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أشهر المنحنيات والمجسمات

أشهر المنحنيات والمجسمات

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أشهر المنحنيات والمجسمات

  1. 1. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬1
  2. 2. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬2
  3. 3. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬3 ‫اﻟﺮﺣﻴﻢ‬ ‫اﻟﺮﺣﻤﻦ‬ ‫اﷲ‬ ‫ﺑﺴﻢ‬ ‫و‬ ، ‫اﻟﺮﻳﺎﺿﻴﺎت‬ ‫ﻓﻲ‬ ‫اﻟﺸﻴﻘﺔ‬ ‫و‬ ‫اﻟﺠﺬﺁﺑﺔ‬ ‫اﻟﻤﻮاﺿﻴﻊ‬ ‫ﻣﻦ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫رﺳﻢ‬‫اﻟﺨﻠﻔﻴﺔ‬ ‫اﻟﻰ‬ ‫اﻟﺠﺬﺁﺑﻴﺔ‬ ‫هﺬﻩ‬ ‫ﺗﺮﺟﻊ‬‫و‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻟﺤﺴﺎﺑﻴﺔ‬‫ﺗ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻔﻨﻴﺔ‬ ‫و‬‫ه‬ ‫ﺑﻬﺎ‬ ‫ﺘﻤﺘﻊ‬‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﺬﻩ‬.‫اﻟﺪوال‬ ‫ﺟﻤﻴﻊ‬ ‫رﺳﻢ‬ ‫ﻳﻤﻜﻦ‬)‫اﻟﺘﻮاﺑﻊ‬(‫ﺑﺸ‬ ‫اﻟﺮﻳﺎﺿﻴﺔ‬‫ﻣﻨﺤﻨﻴﺎت‬ ‫ﻜﻞ‬ ‫ﺛﻼﺛﺔ‬ ‫ذات‬ ‫آﺎﻧﺖ‬ ‫إذا‬ ‫أﺣﺠﺎم‬ ‫و‬ ‫ﺳﻄﻮح‬ ‫أو‬ ‫ﻣﻨﺤﻨﻴﺎت‬ ‫ﺑﺸﻜﻞ‬ ‫اﻟﻔﻀﺎء‬ ‫ﻓﻲ‬ ‫و‬ ‫ُﻌﺪﻳﻦ‬‫ﺑ‬ ‫ذات‬ ‫آﺎﻧﺖ‬ ‫إذا‬ ‫اﻟﺼﻔﺤﺔ‬ ‫ﻓﻲ‬ ‫ﻣﺴﻄﺤﺔ‬ ‫أﺑﻌﺎد‬.‫ﻣﺸﺎهﺪة‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫داﻟﺔ‬ ‫و‬ ‫راﺑﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺒﺤﺚ‬ ‫ﺳﻬﻮﻟﺔ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻔﻀﺎﺋﻲ‬ ‫أو‬ ‫اﻟﻤﺴﻄﺢ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺷﻜﻞ‬ ‫ﻳﺴﺎﻋﺪ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻄﺮأ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺘﻐﻴﺮات‬‫اﻟﻤﻘﺎدﻳ‬ ‫ﺗﻐﻴﺮ‬ ‫ﻧﺘﻴﺠﺔ‬ ‫اﻟﺪاﻟﺔ‬ ‫ﻣﺘﻐﻴﺮات‬‫ﻋﺪدﻳﺔ‬ ‫ﻓﻮاﺻﻞ‬ ‫ﻓﻲ‬ ‫اﻟﻌﻼﺋﻢ‬ ‫و‬ ‫ﺮ‬‫ﺣﺘﻰ‬ ‫آﺒﻴﺮة‬ ‫أو‬ ‫ﺻﻐﻴﺮة‬ ‫ﻧﻬﺎ‬ ‫اﻟﻼ‬ ‫ﻓﻲ‬ُ‫ﻣ‬ ‫ﻋﻨﺪ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺳﻠﻮك‬ ‫و‬ ‫اﻟﻌﻈﻤﻰ‬ ‫و‬ ‫اﻟﺼﻐﺮى‬ ‫ّﻢ‬‫ﻴ‬‫اﻟﻘ‬ ‫ﻧﻘﺎط‬ ‫ﻣﺸﺎهﺪة‬ ‫ﻳﻤﻜﻦ‬ ‫آﺬﻟﻚ‬ ، ‫ﻳﺔ‬ّ‫ﻮ‬‫ﺗﻘ‬ ‫و‬ ‫ﻘﺎرﺑﻪ‬‫و‬ ‫ﺳﻪ‬ ‫هﻨﺪﺳﻴﺔ‬ ‫و‬ ‫ﺟﺒﺮﻳﺔ‬ ‫أﺧﺮى‬ ‫أﺳﺎﺳﻴﺔ‬ ‫ﻣﻮاﺿﻴﻊ‬ ‫و‬ ‫ﻃﻮﻟﻪ‬ ‫و‬ ‫ﻣﺴﺎﺣﺘﻪ‬. ‫ﺑﻴﻦ‬ ‫ﻣﻦ‬ّ‫ﺪ‬‫ﻋ‬ ‫ﻳﻤﻜﻦ‬ ‫ﻻ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﺟﻤﻴﻊ‬‫ﺑﻌﺾ‬ ‫ﺗﻮﺟﺪ‬ ‫ﺣﺼﺮهﺎ‬ ‫و‬ ‫هﺎ‬‫و‬ ‫اﻟﺮﻳﺎﺿﻴﺎت‬ ‫ﻓﻲ‬ ‫ﺑﺎﻟﻐﺔ‬ ‫أهﻤﻴﺔ‬ ‫ﻟﻬﺎ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫و‬ ‫ﻣﻬﻤﺔ‬ ‫ﻣﻌﺎدﻻت‬ ‫و‬ ‫ﻣﺴﺎﺋﻞ‬ ‫أﺟﻮﺑﺔ‬ ‫هﻲ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫هﺬﻩ‬ ‫ﺑﻌﺾ‬ ‫دوال‬ ‫و‬ ‫رواﺑﻂ‬ ‫ﻷن‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻟﻌﻠﻮم‬ ‫و‬ ‫اﻟﻔﻠﻚ‬ ‫و‬ ‫اﻟﻔﻴﺰﻳﺎء‬ ‫ذات‬ ‫ﺑﻌﻀﻬﺎ‬‫ﻣﺴﺎﺋﻞ‬ ‫ﺣﻞ‬ ‫وراء‬ ‫اﻟﺴﺒﺐ‬ ‫آﺎﻧﺖ‬ ‫و‬ ، ‫اﻟﺮﻳﺎﺿﻴﺎت‬ ‫ﻋﻠﻤﺎء‬ ‫أﺑﺮز‬ ‫ﻋﻠﻴﻬﺎ‬ ‫ﻋﻤﻞ‬ ‫هﻨﺪﺳﻲ‬ ‫و‬ ‫ﺣﺴﺎﺑﻲ‬ ‫ﻃﺎﺑﻊ‬ ‫رﻳﺎﺿ‬ ‫ﻣﺴﺎﺋﻞ‬ ‫ﻇﻬﻮر‬ ‫و‬ ‫ﻣﺘﻨﻮﻋﺔ‬‫دور‬ ‫ﻟﻌﺐ‬ ‫اﻟﺬي‬ ‫اﻟﻤﻤﺎﺳﺎت‬ ‫ﻣﺘﺴﺎوي‬ ‫آﻤﻨﺤﻨﻲ‬ ‫ﻣﻬﻤﺔ‬ ‫أﺧﺮى‬ ‫ﻴﺔ‬ً‫ا‬‫ﻻ‬ ‫هﻨﺪﺳﺔ‬ ‫ﺑﻨﺎء‬ ‫ﻓﻲ‬ ً‫ﺎ‬‫ﻣﻬﻤ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫آﺎن‬ ‫ﺣﻴﺚ‬ ‫اﻟﻬﺬﻟﻮﻟﻴﺔ‬ ‫آﺎﻟﻬﻨﺪﺳﺔ‬ ‫إﻗﻠﻴﺪﻳﺔ‬)‫ُﻘﺎرﺑﻪ‬‫ﻣ‬ ‫ﺣﻮل‬ ‫دوراﻧﻪ‬ ‫ﻣﻦ‬ ‫اﻟﻨﺎﺗﺞ‬ ‫اﻟﺴﻄﺢ‬ ‫و‬(‫ﻟ‬ ً‫ﺎ‬‫ﻧﻤﻮذﺟ‬‫ﻠ‬‫ﺨﻂ‬)‫و‬ ‫اﻟﺴﻄﺢ‬(‫اﻟﺠﺴ‬ ‫و‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻟﻤﺴﺎﺋﻞ‬ ‫ﻓﻲ‬ ‫اﻟﺴﻠﺴﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫أهﻤﻴﺔ‬ ‫آﺬﻟﻚ‬ ، ‫اﻟﻬﺬﻟﻮﻟﻴﺔ‬ ‫اﻟﻬﻨﺪﺳﺔ‬ ‫ﻓﻲ‬ُ‫ﻤ‬‫اﻟ‬ ‫ﻮر‬‫ﻌﻠﻘﺔ‬.‫ﻣﻨﺤﻨﻴﺎت‬ ‫و‬ ‫ﻣﻨﺤﻨﻴﺎت‬ ‫و‬ ‫ﺣﻠﻬﺎ‬ ‫ﻳﻤﻜﻦ‬ ‫ﻻ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﻌﺎدﻻت‬ ‫ﺟﺬور‬ ‫ﻋﻠﻰ‬ ‫اﻟﺤﺼﻮل‬ ‫و‬ ‫اﻟﺠﺒﺮﻳﺔ‬ ‫اﻟﻤﻌﺎدﻻت‬ ‫ﺣﻞ‬ ‫ﻓﻲ‬ ‫ﺳﺎﻋﺪت‬ ‫أﺧﺮى‬ ‫اﻟﻤﻜﻌﺐ‬ ‫ﺗﻀﻌﻴﻒ‬ ‫و‬ ‫اﻟﺰاوﻳﺔ‬ ‫ﺗﺜﻠﻴﺚ‬ ‫و‬ ‫اﻟﺪاﺋﺮة‬ ‫آﺘﺮﺑﻴﻊ‬ ‫ﻣﻬﻤﺔ‬ ‫هﻨﺪﺳﻴﺔ‬ ‫ﻟﻤﺴﺎﺋﻞ‬ ‫ﺗﻘﺮﻳﺒﻴﺔ‬ ‫ﺣﻠﻮل‬ ‫أﻋﻄﺎء‬ ‫ﻋﻠﻰ‬ ‫ﺳﺎﻋﺪت‬. ّ‫ﻌ‬‫ﺳ‬‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫أﺷﻬﺮ‬ ‫أﺟﻤﻊ‬ ‫أن‬ ‫اﻟﻜﺘﺎب‬ ‫هﺬا‬ ‫ﻓﻲ‬ ‫ﻴﺖ‬‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻷﺷﻜﺎل‬ ‫ﺑﻌﺾ‬ ‫و‬ ‫اﻟﻔﻀﺎﺋﻴﺔ‬ ‫و‬ ‫اﻟﻤﺴﻄﺤﺔ‬‫ﻣﺮﻓﻮﻗﺔ‬ ‫اﻟﻤﻬﻤﺔ‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫ﺧﺼﺎﺋﺼﻬﺎ‬ ‫ﺑﻌﺾ‬ ‫و‬ ‫رﺳﻤﻬﺎ‬ ‫ﻣﻌﺎدﻻت‬ ‫ذآﺮ‬ ‫ﻣﻊ‬ ‫ﺑﺼﻮرة‬.