Abbiamo visto nella lezione precedente come [b]una successione di potenze di un numero complesso generi una SPIRALE. [/b]Ma una spirale di che tipo?[br]L'argomento sarebbe degno di approndimento, ma si rischierebbe di perderci nei [b]"vortici " delle spirali e [/b]di distogliere la nostra attenzione dal "[b]mondo dei numeri complessi". [br][/b][br][br][b][i]Lasciamo a te, anche in futuro , la possibilità di approfondire l'argomento. [br]Vogliamo solo sottoporre alla tua attenzione le definizioni delle spirali più note in matematica. [br][br][/i][/b][b][i] 1. Spirali a crescita lineare Esempio: Spirale di Archimede[/i][/b][br][list] [*][b][i]La distanza dal centro aumenta “a passi uguali” [/i][/b][/*][*] [b][i]I giri sono equidistanti tra loro [/i][/b][/*][*][b][i]Ha un andamento regolare e uniforme [/i][/b][/*][/list][b][i]È la spirale più semplice da immaginare.[/i][/b][b][i][br][img]data:image/png;base64,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[/img][br][br]2. Spirali a crescita esponenziale : la Spirale logaritmica [/i][/b][br][list] [*][b][i]La distanza cresce sempre più velocemente [/i][/b][/*][*][b][i]I giri si allargano progressivamente [/i][/b][/*][*][b][i]La forma resta sempre simile a sé stessa (autosimile) [/i][/b][/*][/list][b][i]È quella che compare spesso in natura (Nautilus, Nei[br]girasoli, nelle pigne , Uragani e alcune Galassie....) [br][img]data:image/png;base64,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[/img][br][/i][/b][b][i][br]3. Spirali che si avvicinano al centro : Spirale iperbolica [/i][/b][br][br][list][*][b][i]La curva gira avvicinandosi sempre di più al centro [/i][/b][/*][*][b][i]Non lo raggiunge mai [/i][/b][/*][*][b][i]I giri diventano sempre più stretti [/i][/b][/*][/list][img]https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQG4rTgGWZKj2Sk1eYFq9cr0SuwO15UGunD04vzBAG1Qw&s[/img][br][br][b][i]4. Spirali definite da leggi particolari : [/i][/b][list] [*][b][i]Spirale di Fermat [/i][/b][/*][*][b][i]Spirale di Fibonacci [/i][/b][/*][/list][b][i]Qui la crescita segue leggi specifiche (radice,[br]successioni, ecc.).