

      Abstract<\/strong>.<\/strong> In the StruniMa system there is a functionality for visualizing square grid in 3D \u2013 using it one can find the orthogonal projections of shapes that lie inside the grid or consisting of unit cubes with vertices that coincide with the vertices of the grid. Also there is a part of the system which consists of competitive levels which require finding the third projection when only the first and the second are given.<\/p>\n

      Keywords<\/em>: <\/em>StruniMa; technical drawing; projections; vertices; edges; faces<\/p>\n

       <\/p>\n

      Dr. <\/strong>Mladen Valkov<\/strong><\/p>\n

      \u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043f\u043e \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0418\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430<\/p>\n

      \u0415-mail: mladen.vulkov@math.bas.bg<\/a><\/p>\n

      strunimaofficial@gmail.com<\/a><\/p>\n

      mlado1992@abv.bg<\/a><\/p>\n

      https:\/\/strunima.valkovbg.com\/download.html<\/p>\n

       <\/p>\n

      >> Download the article as a PDF file <<<\/a><\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>","protected":false},"excerpt":{"rendered":"

      \u041c\u043b\u0430\u0434\u0435\u043d \u0412\u044a\u043b\u043a\u043e\u0432 \u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0411\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0430 \u0430\u043a\u0430\u0434\u0435\u043c\u0438\u044f \u043d\u0430 \u043d\u0430\u0443\u043a\u0438\u0442\u0435 (\u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f) https:\/\/doi.org\/10.53656\/voc25-3-4-13 \u0420\u0435\u0437\u044e\u043c\u0435. \u0412 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u0421\u0442\u0440\u0443\u043d\u0438\u041c\u0430\u201c \u0435 \u0438\u0437\u0433\u0440\u0430\u0434\u0435\u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e\u0441\u0442 \u0437\u0430 \u0432\u0438\u0437\u0443\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u0430 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[…]<\/p>","protected":false},"author":124332423427287,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[1],"tags":[15825,15826,15827,15824,12805],"yoast_head":"\n\u0418\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u0440\u044a\u0431\u0435\u0441\u0442\u0438 \u0442\u0435\u043b\u0430 \u0432 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u0421\u0442\u0440\u0443\u043d\u0438\u041c\u0430\u201c - \u0410\u0437-\u0431\u0443\u043a\u0438<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathinfo.azbuki.bg\/en\/uncategorized\/izsledvane-na-proekczii-na-rabesti-tela-v-sistemata-strunima\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u0418\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0446\u0438\u0438 \u043d\u0430 \u0440\u044a\u0431\u0435\u0441\u0442\u0438 \u0442\u0435\u043b\u0430 \u0432 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u0421\u0442\u0440\u0443\u043d\u0438\u041c\u0430\u201c - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"\u041c\u043b\u0430\u0434\u0435\u043d \u0412\u044a\u043b\u043a\u043e\u0432 \u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0411\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0430 \u0430\u043a\u0430\u0434\u0435\u043c\u0438\u044f \u043d\u0430 \u043d\u0430\u0443\u043a\u0438\u0442\u0435 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