‫ﻋﻠﻰ‬ ‫أﻋﺜﺮ‬ ‫ﻟﻢ‬ ‫اﻷﺷﻜﺎل‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﺑﻌﺾ‬ ‫ﻟﻠﻤﺼﻄﻠﺢ‬ ‫اﻟﻠﻐﻮﻳﺔ‬ ‫اﻟﺘﺮﺟﻤﺔ‬ ‫ﻳﻨﺎﺳﺐ‬ ‫ﺑﺎﻟﻌﺮﺑﻲ‬ ‫أﺳﻢ‬ ‫ﻟﻬﺎ‬ ‫أﻧﺘﺨﺐ‬ ‫أن‬ ‫ﻓﺄﺿﻄﺮﻳﺖ‬ ‫اﻟﻌﺮﺑﻴﺔ‬ ‫اﻟﻠﻐﺔ‬ ‫ﻓﻲ‬ ‫ﻣﻌﺎدﻟﻬﺎ‬ ‫أو‬ ‫ﺗﻌﺮﻳﺒﻬﺎ‬ ‫أو‬ ‫اﻹﻧﺠﻠﻴﺰي‬‫اﻟﻜﻠﻤﺔ‬ ‫ﻟﺘﻠﻚ‬ ‫اﻟﺮﻳﺎﺿﻲ‬ ‫اﻟﻤﻔﻬﻮم‬ ‫ﻳﻨﺎﺳﺐ‬. ‫أﺿﻔﺘﻬﺎ‬ ‫اﻟﻤﺪرﺳﺔ‬ ‫ﻓﻲ‬ ‫دراﺳﺘﻲ‬ ‫أﻳﺎم‬ ‫ﻋﻠﻴﻬﺎ‬ ‫ّهﻨﺖ‬‫ﺮ‬‫ﺑ‬ ّ‫ﺪ‬‫ﻗ‬ ‫آﻨﺖ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﺑﻌﺾ‬ ‫ﻟﻤﻌﺎدﻻت‬ ‫ﺑﺮاهﻴﻦ‬ ‫أوراﻗﻲ‬ ‫ﺑﻴﻦ‬ ‫آﺎﻧﺖ‬ ‫آﻲ‬ ‫ﻟﻠﻜﺘﺎب‬‫ﻳ‬‫ﺑﻌﺾ‬ ‫ﻣﻦ‬ ‫ﻟﻬﺎ‬ ‫اﻟﻮﺻﻮل‬ ‫آﻴﻔﻴﺔ‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﻣﻌﺎدﻻت‬ ‫و‬ ‫رواﺑﻂ‬ ‫إﺳﺘﻨﺘﺎج‬ ‫ﻃﺮﻳﻘﺔ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻘﺎرئ‬ ‫ﺘﻌﺮف‬ ‫اﻟﻔﻴﺰﻳﺎﺋﻴﺔ‬ ‫و‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻟﺨﺼﺎﺋﺺ‬‫اﻟﺘﺤ‬ ‫ﺑﻌﺾ‬ ‫و‬‫ﻮ‬‫اﻟﻤﺜﻠﺜﺎﺗ‬ ‫ﻳﻼت‬‫إﺳﺘﻨﺘﺎج‬ ‫ﻋﻠﻰ‬ ‫ﻳﺴﺎﻋﺪﻩ‬ ‫ﻣﺎ‬ ‫اﻟﻘﺎرئ‬ ‫ﻓﻴﻬﺎ‬ ‫ﻳﺠﺪ‬ ‫أن‬ ‫ﻋﺴﻰ‬ ‫ﻴﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﺳﺎﺋﺮ‬ ‫ﻣﻌﺎدﻻت‬.
  4. 4. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬4 ‫ﻋﻠﻰ‬ ‫أآﺜﺮ‬ ‫ﻟﻠﺘﻌﺮف‬ ‫زﻳﺎرﺗﻬﺎ‬ ‫أو‬ ‫ﻣﺮاﺟﻌﺘﻬﺎ‬ ‫ﻳﻤﻜﻦ‬ ‫ﻣﻌﺘﺒﺮة‬ ‫ﻣﻮاﻗﻊ‬ ‫و‬ ‫ﻣﺮاﺟﻊ‬ ‫ﻣﻦ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﻣﻦ‬ ‫اﻟﻤﺠﻤﻮﻋﺔ‬ ‫هﺬﻩ‬ ‫ﺟﻤﻌﺖ‬ ‫ﺧﺼﺎﺋﺼﻬﺎ‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫هﺬﻩ‬.‫ﺑﺮﺳ‬ ‫ﺗﻘﻮم‬ ‫ﺣﺎﺳﻮﺑﻴﺔ‬ ‫ﺑﺮاﻣﺞ‬ ‫اﻟﻴﻮم‬ ‫ﺗﻮﺟﺪ‬‫و‬ ‫اﻟﻔﻀﺎﺋﻴﺔ‬ ‫و‬ ‫اﻟﻤﺴﻄﺤﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﺟﻤﻴﻊ‬ ‫ﻢ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺧﺼﺎﺋﺺ‬ ‫ﺗﻮﺿﺢ‬ ‫ﻣﺘﺤﺮآﺔ‬ ‫رﺳﻮم‬ ‫ﻣﻊ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫أﺷﻬﺮ‬ ‫ﺑﺮﺳﻢ‬ ‫ﺗﻘﻮم‬ ‫اﻷﻧﺘﺮﻧﻴﺖ‬ ‫ﺷﺒﻜﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻣﺒﺎﺷﺮة‬ ‫ﺑﺮاﻣﺞ‬ ‫و‬ ‫ﺟﺎﻣﻌﺔ‬ ‫و‬ ‫ُﺒﺴﻄﺔ‬‫ﻣ‬ ‫ﺑﺼﻮرة‬ ‫و‬ ‫اﻟﻌﺮﺑﻴﺔ‬ ‫ﺑﺎﻟﻠﻐﺔ‬ ‫و‬ ‫واﺣﺪ‬ ‫ﻣﻠﻒ‬ ‫ﻓﻲ‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻷﺷﻜﺎل‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫هﺬﻩ‬ ‫ﻧﺠﻤﻊ‬ ‫أن‬ ‫ﻟﻜﻦ‬ ‫أﻣﻨﻴﺎ‬ ‫ﻣﻦ‬ ‫آﺎن‬ ‫ﺷﺊ‬ ‫ﻣﻨﺎﺳﺒﺔ‬ ‫آﻴﻔﻴﺔ‬‫ﺗﻲ‬‫ﷲ‬ ‫اﻟﺤﻤﺪ‬ ‫و‬ ‫ﺗﺤﻘﻘﺖ‬ ‫ﻗﺪ‬ ‫و‬.‫اﻟﻤﻮاﺿﻴﻊ‬ ‫آﻬﺬﻩ‬ ‫اﻟﻰ‬ ‫اﻟﺤﺎﺟﺔ‬ ‫ﺑﺄﻣﺲ‬ ‫ﻧﺤﻦ‬ ‫اﻟﻴﻮم‬‫و‬ ‫اﻟﻤﺸﺎرﻳﻊ‬‫ﺗ‬ ‫اﻟﺘﻲ‬‫اﻟﺮﻳﺎﺿﻴﺔ‬ ّ‫ﺺ‬‫ﺑﺎﻷﺧ‬ ‫و‬ ‫اﻟﻌﻠﻮم‬ ‫ﻧﺸﺮ‬ ‫و‬ ‫ﺑﺴﻂ‬ ‫ﻋﻠﻰ‬ ‫ﺴﺎﻋﺪ‬‫هﻲ‬ ‫اﻟﺘﻲ‬‫اﻟ‬‫و‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻟﻌﻠﻮم‬ ‫ﻟﺠﻤﻴﻊ‬ ‫ﻠﺒﻨﺔ‬ ‫اﻟﻌﻤﻠﻴﺔ‬،‫و‬‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫اﻟﺴﻄﻮح‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﻣﻮﺿﻮع‬‫إدراك‬ ‫ﻣﻦ‬ ‫أﺳﺘﻄﻌﻨﺎ‬ ‫ﻟﻤﺎ‬ ‫ﻟﻮﻻﻩ‬ ‫و‬ ‫اﻟﺮﻳﺎﺿﻴﺎت‬ ‫ﻣﻮاﺿﻴﻊ‬ ‫أهﻢ‬ ‫أﺣﺪ‬ ‫اﻟﺪوا‬ ‫ﺗﺠﺴﻴﻢ‬ ‫و‬‫اﻟ‬ ‫و‬ ‫اﻟﺠﺒﺮﻳﺔ‬ ‫اﻟﺘﻮاﺑﻊ‬ ‫و‬ ‫ل‬‫ﻬ‬‫ﻨﺪﺳﻴﺔ‬. ‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬ 6-10-2009
  5. 5. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬5 ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫أﻧﻮاع‬ ‫اﻟﻤﺴﻄﺤﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬)Plane Curves( o‫اﻟﻤﻨﺤﻨﻴﺎت‬‫اﻟﻘﺪﻳﻤﺔ‬)Ancient curves( -‫اﻟﺪاﺋﺮة‬)circle( -‫اﻹهﻠﻴﻠﺞ‬)ellipse( -‫اﻟﻬﺬﻟﻮﻟﻲ‬)hyperpola( -‫اﻟﺸﻠﺠﻤﻲ‬)parapola( o‫اﻟﻜﻼﺳﻴﻜﻴﻪ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬)Classical curves( -‫اﻟﻤﻤﺎﺳﺎت‬ ‫ﻣﺘﺴﺎوي‬)tractrix( -َ‫ﺻ‬ ‫ﻣﻨﺤﻨﻲ‬َ‫ﺪ‬‫ﺑﺎﺳﻜﺎل‬ ‫ﻓﺔ‬)limacon( -‫ُﻠﻮي‬‫ﻜ‬‫اﻟ‬ ‫اﻟﻤﻨﺤﻨﻲ‬)nephroid( -‫دﻳﻜﺎرت‬ ‫ﻣﻨﺤﻨﻲ‬)folium( -‫ﻣﻨﺤﻨﻲ‬‫ذو‬‫اﻟﻌﺮوﺗﻴﻦ‬)leminscat( o‫اﻟﺪورﻳﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬)Cycloid Curves( -‫اﻟﻘﻠﺒﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬)cardioid( -‫ُﻠﻮي‬‫ﻜ‬‫اﻟ‬ ‫اﻟﻤﻨﺤﻨﻲ‬)nephroid( -‫اﻟﺪاﻟﻴﺔ‬ ‫ﻣﻨﺤﻨﻲ‬)delta( -‫دو‬ ‫ﻣﻨﺤﻨﻲ‬‫اﻟﻘﺮن‬ ‫رﺑﺎﻋﻲ‬ ‫ﺗﺤﺘﻲ‬ ‫ﻳﺮي‬)astroid( -‫دوﻳﺮي‬)cycloid( -‫هﺎﻳﺒﻮﺗﺮوآﻮﺋﻴﺪ‬)hypotrochoid( o‫اﻟﺤﺪﻳﺜﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬)Modern Curves( -‫اﻟﺠﻴﺐ‬ ‫ﻣﻨﺤﻨﻲ‬)sine( -‫ﻟﻴﺴﺎﺟ‬ ‫ﻣﻨﺤﻨﻲ‬‫ﻮ‬)Lissajous( -‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻣﻨﺤﻨﻲ‬)catenary( -‫آﻠﻮﺛﻮﺋﻴﺪ‬)clothoid( -‫ﺗﺎآﻨﺪال‬)tacnodal( -‫اﻟﺤﻠﺰوﻧﺎت‬)Spirals( -‫أرﺧﻤﻴﺪس‬ ‫ﺣﻠﺰون‬)Archimedean spiral( -‫اﻟﻠﻮﻏﺎرﻳﺜﻤﻲ‬ ‫اﻟﺤﻠﺰون‬‫اﻟﺰواﻳﺎ‬ ‫ﻣﺘﺴﺎوي‬ ‫ﺣﻠﺰون‬ ‫أو‬)logarithm spiral(
  6. 6. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬6 ‫اﻟﻔﻀﺎﺋﻴﻪ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬)Space Curves( o‫اﻟﻜﻼﺳﻴﻜﻴﻪ‬ ‫اﻟﻔﻀﺎﺋﻴﻪ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬)Classical Space Curves( -‫اﻟﻠﻮﻟﺐ‬)helix( o‫اﻟﻤﻨﺤﻨﻴﺎت‬‫اﻟﺠﺒﺮﻳﻪ‬)Algebraic Curves( o‫ﻣﻨﺤﻨﻴﺎت‬‫اﻟﺘﻔﺎﺿﻠﻴﻪ‬ ‫اﻟﻬﻨﺪﺳﻪ‬)Differential Geometry Curves( o‫ُﻘﺪ‬‫ﻌ‬‫اﻟ‬)Knots( ‫اﻟﺴﻄﻮح‬)surfaces( -‫اﻟﻜﺮة‬)sphere( -‫اﻟﻬﺬﻟﻮﻟﻲ‬ ‫ُﺠﺴﻢ‬‫ﻤ‬‫اﻟ‬)hyperboloid( -‫اﻟﺸﻠﺠﻤﻲ‬ ‫ُﺠﺴﻢ‬‫ﻤ‬‫اﻟ‬)paraboloid( -‫اﻟﻄﺎرة‬)torus( ‫اﻟﻮﺟﻮﻩ‬ ‫أو‬ ‫اﻟﺴﻄﻮح‬ ‫ﻣﺘﻌﺪدات‬)Polyhedron( -‫اﻟﻤﻜﻌﺐ‬)cube( -‫اﻟﺴﻄﻮح‬ ‫رﺑﺎﻋﻲ‬)tetrahedron( -‫اﻟﺴﻄﻮح‬ ‫ﺛﻤﺎﻧﻲ‬)octahedron( -‫اﻟﺴﻄﻮح‬ ‫ﻋﺸﺮي‬ ‫إﺛﻨﺎ‬)dodecahedron( -‫ﻋﺸﺮوﻧﻲ‬ ‫ُﺠﺴﻢ‬‫ﻣ‬)icosahedron(
  7. 7. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬7 ‫ﺑﻌﺾ‬‫اﻹ‬‫و‬ ‫ﺻﻄﻼﺣﺎت‬‫اﻟ‬‫ﻤﻔﺎهﻴﻢ‬‫و‬‫ﺑﻌﺾ‬ ‫إﺛﺒﺎت‬‫ﻣﻌﺎدﻻت‬‫ا‬‫ﻟﻤﻨﺤﻨﻴﺎت‬ Minimal surface ‫أﺻﻐﺮي‬ ‫ﺳﻄﺢ‬-‫ﻳﺘﻼﺷﻰ‬ ‫اﻟﺬي‬ ‫اﻟﺴﻄﺢ‬ ‫هﻮ‬)‫ﺻﻔﺮ‬ ‫ﻳﺴﺎوي‬(‫اﻟﻮﺳﻄﻲ‬ ‫ﺗﻘﻮﺳﻪ‬ ً‫ﺎ‬‫ﺗﻄﺎﺑﻘﻴ‬)mean curvature( Fractals ‫آﺴﻮ‬ ‫ﻣﺠﻤﻮﻋﻪ‬‫رﻳﻪ‬-‫ﺻﺤﻴﺢ‬ ‫ﻏﻴﺮ‬ ‫هﺎوﺳﺪورﻓﻲ‬ ‫ُﻌﺪ‬‫ﺑ‬ ‫ذات‬ ‫ﻣﺠﻤﻮﻋﻪ‬ ‫هﻲ‬.‫ﻧﺴﺘﺒﺪل‬ ‫ﺑﺄن‬ ‫ﻣﻨﺘﻈﻢ‬ ‫ﻣﻀﻠﻊ‬ ‫أي‬ ‫ﻣﻦ‬ ‫ﺗﻜﺴﻴﺮي‬ ‫ﻣﻨﺤﻦ‬ ‫ﺑﻨﺎء‬ ‫ﻳﻤﻜﻦ‬ ‫ﻧﻔﺴﻪ‬ ‫اﻷﺳﻠﻮب‬ ‫ﻧﻜﺮر‬ ‫ﺛﻢ‬ ‫ﺿﻠﻊ‬ ‫ﺑﻜﻞ‬ ‫اﻟﻤﻮﻟﺪ‬. Elliptic Function ‫إهﻠﻴﻠﺠﻴﺔ‬ ‫داﻟﺔ‬-‫ﻣﺘﺴﺎﻣﻴﺔ‬ ‫ﻏﻴﺮ‬ ‫داﻟﺔ‬.‫إهﻠﻴﻠﺠﻴﺔ‬ ‫ﺗﻜﺎﻣﻼت‬ ‫ﻣﻌﻜﻮس‬ ‫هﻲ‬ Cusp ‫ﻗﺮﻧﺔ‬-‫ﻟﻤﻨﺤ‬ ‫ﻓﺮﻋﺎن‬ ‫ﻋﻨﺪهﺎ‬ ‫ﻳﻠﺘﻘﻲ‬ ‫ﻧﻘﻄﺔ‬‫ﻓﺮع‬ ‫ﻟﻜﻞ‬ ‫اﻟﻤﻤﺎس‬ ‫ﻧﻬﺎﻳﺘﺎ‬ ‫ﻋﻨﺪهﺎ‬ ‫ﺗﻨﻄﺒﻖ‬ ‫و‬ ‫ﻦ‬. Asymptotic ‫ُﻘﺎرب‬‫ﻣ‬-‫ﺻﻔﺮ‬ ‫ﻧﺤﻮ‬ ‫ُﻘﺎرب‬‫ﻤ‬‫اﻟ‬ ‫ﻣﺴﺘﻘﻴﻤﻪ‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺑﻴﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ﺗﺴﻌﻰ‬ ‫رﺳﻤﻪ‬ ‫اﻟﻤﻄﻠﻮب‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫آﺎن‬ ‫إذا‬( )y f x=.‫ﻗﻴﻤﺔ‬ ‫ﻓﻴﻪ‬ ‫ﺗﺼﺒﺢ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻨﻘﺎط‬ ‫ﺑﻌﺪد‬( )f x‫ُﻘﺎرب‬‫ﻣ‬ ‫ﻳﻮﺟﺪ‬ ‫ﻧﻬﺎﺋﻴﺔ‬ ‫ﻻ‬‫أي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﻬﺬا‬ 0 lim ( ) x x f x → = ∞‫ُﻘﺎرب‬‫ﻤ‬‫اﻟ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬0y x= ‫اﻟﺼﻮرة‬ ‫ﺑﻬﺬﻩ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬2 1 1 y x = − ‫هﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ُﻘﺎرب‬‫ﻣ‬ ‫ﻣﻌﺎدﻟﺔ‬1y =‫و‬1y = − Cartesian Coordinate ‫ﻣﺘﻌﺎﻣﺪة‬ ‫دﻳﻜﺎرﺗﻴﺔ‬ ‫إﺣﺪاﺛﻴﺎت‬-‫اﻟﻤﺘﻌﺎﻣﺪة‬ ‫اﻟﻤﺤﺎور‬ ‫ﻣﻦ‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫ﻃﻮل‬ ‫ﻋﻠﻰ‬ ‫ﻣﻘﻴﺴﺔ‬ ‫أﺑﻌﺎدهﺎ‬ ‫ﺑﺪﻻﻟﺔ‬ ‫ﻓﻀﺎء‬ ‫ﻓﻲ‬ ‫ﻧﻘﻄﺔ‬ ‫ﻟﺘﻤﺜﻴﻞ‬ ‫ﻣﻨﻈﻮﻣﺔ‬ ‫هﻲ‬ ً‫ﺎ‬‫ﺛﻨﺎﺋﻴ‬.‫اﻟﺼﻔﺤﺔ‬ ‫ﻓﻲ‬ ‫ﻟﻠﻨﻘﻄﺔ‬ ‫ﺗﻜﺘﺐ‬( , )x y‫اﻟﻔﻀﺎء‬ ‫ﻓﻲ‬ ‫و‬( , , )x y z Polar Coordinate ‫ﻗﻄﺒﻴﺔ‬ ‫إﺣﺪاﺛﻴﺎت‬-‫اﻟﻄﻮل‬ ‫ﺑﻮاﺳﻄﺔ‬ ‫ﻣﺴﺘﻮى‬ ‫ﻓﻲ‬ ‫ﻧﻘﻄﺔ‬ ‫ﻣﻮﺿﻊ‬ ‫ﺗﺤﺪد‬ ‫إﺣﺪاﺛﻴﺎت‬ ‫زوج‬r‫اﻟﺰاوﻳﺔ‬ ‫و‬θ‫ﺗﻜﺘﺐ‬ ‫و‬( , )r θ Parametric Equation ‫وﺳﻴﻄﻴﺔ‬ ‫ﻣﻌﺎدﻻت‬–‫ﺻﺮﻳﺤﺔ‬ ‫آﺪوال‬ ‫آﻤﻴﺎت‬ ‫ﻣﻦ‬ ‫ﻋﺪد‬ ‫ﻋﻦ‬ ‫ﺗﻌﺒﺮ‬ ‫ﻣﻌﺎدﻻت‬ ‫ﻣﺠﻤﻮﻋﺔ‬ cos sin x R t y R t =⎧ ⎨ =⎩ ‫اﻟﺼﻔﺤﺔ‬ ‫ﻓﻲ‬ cos sin x R t y R t z bt =⎧ ⎪ =⎨ ⎪ =⎩ ‫اﻟﻔﻀﺎء‬ ‫ﻓﻲ‬