[/i][/b][br][br][img]data:image/jpeg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wCEAAkGBxITEhURExIQFhUXFRoYEhUVFxYWFRgWGRcXGRgWGBoYHiggGBolGxcaITEhJSkrLi4uGCAzODMtNygtLisBCgoKDg0OGxAQGy0mICU3NystNi0vKy0tNy01Ky03Li0tLisuLS0uLS0tLSstLS0rNS0uLS0vKy0vLS0tKystLf/AABEIALcBEwMBIgACEQEDEQH/xAAcAAEAAgMBAQEAAAAAAAAAAAAABAUBAgMGBwj/xABNEAACAQICBQcGCwUFBwUAAAABAgADEQQhBRIxQVETFGFxgZHRBiIykrHBByMzQlJTYnJzobNjgrLh8CRDotLxCBU1dJOjwhY0ZIOE/8QAFwEBAQEBAAAAAAAAAAAAAAAAAAIBA//EACQRAQEAAgEEAgIDAQAAAAAAAAABAhExITJBURLwInFhoeFC/9oADAMBAAIRAxEAPwD7jERAREQEREBERAREQEREBERAREQEREBERAREQEREBERAREQEREBERAREQEREBERASJSxwYBlSqQdhsBccczJciaNUcjSO8U1sf3RA250fq6vcPGOdH6ur3Dxna5G3MWzPiJsDAj86P1dXuHjHOj9XV7h4yTNGqbtp4D38IHHnR+rq9w8YOLttp1O5fGdtUnabdA8f9JlUA3du/vgRuffs6vYoPsM3GJP1dTuHjJEQIzYu393V7FB9hmvPvsVO0Ae+S5q7AAk2A332QOHOj9XV7h4xzo/V1e4eMrzpqk2WHWrXPGiByfbUYin2Ak9EyVxr7Wo0RvCA1qltx1m1VB/daX8L56I+c8dU/nR+rq9w8ZHxWmaVPOodT77IvtacV0IjZ1KuIrcdeoyr2pT1U7xJWE0Th6XydCinSqKD2kDONYm8vSvPlZhvmmq/wCFTar+mDO1PyhptspYztw1ce1JbxG8fX9/4ay9/wBf6qv9+re3I4u9r/Ivs7pyreUtJfSpYwf/AJq5HeEli3yw/DP8SyTM3j6brL2pqPlNhmOqHGt9ElVb1WIP5SwGLP1dXuHjNsVTRhquivf5rAMO2+6Vw8naQ+SL0DwoMUQf/XmjdqzfxrPyiacb+zqdVhf2zdcVf+7q9oA98rxzujup4hPs2pVgOonk3PaklYDStKqSqkhx6VNwUqL0lGzt0jI7jMuN5jZnOHVsXb+7q9ig++YXG32U6ncvjJU1ZQdokqcOdH6ur3DxjnR+rq9w8Z1sRszHA7ew+M2RgdkDhzo/V1e4eM6YeuHvYMCpswIsQbA+wibud3HptlxkfBCzVR+0H6VOBKiIgIiICIiAkXRfyNL8NP4RJUi6L+Rpfhp/CIEqaMu0jb03t3TecvS+7u6enqgYD63EDjvbq6Pz9s6qoGyCJqAR0jO99o6uMDeJhWuL/wApmAmtSoFBZiAALkk2AHEk7JAx+lQjclTU1axFxTU2sDsao2ymvSczY2BOU40tDmoRUxTCqwN1pgWoId1lPpsPpNfiAuyVMfNRcvEa/wC9qlbLC0wy/X1LrR60HpVeyyn6U2TQSsdbEO2IbbZ8qQP2aQ83LcW1m6ZbxN+eu3ofDfd1YVQMhs3TMRIW0ZN4yPHx4wj7jkfyPSJvNXS/9bIG0TRG3Hb7Rxm8CMflh+Gf4lnZ23DadnjOLH40fht/Es7UxvO0+zcIGUS3vPGbREBImkNG0qwAqLcjNGBKuh4o62ZT1GS4my2cMsl6VS85rYb5bWrUfrlX41B+1RRZl+2o612tLejVVlDKQykXVgbgg7CCNom8pa2CfDk1cMushN62HGw32vR3K+8rsboJua6ZftPXH9Lqc6ot52wgZ8COBmmCxaVUFSm2srbDs6CCDmCDkQcwRadL3PVty29H9dEizS5dsUs88r7OwE2BnHCenW/EH6VOdmyN9xyPXuPu7pxwnp1vxB+lTgSoiICIiAiIgJF0X8jS/DT+ESVImjmtQpn9mn8Igd3zOru+d4f1750mtNbDp39c2gIiDA1ZN++1rylq6QeuWp4dgqKdWribAgHYVog5MwORY3VftG4GDUbGEqhZcKDZnFw1cjaqHaKXFhm27LM3VGkqqFUBVAAVQLAAbAANgl9vPLn3ccOGj8BTorq0xa5uzElmZt7OxzZjxMlTTV4cc73PdwmVbs6DJt3yuTXSNoiJjSIiAiIgaVFv1jYZlGuPaOBm05nI33HI9e4+7ugR63yqjih7tZSfZJkjMvxyn9mw72XwkmAiIgIieI0r5G4jEo1ZtIY6jiTdqYo1dTD0jtSnya5OBkGYm7EE5DKB7eJ5b4OtLV8Zo2jWrm1Yh0d1C5lHZOUAI1bnVvste+VspD+DevWZ9IpWr1axp496aNUIuEVEsAFAVepQBe+UC7x9I4dziaSkoc8VSUXuPrkA+eBtA9IdIEtsO6sqsrBlIBVgbgg5g36ZlRc37s92Wcp8L/Zq3I/3NYk0OCVc2ej1MLuvU4+iJfdP5R23+F063FpFwJu1U/tB+lTkuRMEtmrfiXH/AE6chaXERAREQEREBIWjc6VEcKak+qLe/uk2Q9EfI0z9hPyUe+8CZERASjrscW7UlJGHQlazDLlnG2ip+gDkxG0+b9KddK13qOMJSJUka1eoNtOkcrA/WPYgcAC24A2WFw6U0WmihVUAKo2ADYJc/Gb8ud/K68N0UAAAAACwAyAHATaIkOhMFR4TwumNNaWxLMui6OHWihK85xJIFVlJDCio+YCLa5Fm3ZZmh8k/hIxqaQGi9K0Ep1XOrTqILDWIulxchlbYGXfYW22D6tmOkW27+4DObAzMqPKfTVPBYepiXvlYKqi7VHJslNRvZjl/pAt4ng/gj8ra+kMNWr4hqdxiCiBF1Qq6iMF6TdjnPeQEREBNXW4tNogRRUvVXiabHuZb+2SpBOVYdCnuZl/8vZJ0BERASn05Xd74Sg1qrr59QWPIUzcGqftnMIN7C+xWtbsMttunhPFp5AOC7DS2lgXcu9qlIXY2F8qXAAdAAAyED1ei8BTw9Knh6S6tOmioi8FUWF+Jy2755D4O862lRl/xKpcHhqJPRaE0PzWiyCtiKrMxZq1ZhUqsSABfICwAAAtunHye8mxhHruteq/L1TWqhwluUYAErqqCBkMs9kC9kTSuBFak1O5UmxRxtR1N0cdIYA9klxNl1dss3NVB0NjTVpBmGq4JSqv0ainVcDouLg7wQd86YNrtV6Klv+3TkEfE4s7kxCa3QK1IAHtanb/omTMBtq/iC/WaVMzcp16MwvTqmRESVEREBERASJolbUad9uovsElyPo/5Kn9xf4RAkSFpbHcjTLAazkhaSb3qNkq9AvmTuAJ3SbKbBDl67Vz8nSLU8PwL7KtXvHJj7r7mlYzzU5W8TlL0RgOSTzjrVHOvWe1teodp6FAAUDcFAk6ImW7u62TU1CIiY1pSpKqhVAVVACqoAAAFgABsAG6fOdJ6EXSGnKOIQqaWAQCvUBB1q+szpQFturrBjwuRtM93pjRaYmmaTvXUHaaVV6Td6EEjoOU+B+WHkzX8nsTRx2DrVGoNU1SGNjfaaNW2TqyhrG2VuIBgfoljbM7N88Rg1OkKz6Qa/NqC1E0ep2VGsVfFkd6p0Xb50gfCH5W4bXoaPq1xQpYimKuLqNrX5sbgUU1QfPqEFSdyht5EnVPhI0MtE06eMogCmVRQtQAALYADVyEDzv8As3/+wxH/ADR/Spz6wFI2bN9737J8P+AXyqweGw1TD166pVqYr4tCGJbWSmotYEZsLT7nA1Vr+B2zaasvfa194mC1tuzj4jdA3iIgQ3W9boNIj/EPGSqbXGe0ZHrnI/Kj7h9qzo4t5w7RxHjA3iYU3zEzARE1Y8Nvu3mBjad+XcZvMKthYbpmAiJq7W9w4wKnynypCrbOiy1upEPxnfTLjtk7Bnzqv4g/SpzpUoBlZWz1wQ3URa3VaVfklULUAWzYaquftJSpo3+JTK5xRxkuoiJKyIiAiIgJE0SfiKV/oL7BJchaJ+Spj9mh70HgYHLT2JZaYp0zarWYU6R+iSCWf91Azfu23yZg8MtJFpoLKihVHQBYSuwvxuKqVfm0RyNP77ar1W6ctRegq4lvLy6TSMettIiJCyaVagVSzEBVBLE7AALkzeYIvke2BReTvllgMdfm2JpuRtU3R+vUcBiOm1p5f4TaS6SNHRNAh35ZKmLdbEYeioa5Y7FqNfzVOZz3Zy8qfBxolm1zgMPfoBVfVUhfynoNHaOo4dBToUqVJBsWmqot95so29MCFpDyXwVdlathMNVZVCK1SmrsFF7LcjZme+V+M8h9GCm5Gj8FcK1viafA9E9PMMoIIIuDkR0QPiv+z9oLC18JVrVsPQqVExXxbuis62p02GqSLixN59rkLRWiMPhlKYejSoqTrMtNFQFrAXIUbbAd0mwEREDTU4ZZ57/9JkPuOR3A2ztwm0EQIh+XH4R/iElysqawqAqVtqN6QLHJlAtYi+ya4zSRpAGo9O7ZU0VHao522VAST7t9pslvDLZOtWRS2Y7RuPgZlXBy2Hgdv85SUcZjmI/s9BFOzlahD3+7TDgetOVTSmKWmtZ6NA0iAWKOS4vbPVfVBtfc1+AMr4X7U/OfY9CTbOaoN5t0dUqMJjnq6nJ1KZVrm+qxHm5EHWNwwORBsQRaWKLW3tSPUrD/AMjIs0uXaTEhua17K1Lp8xj+euAIWlWPpNTPY3jAkmpuGZ/IdZhE3nM8fcOE4MtYbDSt91v80wDV2a1MHgUb/NnAlyl8l6Wqtcf/ACqxHUXJHtljq1vpUvVb/NIXk9e1e9r84qbMh82XO2ovdFtERIWREQEREBKoYoUcKtY3slAFgNpAQGw4m4t2y1lBihr0cJQz+MamWt9CmvKnsJRV/elYzdTldRYaEwhpUERra9i1U8ajkvUPrMZPmlNuO0ZGbzLd3bZNTRERMaREQEREBERAREQEREBNajWHs6902nPaege3+Q9sCu0riRRtUsWK09VEG16jMioo6Sxt233Sq0joHEGiWU0amIqEHEF3qU1ZBcihSdAWpoG1R0gG9yTJ2k6RbG4YbQlOtUI4sppov6jHrAk/F0GqFClepT1SdYIKZ1gRbVYOptbblYy70x/aJ1y36QfJWuj4VNWk9Eazo1N2LslRXZailyTrWcNY3sRa07DzsMtPzdY0gp87VsCtiy3B48JLw1Faa8mnEkn7TEszE/SLEnrM8f5Lph1YOwoK4puKzMRr31hYHXA1Bq21d2rq8bzJjuWtuWrIt8HQ5LFoFYMtWkzVLAActT5NS4A2aytmNnxYl+c8t2/p6J5jRVbla9OogAprTqKrlUBqO7KdZdXJkRVtrDJixtsnqlFspufjfLMPOuACZiVWD09TqYqrg1WqHpU1qOWQqpVyQNXWzbNTna2W2QtazDLfIzzOkfLSnQdOUw+LFB6gpjFai8hrMbAnztcITkH1dU3BBIIM9PA0U2Nj2H3HplN5JOWpVHPz8RUdfuudZf8ACRJflBiClByvptZKX4jkIn+JgeoGNC4daatTX0UKKOpaNIe6X/yjnJYxESFkREBERASg0X59en+xwadV6xH5gUB60v5Q+SouKr/apoOpKFLL1i8rHi1GXMi6cW84dvSPETcG+czOR83Pdv6OnqkrdYgGICIiAiIgIiICIiAiJq726TuH9boGKjbhtOzxmUWwtMItsztO3w6pvAo9IanPqCuqm+Hr21gDmHw5377XliuAp7eTpjoCr+ZtKzyj8x6WItlQ89/wyQlTuRy/7gl4WyvutfLOXlxKjHmxBqU8MrClbDBjsQhASN9l3zZ9E0Lg8jRuNl0U92WXZPnbaSNPnmh8RTAxWJqVWwtZ2HJ1xWLGnUNT5j0xYavpAU1C3Np9NwlMqiKzFmVQGY7WIABY9J2yNr0ic1piohVFU2ceaAp2AWuP62SbcjbmPz7hI2IuKtPgda/cAP66JMgJ5TB/8bxH/IYf9bET1JThlnc9PXKHDaIqrj6mMapTtUorRFLVNwKbO4bX1rEnXOWqNnRArPhLxNenQWoMNTr4Wk6VcWvKFapSmwcaq6uqQGUMbtmFtaxM9hRqBlDDYwBHURcSgbRGLrUXw2Lr4d6b3Wo1Gk9N3pH0kzqMEuMiRc2JtY+dJ+lcaU1aFEKazi1MW81FGRquB8xeG82G+bJu6ZbqbcmPL4oAfJ4bMnc1dlsB+4jEnpqLvEnYNbNV6al/+3TjRuCWjTWmtza5LH0mYm7Ox3szEk9cxgT51X8TLq5Onabld8MxmuUuIiSoiIgIiICUfkYDzYE7TUqk9lV1H5AS8lV5Lj+zU+kue+ox98udt++0Xun30tYiJC3MqRmO0eHCbI4PvG+bTVkB8d8DaJp5w6fyPgfyjlBvuOvx2QN4mAbzMBETUuBvEDaJprncD25Dx/KY1L7T2DIfzgDUvkufTuHj1TZEt0neZsBEBERAiVbGqFIBBpsDffmuVuq8rtGVDQcYOoTYX5q5OToB8kT9NBu3qAc7NawrD40W26ht6y5xi8HTqpyVRbg57SDrA3DK20MDmCDcGVjZxeE5S8zl4upoUVMDicJjKFapXatXqK6ozF3Z2OHq0qnoqQnJoLkaoSxsJ7PQtGqmHopWfXqrSRar/SqBQHbta5kNauJw+Tq2IpDZUQDl1H7RMhU+8mf2d8k4XTNCpklRS30D5lQfeRrMvaIuF8Eznl1xBvURehr8d1vfJNNuO0bZGdfjKfHzr9wnXEVFXzmZVtvYgDtvukqdolQfKKicqetWbZagOUF/vjzF/eYTBo4qv6bDD0/o0yGrMOBqejT6QoJ4MJfwvnoj5zx1bY7SZVuQoLyta3o3OrTB2PVe/mjgvpHdvI7aK0eKeszMXqvY1ahyJ4KB8xBuXdntJJPfBYKnRTUpqFW9zxJO1mJzZjvJzM6Ob+jt+l/PfMuXTUJjd7raodw2n8hxnDBjzqv4g/SpzsuW3eduZ7+E44T0634g/SpyVpUREBERAREQEi09HUlFlWw3AFgOwAyVECNzFOB9ZvGOYpwPrN4yTECNzFOB9ZvGOYpwPrN4yTECNzFOB9ZvGOYpwPrN4yTECKdH0/o/m3jNhg04H1m8ZIiBGbAUztUnrZvGBgKfA+s3jJMQI3MU4H1m8Y5inA+s3jJMQI3MU4H1m8Y5inA+s3jJMQI3MU4H1m8Y5inA+s3jJMQIv+76d76pva19Zr24Xv0DumTgafA+s3jJMQI4wSWtY2+83jI+K0Jh6g1alFHG4ONYdzSwibLrhlm1UPJzCbsPSHULDuEx/wCmcFcNzWhrDY2oLjt2y2ib88vbPhj6RhgKeyxt95vGFwNMbAfWbxkmJKkZsBTOZU+s3jHMU4H1m8ZJiBG5inBvWbxnWhQVBZQACbnpPE8Z0iAiIgIiICIiAiIgIiICIiAiIgIiICIiAiIgIiICIiAiIgIiICIiAiIgIiICIiAiIgIiIH//2Q==[/img][br]
