Derivation of the equation of caustics in Cartesian coordinates with Maple
The well-known equation of caustics about crack tips in experimental fracture mechanics is usually expressed in the form of two equations for the Cartesian coordinates (x,y) with a parameter (an appropriate polar angle playing the role of this parameter). Here we derive the equivalent unique equatio...
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| Γλώσσα: | English |
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Pergamon Press (Elsevier Science)
2023
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| Διαθέσιμο Online: | https://hdl.handle.net/10889/26280 https://doi.org/10.1016/0013-7944(94)90151-1 |
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nemertes-10889-262802023-12-04T11:09:00Z Derivation of the equation of caustics in Cartesian coordinates with Maple Ioakimidis, Nikolaos Anastasselou, Eleni The well-known equation of caustics about crack tips in experimental fracture mechanics is usually expressed in the form of two equations for the Cartesian coordinates (x,y) with a parameter (an appropriate polar angle playing the role of this parameter). Here we derive the equivalent unique equation (but without a parameter) also in Cartesian coordinates with the help of the computer algebra system Maple V by using the available Gröbner-bases algorithm. The obtained Cartesian equation is of the sixth degree and it can be solved in closed form with respect to y yielding an explicit result y =y(x). The same equation is also checked in two special cases, where it gives the same results as the equivalent pair of parametric equations. Analogous more general results can also be derived. 2023-11-24T13:14:21Z 2023-11-24T13:14:21Z 1994-05 https://hdl.handle.net/10889/26280 https://doi.org/10.1016/0013-7944(94)90151-1 en application/pdf Pergamon Press (Elsevier Science) |
| institution |
UPatras |
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Nemertes |
| language |
English |
| description |
The well-known equation of caustics about crack tips in experimental fracture mechanics is usually expressed in the form of two equations for the Cartesian coordinates (x,y) with a parameter (an appropriate polar angle playing the role of this parameter). Here we derive the equivalent unique equation (but without a parameter) also in Cartesian coordinates with the help of the computer algebra system Maple V by using the available Gröbner-bases algorithm. The obtained Cartesian equation is of the sixth degree and it can be solved in closed form with respect to y yielding an explicit result y =y(x). The same equation is also checked in two special cases, where it gives the same results as the equivalent pair of parametric equations. Analogous more general results can also be derived. |
| author |
Ioakimidis, Nikolaos Anastasselou, Eleni |
| spellingShingle |
Ioakimidis, Nikolaos Anastasselou, Eleni Derivation of the equation of caustics in Cartesian coordinates with Maple |
| author_facet |
Ioakimidis, Nikolaos Anastasselou, Eleni |
| author_sort |
Ioakimidis, Nikolaos |
| title |
Derivation of the equation of caustics in Cartesian coordinates with Maple |
| title_short |
Derivation of the equation of caustics in Cartesian coordinates with Maple |
| title_full |
Derivation of the equation of caustics in Cartesian coordinates with Maple |
| title_fullStr |
Derivation of the equation of caustics in Cartesian coordinates with Maple |
| title_full_unstemmed |
Derivation of the equation of caustics in Cartesian coordinates with Maple |
| title_sort |
derivation of the equation of caustics in cartesian coordinates with maple |
| publisher |
Pergamon Press (Elsevier Science) |
| publishDate |
2023 |
| url |
https://hdl.handle.net/10889/26280 https://doi.org/10.1016/0013-7944(94)90151-1 |
| work_keys_str_mv |
AT ioakimidisnikolaos derivationoftheequationofcausticsincartesiancoordinateswithmaple AT anastasseloueleni derivationoftheequationofcausticsincartesiancoordinateswithmaple |
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1799945006193049600 |