  8. 8. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬8 Symmetry ‫ﺗ‬‫ﻨﺎﻇﺮ‬–‫ﻣﺴﺘﻮي‬ ‫أو‬ ‫ﺗﻨﺎﻇﺮ‬ ‫ﻣﺮآﺰ‬ ‫أو‬ ‫ﺗﻨﺎﻇﺮ‬ ‫ﻣﺤﻮر‬ ‫ﺣﻮل‬ ً‫ا‬‫ﻣﺘﻨﺎﻇﺮ‬ ‫ﺑﻜﻮﻧﻪ‬ ‫هﻨﺪﺳﻲ‬ ‫ﺗﺸﻜﻴﻞ‬ ‫ﺧﺎﺻﻴﺔ‬)‫ﺻﻔﺤﺔ‬(‫ﺗﻨﺎﻇﺮ‬. Squircle ‫ﻣﺮﺑﻌﺔ‬ ‫داﺋﺮة‬–‫اﻟﻔﺎﺋﻖ‬ ‫اﻹهﻠﻴﻠﺞ‬ ‫ﻣﻦ‬ ‫ﺣﺎﻟﺔ‬ ‫هﻮ‬ ‫و‬ ، ‫اﻟﻤﺮﺑﻊ‬ ‫و‬ ‫اﻟﺪاﺋﺮة‬ ‫ﺑﻴﻦ‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬ ‫ﺧﻮاﺻﻪ‬ ‫ﻣﻨﺤﻨﻲ‬)superellipse(‫اﻟﺪاﺋﺮة‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫اﻟﻤﺮﺑﻌﺔ‬4 4 4 ( ) ( )x a y b R− + − = Hippopede ‫اﻟﻔﺮس‬ ‫ﻣﺮﺑﻂ‬–‫ﺗﻌﻨﻲ‬ ‫ﻳﻮﻧﺎﻧﻴﺔ‬ ‫اﻟﻜﻠﻤﺔ‬horse fetter‫ﻣﻌﺎدﻟﺘﻪ‬ ‫ﻣﺴﻄﺢ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫هﻲ‬ ، ‫اﻟﻔﺮس‬ ‫ﻣﺮﺑﻂ‬ ‫أﺳﻢ‬ ‫ﻟﻬﺎ‬ ‫أﻧﺘﺨﺒﺖ‬ 2 2 2 2 2 ( )x y cx dy+ = + Genus ‫ﻧﻮع‬–‫ﻟﺘﺮاﺑﻂ‬ ‫ﻗﻴﺎس‬ Surface of Revolution ‫اﻟﺪوراﻧﻲ‬ ‫اﻟﺴﻄﺢ‬-‫ﻣﺴ‬ ‫ﻣﻨﺤﻦ‬ ‫دوران‬ ‫ﻣﻦ‬ ‫اﻟﻨﺎﺗﺞ‬ ‫اﻟﺴﻄﺢ‬ ‫هﻮ‬‫ﻣﺤﻮر‬ ‫ﺣﻮل‬ ‫ﻄﺢ‬.‫هﻲ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﻣﻦ‬ ‫اﻟﻨﻮع‬ ‫هﺬا‬ ‫ﺑﻬﺎ‬ ‫ﻳﺮﺳﻢ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺪاﻟﺔ‬: ( ) cos ( ) sin ( ) x f u y f u z h u ν ν = = = ⎧ ⎪ ⎨ ⎪⎩ ‫اﻟﻤﺘﻐﻴﺮات‬ ‫هﺬﻩ‬ ‫أو‬ ( ) cos ( ) sin ( ) x f y f z h θ φ θ φ θ = = = ⎧ ⎪ ⎨ ⎪⎩ ‫ﻣﺜﻞ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫دوران‬ ‫ﻣﻦ‬ ‫اﻟﻨﺎﺗﺞ‬ ‫اﻟﺤﺠﻢ‬ ‫و‬ ‫اﻟﺴﻄﺢ‬( )y f x=‫و‬a x b≤ ≤‫ﻣﺤﻮر‬ ‫ﺣﻮل‬x‫ﻳﺴﺎو‬‫ي‬: ‫اﻟﺴﻄﺢ‬2 2 1 [ ( )] b a A y f x dxπ ′= + ∫ ‫اﻟﺤﺠﻢ‬2 [ ( )] b a V f x dxπ= ∫ ‫ﻧﻮع‬3‫ﻧﻮع‬2‫ﻧﻮع‬1‫ﻧﻮع‬0
  9. 9. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬9 ‫ﺑﺼﻮرة‬ ‫داﻟﺘﻪ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﺑﻴﻦ‬ ‫اﻟﻤﺴﺎﺣﺔ‬( )y f x=‫اﻟﻔﺎﺻﻠﺔ‬ ‫ﻓﻲ‬ ‫اﻷﻓﻘﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫و‬a x b≤ ≤‫ﺗﺴﺎوي‬:( ) b a A f x dx= ∫ ‫ﺑﺼﻮرة‬ ‫ﻣﻌﺎدﻟﺘﻪ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻦ‬ ‫ﻗﻄﻌﺔ‬ ‫ﻃﻮل‬( )y f x=‫اﻟﻔﺎﺻﻠﺔ‬ ‫ﻓﻲ‬a x b≤ ≤‫ﺗﺴﺎوي‬:2 1 [ ( )] b a L f x dx′= + ∫ ‫ﺑﺼﻮرة‬ ‫ﻣﻌﺎدﻟﺘﻪ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻦ‬ ‫ﻗﻄﻌﺔ‬ ‫ﻃﻮل‬ ( ) ( ) x t y t ⎧ ⎨ ⎩ ‫اﻟﻔﺎﺻﻠﺔ‬ ‫ﻓﻲ‬a t b≤ ≤‫ﺗﺴﺎوي‬:2 2 [ ( )] [ ( )] b a L x t y t′ ′= + ∫ ‫اﻟﻘﻄﺒﻴﺔ‬ ‫ﻣﻌﺎدﻟﺘﻪ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻦ‬ ‫ﻗﻄﻌﺔ‬ ‫ﻃﻮل‬( )r f θ=‫اﻟﻔﺎﺻﻠﺔ‬ ‫ﻓﻲ‬a bθ≤ ≤‫ﺗﺴﺎوي‬:2 2 ( ) b a dr L r d d θ θ = + ∫ Pedal curve ‫اﻷﻋﻤﺪة‬ ‫ﻣﻮاﻗﻊ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫أو‬ ‫اﻟﻘﺪﻣﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬–‫اﻟﻌ‬ ‫ﻟﻘﺪم‬ ‫اﻟﻬﻨﺪﺳﻲ‬ ‫اﻟﻤﺤﻞ‬ ‫هﻮ‬‫ﻣﻌﻠﻮم‬ ‫ﻟﻤﻨﺤﻦ‬ ‫ﻣﺘﻐﻴﺮ‬ ‫ﻣﻤﺎس‬ ‫ﻋﻠﻰ‬ ‫ﺛﺎﺑﺘﺔ‬ ‫ﻧﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﻤﻮد‬. ‫ُﻌﻄﻰ‬‫ﻣ‬ ‫ﻣﻨﺤﻨﻲ‬)‫اﻟﺸﻜﻞ‬ ‫هﺬا‬ ‫ﻓﻲ‬ ‫اﻟﺪاﺋﺮة‬ ‫ﻣﺜﻞ‬(‫ﻣﺜﻞ‬ ‫ﺛﺎﺑﺘﺔ‬ ‫ﻧﻘﻄﺔ‬ ‫و‬O‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻌﻤﻮد‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﻤﺎس‬ ‫اﻟﺨﻂ‬ ،P‫اﻟﺨﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻣﻦ‬ ‫اﻟﻤﺎر‬O‫اﻟﻨﻘﻄﺔ‬ ،P‫اﻟﻘﺪﻣﻲ‬ ‫ﻟﻠﻤﻨﺤﻨﻲ‬ ‫اﻟﻬﻨﺪﺳﻲ‬ ‫اﻟﻤﺤﻞ‬ ‫هﻲ‬ ‫و‬ ‫أآﺜﺮ‬ ‫ﻻ‬ ‫واﺣﺪة‬. ‫اﻟﻘﺪم‬ ‫ﻧﻘﻄﺔ‬ ‫آﺎﻧﺖ‬ ‫إذا‬)pedal point(‫هﻲ‬0 0( , )x y‫ُﻌﻄﻰ‬‫ﻤ‬‫اﻟ‬ ‫ﻟﻠﻤﻨﺤﻨﻲ‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫و‬ ( ) ( ) x f t y g t =⎧ ⎪ ⎨ ⎪ =⎩ ‫هﻲ‬ ‫اﻟﻘﺪﻣﻲ‬ ‫ﻟﻠﻤﻨﺤﻨﻲ‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬: 2 2 0 0 2 2 2 2 0 0 2 2 ( ) ( ) P P x f fg y g f g x f g y g gf x f f g y f g ′ ′ ′ ′⎧ + + − =⎪ ′ ′+⎪⎪ ⎨ ⎪ ′ ′ ′ ′+ + −⎪ = ′ ′+⎪⎩ Critical Points ‫اﻟﺤﺮﺟﺔ‬ ‫اﻟﻨﻘﺎط‬–‫ﻣﺜﻞ‬‫ﻧﻘﺎط‬‫اﻟﻨﻬﺎﻳﺔ‬‫اﻟﻌﻈﻤﻰ‬)maximum(‫اﻟﻨﻬﺎﻳﺔ‬ ‫و‬‫اﻟﺼﻐﺮى‬)minimum(‫ﻟﻠﻤﻨﺤﻨﻲ‬‫اﻟﻨﻘﺎط‬ ‫هﺬﻩ‬ ‫ﻋﻠﻰ‬ ‫ﻧﺤﺼﻞ‬ ، ‫اﻟﺠﺬور‬ ‫ﻋﻼﺋﻢ‬ ‫ﻓﻲ‬ ‫اﻟﺒﺤﺚ‬ ‫و‬ ‫اﻟﺤﺎﺻﻠﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺟﺬور‬ ‫ﺣﺴﺎب‬ ‫ﺛﻢ‬ ‫ﻣﻦ‬ ‫اﻟﺼﻔﺮ‬ ‫ﻣﻊ‬ ‫ﻟﻠﺪاﻟﺔ‬ ‫اﻷول‬ ‫اﻹﺷﺘﻘﺎق‬ ‫ﺗﺴﺎوي‬ ‫ﻣﻦ‬. ‫ﻣﺜﺎل‬:5 ( ) 5 3f x x x= − +‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫آﻤﺎ‬ ‫اﻟﺼﻐﺮى‬ ‫و‬ ‫اﻟﻌﻈﻤﻰ‬ ‫اﻟﻨﻘﺎط‬ ‫اﻷول‬ ‫اﻹﺷﺘﻘﺎق‬ ‫آﺎن‬ ‫إذا‬‫ﺣﺮﺟﺔ‬ ‫ﻧﻘﻄﺔ‬ ‫آﺬﻟﻚ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫هﺬﻩ‬ ‫ﺗﻌﺘﺒﺮ‬ ، ‫ﻻﻧﻬﺎﺋﻲ‬ ‫ﻟﻠﺪاﻟﺔ‬. f ′‫و‬g ′‫إﺷﺘﻘﺎق‬f‫و‬g‫اﻟﻰ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬t