A livello intuitivo, ma anche riflettendo su [b]come cresce il modulo e su come varia l'angolo [b]delle potenze successive del numero complesso [math]z[/math], [/b][/b]a quale tipo tra quelli elencati, ti sembra appartenere la spirale delle potenze dei numeri complessi? [br]Prova a dare una risposta e se vuoi motivala:
Nella [b]creazione e nella forma di una spirale delle potenze di numeri complessi [/b] scritti in [b]forma esponenziale o [/b] [b]trigonometrica [/b] , [math]z=r\cdot e^{i\alpha}[/math] o [math]z=r\cdot\left(cos\alpha+i\cdot sin\alpha\right)[/math], un [b]ruolo fondamentale[/b] assumono la quantità [math]r[/math], ovvero il[b] modulo [/b]del numero complesso ( la distanza di [math]z[/math] dall'Origine) e l'angolo [math]\alpha[/math], l'[b]argomento[/b] ( ovvero la direzione del vettore corrispondente al numero complesso [math]z[/math]).[br][br]In questa attività guidata ti verrà chiesto di [b]scoprire [/b] come le potenze di numeri complessi generano comportamenti "diversi"[b] (spirali divergenti, convergenti o circonferenze)[/b] e che ruolo hanno i due "protagonisti" principali, modulo [math]r[/math] e argomento [math]\alpha[/math]
La prima applet di Geogebra che ti viene proposta è finalizzata a scoprire il ruolo di [math]r[/math], modulo del numero z[br] [br]1. Apri GeoGebra e inserisci un numero complesso, [math]z_1=r\cdot\left(cos\left(18°\right)+i\cdot sin\left(18°\right)\right)[/math][br] avente un argomento fissato (18°) e modulo variabile [math]r:[/math] [br][br]Automaticamente sarà generato uno slider [math]r,[/math] che devi impostare come numero intero, che varia da 0 a 5 [br][br] 2. Inserisci la retta passante per l'origine e il punto [math]z[/math][list]3. Inserisci la circonferenza avente centro nell'origine e raggio unitario. . .4. Genera le potenze di [math]z_1[/math] attraverso il comando : [math]Successione((z_1)^n,n,1,40,1)[/math] . -Ti ricordiamo che Il comando [b]Successione[/b] di GeoGebra (in inglese [i]Sequence[/i]) serve per [b]costruire una lista di oggetti, ripetendo una stessa regola al [/b][b]variare di un indice[/b]. [/list][list][*] Struttura generale[br] Il comando ha questa forma: [/*][math]Successione(espressione,variabile,valore_{ }iniziale,valore_{ }finale,incremento)[/math][b][br]Successione [/b] significa: [/list][list][list][*]prendi una [b]espressione che contenga una variabile[/b] [/*] [*]fai variare la [b]variabile[/b] (di solito la variabile è un indice, come [math]n)[/math] [/*] [*]dai all'indice un valore iniziale a uno finale [/*] [*]scegli l'incremento della variabile e raccogli tutti gli [math]n[/math] risultati in una [b]lista. [/b][/*][/list][/list][b][br]Vedrai comparire accanto al comando la lista costituita da n valori. [/b][br][list][list] [/list][/list]
