Academic Journal
NORMALIZATION OF PARAMETRIC EQUATIONS FOR GAUSSIAN-MODIFIED CYCLOIDAL CURVES ; НОРМАЛІЗАЦІЯ ПАРАМЕТРИЧНИХ РІВНЯНЬ ГАУСІВСЬКО-МОДИФІКОВАНИХ ЦИКЛОЇДАЛЬНИХ КРИВИХ
| Τίτλος: | NORMALIZATION OF PARAMETRIC EQUATIONS FOR GAUSSIAN-MODIFIED CYCLOIDAL CURVES ; НОРМАЛІЗАЦІЯ ПАРАМЕТРИЧНИХ РІВНЯНЬ ГАУСІВСЬКО-МОДИФІКОВАНИХ ЦИКЛОЇДАЛЬНИХ КРИВИХ |
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| Συγγραφείς: | Котляр, Дмитро |
| Πηγή: | APPLIED GEOMETRY AND ENGINEERING GRAPHICS; No. 110 (2026): APPLIED GEOMETRY AND GRAPHICS; 128-138 ; Прикладна геометрія та інженерна графіка; № 110 (2026): ПРИКЛАДНА ГЕОМЕТРІЯ ТА ІНЖЕНЕРНА ГРАФІКА; 128-138 ; 1027-3050 ; 0131-579X ; 10.32347/0131-579x.2026.110 |
| Στοιχεία εκδότη: | Kyiv National University of Construction and Architecture |
| Έτος έκδοσης: | 2026 |
| Θεματικοί όροι: | граничні умови, аеродинамічний обвід, нормалізація профілю, функція Гауса, модифікована циклоїда, геометричне моделювання, boundary conditions, aerodynamic contour, profile normalization, Gaussian function, modified cycloid, geometric modeling |
| Περιγραφή: | The article is devoted to a relevant scientific and practical problem of geometric modeling for technical shape contours operating under complex gas-dynamic conditions. The object of the study is the formation process of high-smoothness parametric lines for designing the nose cones of aviation and rocket technology objects. The scientific problem lies in the emergence of a geometric contradiction during attempts at local profile modification: the introduction of additional functions for curvature control leads to uncontrolled deviation of terminal point coordinates from the specified overall dimensions (projectile caliber and length). A mathematical model is proposed, based on the parametric equations of a cycloid modified by the approximation apparatus of a Gaussian distribution. The use of a Gaussian function allows for precise local control of the generating circle radius while ensuring the preservation of a high class of curve smoothness (C3). Primary focus is placed on the development and analytical substantiation of a method for normalizing the resulting equations. The author derives a normalization operator N, the integration of which into the radius-function structure ensures the invariance of the profile's terminal point position, regardless of the modification parameters' variation (amplitude α, localization σ, and peak position μ). The validity of the developed approach is confirmed by a series of computational experiments. Based on quantitative data analysis, it was established that without the normalization procedure, the terminal point coordinate error can reach significant values: from 1.31 % for amplitude α variation to 4.36 % when shifting the modification peak μ from the curve's starting point to the target interval limit (μ → π). For small-scale technical shapes, such deviations (exceeding 1.2 mm for a length of 26.5 mm) fall outside the technological tolerances of precision mechanical engineering. It is proved that the implementation of the normalization operator N completely eliminates these ... |
| Τύπος εγγράφου: | article in journal/newspaper |
| Περιγραφή αρχείου: | application/pdf |
| Γλώσσα: | Ukrainian |
| Relation: | https://ageg.knuba.edu.ua/article/view/363470/348962; https://ageg.knuba.edu.ua/article/view/363470 |
| DOI: | 10.32347/0131-579X.2026.110.128-138 |
| Διαθεσιμότητα: | https://ageg.knuba.edu.ua/article/view/363470 https://doi.org/10.32347/0131-579X.2026.110.128-138 |
| Rights: | http://creativecommons.org/licenses/by/4.0 |
| Αριθμός Καταχώρησης: | edsbas.3A115474 |
| Βάση Δεδομένων: | BASE |
| DOI: | 10.32347/0131-579X.2026.110.128-138 |
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