  10. 10. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬10 Extremum ‫ﻗﺼﻮى‬ ‫ﻧﻬﺎﻳﺔ‬–‫ﻣﺤﻠﻴﺔ‬ ‫ﺗﻜﻮن‬ ‫اﻟﺘﻲ‬ ‫و‬ ‫ﺻﻐﺮى‬ ‫ﻧﻬﺎﻳﺔ‬ ‫أو‬ ‫ﻋﻈﻤﻰ‬ ‫ﻧﻬﺎﻳﺔ‬ ‫ﻋﻨﺪهﺎ‬ ‫ﻟﺪاﻟﺔ‬ ‫ﺗﻜﻮن‬ ‫ﻧﻘﻄﺔ‬ ‫هﻲ‬)local(‫ﺷﺎﻣﻠﺔ‬ ‫أو‬)global( Saddle Point ‫ﺳﺮﺟﻴﺔ‬ ‫ﻧﻘﻄﺔ‬–‫ﻧﻬﺎ‬ ‫و‬ ، ٍ‫ﻮ‬‫ﻣﺴﺘ‬ ‫ﻣﺴﺘﻌﺮض‬ ‫ﻣﻘﻄﻊ‬ ‫ﻓﻲ‬ ‫ﻋﻈﻤﻰ‬ ‫ﻧﻬﺎﻳﺔ‬ ‫ﺗﻜﻮن‬ ، ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ٍ‫ﻮ‬‫ﻣﺴﺘ‬ ‫ﻣﺴﺘﻌﺮض‬ ‫ﻣﻘﻄﻊ‬ ‫ﻓﻲ‬ ‫ﺻﻐﺮى‬ ‫ﻳﺔ‬ ‫ﺁﺧﺮ‬. ً‫ﻼ‬‫ﻣﺜ‬‫اﻟﻤﻨﺤﻦ‬ ‫هﺬا‬ ‫ﻣﻌﺎدﻟﺔ‬4 4 z x y= − Convex and Concave ‫اﻟﺘﻘﻌﺮ‬ ‫و‬ ‫اﻟﺘﺤﺪب‬-‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻳﻌﺘﺒﺮ‬0x‫ﻣﺤﺪب‬)convex(‫ﻣﻦ‬ ‫أآﺒﺮ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﺪاﻟﺔ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫اﻹﺷﺘﻘﺎق‬ ‫آﺎن‬ ‫إذا‬ ‫اﻟ‬‫ﺼﻔﺮ‬‫أي‬:0( ) 0f x′′ > ‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻳﻌﺘﺒﺮ‬0x‫ﻣﻘﻌﺮ‬)concave(‫اﻟﺼﻔﺮ‬ ‫ﻣﻦ‬ ‫أﺻﻐﺮ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﺪاﻟﺔ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫اﻹﺷﺘﻘﺎق‬ ‫آﺎن‬ ‫إذا‬‫أي‬: 0( ) 0f x′′ < Point of Inflection ‫ﻧﻘﻄﺔ‬‫إﻧﻌﻄﺎف‬–‫ﻳﺘﻐﻴﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻨﻘﻄﺔ‬‫ﺑﺎﻟﻌﻜﺲ‬ ‫أو‬ ‫ﺗﻘﻌﺮ‬ ‫اﻟﻰ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺗﺤﺪب‬ ‫ﻓﻴﻬﺎ‬‫إﻧﻌﻄﺎف‬ ‫ﺑﻨﻘﻄﺔ‬ ‫ﺗﻌﺮف‬‫اﻹﺷﺘﻘﺎق‬ ‫ﺗﺴﺎوي‬ ‫ﻣﻦ‬ ‫ﺗﺤﺴﺐ‬ ‫و‬ ‫أي‬ ‫اﻟﺼﻔﺮ‬ ‫ﻣﻊ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﺪاﻟﺔ‬ ‫اﻟﺜﺎﻧﻲ‬:0( ) 0f x′′ = Tangent line ‫اﻟﻤﻤﺎس‬ ‫ﺧﻂ‬–‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﻤﺎس‬ ‫ﺧﻂ‬ ‫ﻣﻌﺎدﻟﺔ‬( )y f x=‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬0 0( , )x y‫هﻲ‬:0 0( )y m x x y= − +‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬0( )m f x′=‫ﺟﻬﺘﻬﺎ‬ ‫اﻟﻤﻮﺟﺒﺔ‬ ‫اﻟﺰاوﻳﺔ‬ ، ‫اﻷﻓﻘﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫و‬ ‫اﻟﻨﻘﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﻤﺎس‬ ‫ﺧﻂ‬ ‫ﺑﻴﻦ‬ ‫اﻟﺰاوﻳﺔ‬ ‫ﻇﻞ‬ ‫اﻟﺴﺎﻋﺔ‬ ‫ﻋﻘﺎرب‬ ‫ﺧﻼف‬. Ascending and Descending Function ‫اﻟﺘ‬ ‫و‬ ‫اﻟﺘﺼﺎﻋﺪﻳﺔ‬ ‫اﻟﺪاﻟﺔ‬‫ﻨﺎزﻟﻴﺔ‬–‫ﺗﺼﺎﻋﺪﻳﺔ‬ ‫اﻟﺪاﻟﺔ‬ ‫اﻟﺼﻔﺮ‬ ‫ﻣﻦ‬ ‫أآﺒﺮ‬ ‫ﻟﺪاﻟﺔ‬ ‫اﻷول‬ ‫اﻹﺷﺘﻘﺎق‬ ‫ﻗﻴﻤﺔ‬ ‫آﺎﻧﺖ‬ ‫إذا‬)ascending(‫آﺎﻧﺖ‬ ‫إذا‬ ‫و‬ ‫ﺗﻨﺎزﻟﻴﺔ‬ ‫اﻟﺼﻔﺮ‬ ‫ﻣﻦ‬ ‫أﺻﻐﺮ‬)descending. (‫ﻣﺜﻞ‬ ‫داﻟﺔ‬ ‫ﻟﻜﻞ‬ ‫أي‬( )y f x=‫اﻟﻔﺎﺻﻠﺔ‬ ‫ﻓﻲ‬a x b≤ ≤ ( ) 0f x′ >‫ﺗﺼﺎﻋﺪ‬ ‫اﻟﺪاﻟﺔ‬‫اﻟﻔﺎﺻﻠﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫ﻳﺔ‬ ( ) 0f x′ <‫اﻟﻔﺎﺻﻠﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫ﺗﻨﺎزﻟﻴﺔ‬ ‫اﻟﺪاﻟﺔ‬
  11. 11. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬11 Curve Curvature ‫إﻧﺤﻨﺎء‬)‫ّس‬‫ﻮ‬‫ﺗﻘ‬(‫اﻟﻤﻨﺤﻨﻲ‬–‫اﻟﻘﻮس‬ ‫ﻃﻮل‬ ‫اﻟﻰ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫ﻟﻤﻨﺤﻦ‬ ‫ﻣﻤﺎس‬ ‫إﻧﺤﻨﺎء‬ ‫ﻓﻲ‬ ‫اﻟﺘﻐﻴﺮ‬ ‫ﻣﻌﺪل‬. ‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﻤﺎس‬ ‫ﺧﻄﻲ‬ ‫ﺑﻴﻦ‬ ‫اﻟﺰاوﻳﺔ‬M‫اﻟﻨﻘﻄﺔ‬ ‫و‬N‫هﻲ‬α‫ﺑﻴﻦ‬ ‫اﻟﻘﻮس‬ ‫ﻃﻮل‬ ‫و‬M‫و‬N‫ﻳﺴﺎوي‬SΔ‫إﻧﺤﻨﺎء‬ ، ‫اﻟﺰاوﻳﺔ‬ ‫ﺗﻐﻴﺮات‬ ‫ﻧﺴﺒﺔ‬ ‫ﻧﻬﺎﻳﺔ‬ ‫هﻮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬α‫ﺗﺴﻌﻰ‬ ‫ﻋﻨﺪﻣﺎ‬ ‫اﻟﻘﻮس‬ ‫ﻃﻮل‬ ‫ﺗﻐﻴﺮات‬ ‫اﻟﻰ‬ N‫ﻧﺤﻮ‬M‫أي‬: 0 lim S S α κ Δ → Δ = Δ ‫ﻣﻨﻬﺎ‬ ‫و‬ d ds α κ = ‫ﺑﺼﻴﻐﺔ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫آﺎﻧﺖ‬ ‫إذا‬( )y f x=‫ﻳﺤﺴﺐ‬ ‫ﻧﻘﻄﺔ‬ ‫آﻞ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ّس‬‫ﻮ‬‫ﺗﻘ‬‫اﻟﺮاﺑﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻣﻦ‬ 