Fai variare lo slider[math]r[/math] per capire cosa cambia nella Successione di punti che formano la spirale ( o meglio come si distribuiscono, verso cosa tendono....).[br]Cosa puoi dire in generale?[br][br]Fai una analisi più accurata considerando variando r e descrivi quello che vedi :[br][br][list=1][*] [b] 0<[math]r[/math]<1[/b][/*][*][b] [math]r[/math]=1[/b][/*][*][b] [math]r[/math]>1[/b][/*][/list]Cosa puoi dire? cosa noti al variare di[math]r[/math]?
Avrai osservato che se [math]r=1,[/math] la spirale non si viene a formare. [br][list][*]Che tipo di curva descrive la successione di punti generata dalle potenze del numero, o meglio su quale curca si distriduiscono le potenze del numero complesso [math]z_1=r\cdot\left(cos\left(18°\right)+i\cdot sin\left(18°\right)\right)[/math] ?[/*][*]Sapresti scrivere l'equazione di quella particolare "curva " nel piano complesso e paragonarla all'equazione dellla stessa nel piano cartesiano?[/*][/list]
Caso [math]r\ne1[/math] [br]Leggi con attenzione le due definizioni:[br][list][*][b]SPIRALE CONVERGENTE[/b]: Una spirale si definisce convergente quando, procedendo lungo la curva, la distanza dal polo ( il polo di una spirale è il punto centrale , spesso indicato con [img]data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==[/img] O, attorno al quale la curva si avvolge indefinitamente) diminuisce progressivamente. Le spire si avvolgono su se stesse, restringendosi man mano che ci si avvicina al centro.[/*][/list][list][*][b]SPIRALE DIVERGENTE:[/b] Una spirale si definisce divergente quando, procedendo lungo la curva, la distanza dal polo (centro) aumenta progressivamente. Le spire si allargano sempre di più man mano che si allontanano dal centro.[/*][/list][br]In base a quanto osservato dalla precedente applet di Geogebra, puoi affermare che :
1. SE [math]r=1,[/math] la Spirale .......................................[br]2. SE [math]r<1[/math], la Spirale è .......................................[br]3. SE [math]r>1[/math] la Spirale è ........................................
