BARBORA POKORN´POKORN´ Collision-free Piecewise Quadratic Spline with Regular Quadratic Obstacles Collision-free Piecewise Quadratic Spline with Regular Quadratic Obstacles

A Pavel, Chalmoviansk´ Chalmoviansk´
unpublished
We classify mutual position of a quadratic Bézier curve and a regular quadric in three dimensional Euclidean space. For given first and last control point, we find the set of all quadratic Bézier curves having no common point with a regular quadric. This system of such quadratic Bézier curves is represented by the set of their admissible middle control points. The spatial problem is reduced to a planar problem where the regular quadric is represented by a conic section. Then, the set of all
more » ... the set of all middle control points is found for each type of conic section separately. The key issue is to find the boundary of this set. It is formed from the middle control points of the Bézier curves touching the given conic section. Our results are applicable in collision-free paths computation for virtual agents where the obstacles are represented or bounded by regular quadrics. Another application can be found in searching for pointwise space-like curves in Minkowski space. Kvadratni splajnovi, po dijelovima bez kolizija, s regularnim kvadratnim barijerama SAŽETAKSAˇSAŽETAK U trodimenzionalnom euklidskom prostoru klasificiramo med-usobni odnos kvadratne Bézierove krivulje i regu-larne kvadrike. Za danu prvu i zadnju kontrolnu točku, nalazimo skup svih kvadratnih Bézierovih krivulja koje ne-maju zajedničku točku s regularnom kvadrikom. Sustav ovakvih kvadratnih Bézierovih krivulja prikazuje se skupom njihovih dopustivih srednjih kontrolnih točaka. Pros-torni problem svodi se na ravninski problem gdje konika predstavlja regularnu kvadriku. Tada se za svaku vrstu konike zasebno nalazi skup svih srednjih kontrolnih točaka. Glavna zada´cazada´ca je na´cina´ci granicu ovakvog skupa. Spomenutu granicučinegranicučine središnje kontrolne točke Bézierovih krivulja koje diraju koniku. Naši rezultati primjenjuju se u računanju putanja bez kolizija za virtualna sredstva gdje su barijere prikazane ili ograničene regularnim kvadrikama. Drugu primjenu nalazimo u istraživanju točkovnih pros-tornih krivulja u prostoru Minkowski.
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