2 3 [1 ] y y κ ′′ = ′+ ‫ﺑﺼﻴﻐﺔ‬ ‫اﻟﻘﻄﺒﻴﺔ‬ ‫اﻹﺣﺪاﺛﻴﺎت‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫آﺎﻧﺖ‬ ‫إذا‬( )r f θ=‫اﻟﺸﻜﻞ‬ ‫ﺑﻬﺬا‬ ‫ّس‬‫ﻮ‬‫اﻟﺘﻘ‬ ‫راﺑﻄﺔ‬ 2 2 2 2 3 2 [ ] r r rr r r θ θθ θ κ + − = + ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫اﻟﺮاﺑﻄﺔ‬rθ‫اﻟﻰ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫أوﻟﻰ‬ ‫رﺗﺒﺔ‬ ‫إﺷﺘﻘﺎق‬θ‫و‬rθθ‫اﻟﻰ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫ﺛﺎﻧﻴﺔ‬ ‫رﺗﺒﺔ‬ ‫إﺷﺘﻘﺎق‬θ. ‫ﺑﺼﻴﻐﺔ‬ ‫وﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫آﺎﻧﺖ‬ ‫إذا‬ ( ) ( ) x t y t ⎧ ⎨ ⎩ ‫هﻲ‬ ‫اﻟﺘﻘﻮس‬ ‫راﺑﻄﺔ‬ 2 2 3 [ ] x y y x x y κ ′ ′′ ′ ′′− = ′ ′+ ‫اﻟﺮاﺑﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬x ′‫و‬y ′ ‫اﻟﻰ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫اﻷول‬ ‫اﻟﺘﻔﺎﺿﻞ‬t‫و‬x ′′‫و‬y ′′‫اﻟﻰ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫اﻟﺘﻔﺎﺿﻞ‬t. ‫ﻳﺴﺎوي‬ ‫اﻟﺘﻘﻮس‬ ‫ﻗﻄﺮ‬ ‫ﻧﺼﻒ‬ 1 R κ = Bezier Curves ‫ﺑﻴﺰﻳﻴﺮ‬ ‫ﻣﻨﺤﻨﻴﺎت‬–‫اﻟﻔﺮﻧﺴﻲ‬ ‫اﻟﻤﻬﻨﺪس‬ ‫أﺳﺘﺨﺪم‬Pierre Bezier‫اﻟﺴﻴﺎرات‬ ‫ﺑﺪﻧﺔ‬ ‫ﻟﺘﺼﻤﻴﻢ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫هﺬﻩ‬.‫ﺑﺎﻟﻐﺔ‬ ‫أهﻤﻴﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﻟﻬﺬﻩ‬ ‫آﺎﻟﻔﻮﺗﻮﺷﻮب‬ ‫اﻟﺠﺮاﻓﻴﻚ‬ ‫ﺑﺮاﻣﺞ‬ ‫ﻣﻌﻈﻢ‬ ‫ﻓﻲ‬)Photoshope(‫إﻟﺴﺘﺮﻳﺘﻮر‬ ‫أدوب‬ ‫و‬)Adobe Illustrator(‫و‬ ‫اﻟﻤﺤﺎآﺎة‬ ‫ﺑﺮاﻣﺞ‬ ‫ﻓﻲ‬ ‫و‬ ‫اﻟﺤﺮآﺔ‬)Animation(‫اﻟﺤﺮآﺔ‬ ‫ﻓﻲ‬ ‫ﻟﻠﺘﺤﻜﻢ‬ ‫آﺄداة‬. ‫اﻟﺨﻄﻴﺔ‬ ‫ﺑﻴﺰﻳﻴﺮ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬:‫ا‬‫ﻟﻨﻘﺎط‬0P‫و‬1P‫ﺑﻴﻦ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﺧﻂ‬ ‫هﻮ‬ ‫ﺑﻴﺰﻳﻴﺮاﻟﺨﻄﻲ‬ ‫ﻣﻨﺤﻨﻲ‬ ، ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫و‬ ‫اﻟﻨﻔﻄﺘﻴﻦ‬ ‫هﺬﻩ‬: 0 1( ) (1 )B t t P tP= − + ، [0,1]t ∈ ‫اﻟﺘﺮﺑﻴﻌﻲ‬ ‫ﺑﻴﺰﻳﻴﺮ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬: 2 2 0 1 2( ) (1 ) 2(1 )B t t P t tP t P= − + − + ، [0,1]t ∈ ‫اﻟﺘﻜﻌﻴﺒﻴﺔ‬ ‫ﺑﻴﺰﻳﻴﺮ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬: [0,1]t ∈،3 2 2 3 0 1 2 3( ) (1 ) 3(1 ) 3(1 )B t t P t tP t t P t P= − + − + − +
  12. 12. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬12 ‫اﻟﻤﻤﺎﺳﺎت‬ ‫ﻟﻤﺘﺴﺎوي‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬Tractrix ‫اﻟﻤ‬ ‫هﺬا‬ ‫ﻓﻲ‬‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫أي‬ ‫ﺑﻴﻦ‬ ‫اﻟﻤﻤﺎس‬ ‫ﻗﻄﻌﺔ‬ ‫ﻃﻮل‬ ‫ﻨﺤﻨﻲ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﺜﻞ‬ ‫اﻟﻤﻨﺤﻨﻲ‬P‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻘﺎرب‬ ‫و‬)‫اﻟﺸﻜﻞ‬ ‫هﺬا‬ ‫ﻓﻲ‬ ‫ﻣﺤﻮر‬x(‫ﻳﺴﺎوي‬ ‫و‬ ‫ﺛﺎﺑﺖ‬a.‫اﻟﺰاوﻳﺔ‬ ‫ﻧﻔﺮض‬t‫ﻣﺤﻮر‬ ‫ﺑﻴﻦ‬ x‫اﻟﻤﺘﺠﻬﺔ‬ ‫و‬QP → ‫اﻟﺠﻬﺔ‬ ‫ﻣﻌﻴﻨﺔ‬. ‫ﻣﺜﻞ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻟﺪﻳﻨﺎ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻟﻠﺘﻮﺿﻴﺢ‬( )r t‫إﺳﻘﺎط‬‫ﻣﺜﻞ‬ ‫ﻣﺘﺠﻬﺔ‬r‫هﻮ‬ ‫اﻹﺣﺪاﺛﻲ‬ ‫ﻣﺤﻮري‬ ‫ﻋﻠﻰ‬: 2 2 ( ) cos [ ( )] [ ( )] x t x t y t α = + ، 2 2 ( ) sin [ ( )] [ ( )] y t x t y t α = + ‫ﻧﺴﺘﻌﻤﻞ‬ ‫ﻟﻺﺧﺘﺼﺎر‬x‫و‬y‫ﻋﻦ‬ ً‫ﺎ‬‫ﻋﻮﺿ‬( )x t‫و‬( )y t ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬PHQsinPHQ y a tΔ ⇒ = ‫اﻟﺮاﺑﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻣﻦ‬ 2 2 sin y t x y = + ‫ﻋﻠﻰ‬ ‫ﻧﺤﺼﻞ‬ ‫ﻃﺮﻓﻴﻬﺎ‬ ‫ﺗﺮﺑﻴﻊ‬ ‫و‬: 2 2 2 2 2 2 2 2 2 2 2 2 (1 sin ) sin sin cot sin y t x t y t y x x y t t − + = ⇒ = ⇒ = 2 cos cot cos cot sin sin sin t a x y t x a t t x a x a t t t = ⇒ = ⇒ = ⇒ = − + ‫ﻋﻠﻰ‬ ‫ﻧﺤﺼﻞ‬ ‫اﻷﺧﻴﺮة‬ ‫اﻟﺮاﺑﻄﺔ‬ ‫ﺗﻜﺎﻣﻞ‬ ‫ﻣﻦ‬: cos cos ln tan sin 2 dt t x a t a x a t a c t = + ⇒ = + + ∫ ‫ﻟﻮﺿﻌﻴﺔ‬0x =‫أو‬ 2 t π = cos ln tan 2 t x a t a= + ‫اﻟﻤﻌﺎدﻟ‬ ‫إذن‬‫هﻲ‬ ‫اﻟﻤﻤﺎﺳﺎت‬ ‫ﻟﻤﺘﺴﺎوي‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫ﺔ‬: cos ln tan 2 sin t x a t a y a t ⎧ = +⎪ ⎨ ⎪ =⎩