[b]La spirale non “sceglie” il centro: è forzata dal fatto che: [/b][br][br][img]data:image/png;base64,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[/img][br]ovvero [math]r^n[/math] tende a 0 , essendo [math]r<1[/math][br][br][br]Quindi il [b]centro (punto limite)[/b] è sempre [b]l’origine[/b]. [br][br]Il polo di una spirale logaritmica può essere considerato un [b]attrattore [/b] nel contesto dei sistemi dinamici, quando la spirale viene interpretata come [b]la traiettoria di un punto che si muove verso il centro. [/b]
La spirale, creata da [math]\left(z_1\right)^n=r^n\cdot\left(cos\left(\left(n\cdot18°\right)\right)+i\cdot sin\left(n\cdot18°\right)\right)[/math] , [b]diverge [/b] quando [math]r>1[/math][br]In questo caso: [br][list][*][img]data:image/png;base64,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[/img] al crescere di [math]n[/math][br][/*][*]i punti si allontanano sempre di più[br][/*][/list][br]→ ottieni una [b]spirale che diverge verso infinito[/b][br]
[b]La seconda applet di Geogebra che ti viene proposta è finalizzata a scoprire il ruolo di [/b][math]\alpha[/math][b], argomento del numero [/b][math]z[/math][br] [br]1. Apri GeoGebra e inserisci uno slider [math]\alpha[/math] che devi impostare come [b]angolo[/b], che varia da 0 a 360° [br]2. un numero complesso, [math]z_1=1.1\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math] avente un modulo fissato (1.1 ) e argomento variabile [math]\alpha[/math]:[br]3. Genera le potenze di [math]z_1[/math] attraverso il comando : [math]Successione((z_1)^n,n,1,200,1)[/math] . [br][br] Ti ricordiamo che Il comando Successione di GeoGebra (in inglese [i]Sequence[/i]) serve per costruire una lista di oggetti, ripetendo una stessa regola al variare di un indice. [br][br]4. Genera la successione dei vettori [b]punto origine - punto complesso,[/b] per ogni punto che rappresenta una potenza di[math]z[/math]; a tale scopo introduci: [math]Successione(vettore\left(z_1^n\right),),n,1,200,1)[/math]. Tracciando la successione di vettori che rappresentano i punti complessi, puoi osservare meglio come tra due potenze successive l'angolo [math]\alpha[/math] si mantiene costante (rotazione costante).[br]Nelle impostazioni della successione di vettori , riduci lo spessore della linea e usa il tratteggiato. L'immagine risulterà più chiara. [br][img]data:image/png;base64,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[/img][br][br]4. Visualizza anche la [b][i]Spezzata[/i] (poligonale), [/b]ovvero una figura formata da una sequenza ordinata di segmenti consecutivi non allineati. I segmenti congiungono i punti consecutivi che rappresentano le potenze del numero complesso, quindi [math]z_1[/math][math]z_2[/math] , [math]z_2[/math][math]z_3[/math] , [math]z_3[/math][math]z_4[/math] ...... [math]z_n[/math][math]z_{n+1}[/math]) . [br]Utilizza il comando [math]Spezzata(l1)[/math], Questa poligonale ti aiuterà nelle osservazioni. [br]Nelle impostazioni della spezzata puoi variare il colore, lo spessore della spezzata. [br][br]N.B.[br][i]Il comando [code]Spezzata[/code] in GeoGebra genera una poligonale aperta, ovvero una sequenza di segmenti consecutivi definiti da una lista di punti. Si utilizza digitando [code]Spezzata( l1 )[br][/code]nella barra di inserimento, creando segmenti consecutivi dal primo all'ultimo punto indicato.[/i]
Fai variare lo slider [math]\alpha[/math] , da valori piccoli ( equivale a considerare un numero complesso di partenza con argomento piccolo) a valori via via più grandi.[br][list][*]Come sono distribuiti i punti [math]z_n[/math] ?[/*][*]Osserva anche come cambiano i vettori...................................[br][/*][*]Osserva che più piccolo è l'angolo, più la spezzata .......................................[/*][*]Fai tue cosiderazioni nello spazio risposta[/*][/list]
Ora sari tu a creare una spirale:[br][br]1. puoi scegliere di creare uno slider per[math]r[/math], e lasciare fisso [math]\alpha[/math] [br]2. puoi scegliere di creare uno slider per [math]\alpha[/math] e lasciare fisso r[br]3. puoi scegliere di inserire entrambe le grandezze come slider[br]4. puoi cambiare lìaspetto dei punti, dei vettori , della spezzata. [br]5. anima gli slider e fai le tue osservazioni!!