  13. 13. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬13 ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬‫اﻟﻘﺮن‬ ‫رﺑﺎﻋﻲ‬ ‫ﺗﺤﺘﻲ‬ ‫دوﻳﺮي‬ ‫ﻟﻤﻨﺤﻨﻲ‬Astroid (Hypocycloid with four cusps) 3 4 4 a a OO a′ = − = ‫اﻟﻜﺒﻴﺮ‬ ‫اﻟﺪاﺋﺮة‬ ‫ﻣﺤﻴﻂ‬‫اﻟﺼﻐﻴﺮة‬ ‫اﻟﺪاﺋﺮة‬ ‫أﺿﻌﺎف‬ ‫أرﺑﻌﺔ‬ ‫ة‬( 4 ) ( 3 ) 2 2 π π α θ π θ α θ= − − − ⇒ = − − ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬O NP′Δ‫اﻟﻌﻠﻢ‬ ‫ﻣﻊ‬HH PN′ =sin[ ( 3 )] cos3 4 2 4 a a HH π θ θ′ = − − = − ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬OO H′ ′Δ 3 cos 4 a OH θ′ = ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬O NP′Δcos[ ( 3 )] sin3 4 2 4 a a O N π θ θ′ = − − = ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬OO H′ ′Δ 3 sin 4 a O H θ′ ′ = 33 3 cos ( cos3 ) cos (4cos 3cos ) 4 4 4 4 P P a a a a x OH HH xθ θ θ θ θ′ ′= − = − − ⇒ = + − 33 3 sin sin3 sin (3sin 4sin ) 4 4 4 4 P P a a a a y O H O N yθ θ θ θ θ′ ′ ′= − = − ⇒ = − − 3 3 cos sin P P x a y a θ θ ⎧ =⎪ ⎨ =⎪⎩ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬b ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫أﺧﺮى‬ ‫داﺋﺮة‬ ‫داﺧﻞ‬a ‫ﻟﻬﺬا‬ ‫ﺣﺎﻻت‬ ‫ﻋﺪة‬ ‫ﺑﻴﻦ‬ ‫ﻣﻦ‬ ‫ﺣﺎﻟﺔ‬ ‫ﻧﺒﺤﺚ‬ ‫ﻧﺼﻒ‬ ‫ﻓﻴﻬﺎ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫هﻲ‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫أﺿﻌﺎف‬ ‫أرﺑﻌﺔ‬ ‫اﻟﻜﺒﻴﺮة‬ ‫اﻟﺪاﺋﺮﻩ‬ ‫ﻗﻄﺮ‬ ‫أي‬ ‫اﻟﺼﻐﻴﺮة‬ ‫اﻟﺪاﺋﺮة‬ 4 a b =
  14. 14. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬14 ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬‫اﻟﺨﺎرﺟﻲ‬ ‫اﻟﺪﺣﺮوج‬ ‫ﻟﻤﻨﺤﻨﻲ‬Epicycloid ‫اﻟﻤﻘﺎﺑﻠﺔ‬ ‫اﻟﺰاوﻳﺔ‬ ‫ﻳﺴﺎوي‬ ‫اﻟﺪاﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻣﻦ‬ ‫ﻗﻮس‬ ‫ﻃﻮل‬)‫رادﻳﺎن‬(‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬ ‫ﻓﻲ‬ a a b b θ α α θ= ⇒ = ‫اﻟﺮاﺑﻄﺘﻬﺎ‬ ‫و‬ ‫اﻟﺰاوﻳﺔ‬ ‫هﺬﻩ‬ ‫أﺳﺘﻨﺘﺎج‬ ‫ﻳﻤﻜﻦ‬ ‫اﻟﺸﻜﻞ‬ ‫ﻣﻦ‬( ) ( ) 2 2 a a b b b π π φ θ θ φ θ + = − − ⇒ = − − ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬OO H′Δ( )cosOH a b θ= + ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬OO H′Δ( )sinO H a b θ′ = + ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬O NP′Δsin[ ( )] cos 2 a b a b HH NP b HH b b b π θ θ + + ′ ′= = − − ⇒ = − ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬O NP′Δcos[ ( )] sin 2 a b a b O N b O N b b b π θ θ + + ′ ′= − − ⇒ = ( )cos cosp a b x OH HH a b b b θ θ + ′= + = + − ( )sin sinp a b y O H O N a b b b θ θ + ′ ′= − = + − ( )cos cos ( )sin sin p p a b x a b b b a b y a b b b θ θ θ θ +⎧ = + −⎪ ⎪ ⎨ ⎪ + ⎪ = + − ⎩ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬b‫ﻣﻦ‬ ‫ﺗﺘﺪﺣﺮج‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻋﻠﻰ‬ ‫اﻟﺨﺎرج‬ ‫ﻗﻄﺮهﺎ‬a
  15. 15. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬15 ‫اﻟﺪوﻳﺮي‬ ‫ﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬Cycloid ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬PO H′‫أن‬ ‫اﻟﻌﻠﻢ‬ ‫ﻣﻊ‬PH P H′ ′=sinPO H PH a φ′Δ ⇒ = ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬PO H′cosPO H OH a φ′ ′Δ ⇒ = sin ( sin )P P Px OH P H x a a x aφ φ φ φ′ ′ ′= − ⇒ = − ⇒ = − (1 cos )P P Py O H O H y a acos y aφ φ′ ′ ′= − ⇒ = − ⇒ = − ( sin ) (1 cos ) P P x a y a φ φ φ = −⎧ ⎪ ⎨ ⎪ = −⎩ ‫ﻣﺜﻞ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬P ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬a‫ﺗﺘﺪﺣﺮج‬ ‫ﻣﺤﻮر‬ ‫ﻋﻠﻰ‬ ‫إﻧﺰﻻق‬ ‫دون‬x
  16. 16. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬16 ‫اﻟﻤﻌﺎدﻟﺔ‬‫ﻟﻠ‬ ‫اﻟﻘﻄﺒﻴﺔ‬‫ﺴﺘﺮوﻓﻮﺋﻴﺪ‬Strophoid ‫اﻟﻤ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻣﻦ‬‫اﻟﺴﺎﻗﻴﻦ‬ ‫ﺘﺴﺎوي‬OPM‫ﻷن‬OP MP=‫ﻟﺬﻟﻚ‬ 2 MOP PMO π φ∠ = ∠ = −‫اﻟﺰواﻳﺎ‬OMA∠‫و‬MAO∠ ‫ﺗﺴﺎوي‬: 2 OMA π φ∠ = + 2 2 MAO π φ∠ = − ‫ﻓﻲ‬ ‫اﻟﺠﻴﺐ‬ ‫راﺑﻄﺔ‬ ‫ﻧﻜﺘﺐ‬‫اﻟﻤﺜﻠﺚ‬OMA‫اﻟﻨﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﻜﺎن‬ ‫هﻮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻌﻠﻢ‬ ‫ﻣﻊ‬M‫هﻲ‬ ‫ﻟﻬﺎ‬ ‫اﻟﻘﻄﺒﻴﺔ‬ ‫اﻟﻔﺎﺻﻠﺔ‬ ‫و‬ρ‫أي‬OM ρ= sin( ) 2 tan sin a AM a AM π φ φ φ + = ⇒ = sin cos2 cos2 ( )tan sin cossin( 2 ) 2 AM a a φ φ φ ρ φ ρ π ρ φ φφ = ⇒ = ⇒ = − cos2 cos a φ ρ φ = ‫اﻟﻨ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬‫ﻘﺎط‬M‫و‬N‫ﺑﺤﻴﺚ‬ PM PN OP= = ‫و‬ ‫ﻣﻘﺎرب‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﻬﺬا‬OA a=
  17. 17. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬17 ‫آﺎﺳﻴﻨﻲ‬ ‫ﻟﺒﻴﻀﻮﻳﺎت‬ ‫اﻟﻘﻄﺒﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬Ovals of Cassini 2 1 2PF PF b× = ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬ ‫ﺗﻤﺎم‬ ‫اﻟﺠﻴﺐ‬ ‫راﺑﻄﺔ‬1OPF2 2 2 1 2 cos( )PF r a ar π θ= + − − ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬ ‫ﺗﻤﺎم‬ ‫اﻟﺠﻴﺐ‬ ‫راﺑﻄﺔ‬2OPF2 2 2 2 2 cosPF r a ar θ= + − 2 2 4 2 2 2 2 4 1 2 ( 2 cos )( 2 cos )PF PF b r a ar r a ar bθ θ× = ⇒ + + + − = 4 4 2 2 3 3 3 3 2 2 2 4 2 2 cos 2 cos 2 cos 2 cos 4 cosr a a r a r ar a r ar a r bθ θ θ θ θ+ + − − + + − = 4 4 2 2 2 4 2 (1 2cos )r a a r bθ+ + − = ‫أن‬ ‫ﺑﻤﺎ‬2 2 2 2 2 1 2cos (1 cos ) cos sin cos cos2θ θ θ θ θ θ− = − − = − = ‫اﻟﻘﻄﺒﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬: 4 4 2 2 4 2 cos2r a a r bθ+ − = ‫ﻣﺜﻞ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬P ‫ﻣﻦ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﺎﺻﻠﺔ‬ ‫ﺿﺮب‬ ‫ﺣﺎﺻﻞ‬ ‫ﺑﺤﻴﺚ‬ ‫ﺛﺎﺑﺘﺘﻴﻦ‬ ‫ﻧﻘﻄﺘﻴﻦ‬)‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺑﺆرﺗﻲ‬‫ﺑﻴﻨﻬﻤﺎ‬ ‫اﻟﻔﺎﺻﻠﺔ‬a( ‫ﻣﺜﻞ‬ ‫ﺛﺎﺑﺖ‬ ‫ﻣﻘﺪار‬2 b‫ﻟﺤﺎﻟﺔ‬b a> b a< ‫أو‬ b a> ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬b a= ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬
  18. 18. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬18 ‫ﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻘﻄﺒﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬‫َﻓﺔ‬‫ﺪ‬َ‫ﺻ‬‫ﺑﺎﺳﻜﺎل‬Limacon of Pascal ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬b a> ‫اﻟﺰاوي‬ ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬OPM cos cos OQ OQ a a θ θ= ⇒ = ‫إذن‬ cosOP OQ QP b a θ= + = + ‫اﻟﻘﻄﺒﻴﻪ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬: cosr b a θ= + ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬b a<‫أو‬b a> ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬b a= ‫اﻟﺨﻂ‬OQ‫اﻟﻤﺒﺪأ‬ ‫ﻳﻮﺻﻞ‬O‫ﻣﺜﻞ‬ ‫ﻧﻘﻄﺔ‬ ‫ﺑﺄي‬Q ‫ﻗﻄﺮهﺎ‬ ‫داﺋﺮة‬ ‫ﻋﻠﻰ‬a.‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﺜﻞ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬P‫ﻣﻦ‬ ‫ﻓﺎﺻﻠﺘﻬﺎ‬ ‫ﺑﺤﻴﺚ‬Q ‫ﻣﺜﻞ‬ ‫ﺛﺎﺑﺖ‬ ‫ﻣﻘﺪار‬b‫أي‬PQ b=
  19. 19. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬19 ‫ﻟﻠﺪاﺋﺮة‬ ‫اﻟﻤﻨﺸﺄ‬ ‫اﻟﻤﻨﺤﻨﻲ‬Involute of a circle OM a= ‫اﻟﻘﻮس‬ ‫ﻃﻮل‬AP MP aφ= = ‫ﻧﺮ‬‫ﻣﻦ‬ ‫ﻋﻤﻮد‬ ‫ﺧﻂ‬ ‫ﺳﻢ‬P‫اﻟﻤﺤﻮر‬ ‫ﻳﻘﻄﻊ‬x‫ﻓﻲ‬H‫إﺣﺪاﺛﻴﺎت‬P‫اﻟﺰاوﻳﻪ‬ ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻣﻦ‬OPH‫هﻲ‬: sin sinP y yθ ρ θ ρ = ⇒ = cos cosP x xθ ρ θ ρ = ⇒ = ‫اﻟﺰاوﻳﻪ‬ ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻣﻦ‬OMP sin( ) sin cos sin cos MP MP aφ θ ρ φ θ ρ θ φ φ ρ − = ⇒ − = = cos( ) cos cos sin sin a aφ θ ρ φ θ ρ φ θ ρ − = ⇒ + = ‫ﻋﻠﻰ‬ ‫ﻧﺤﺼﻞ‬ ‫اﻟﺮاوﺑﻂ‬ ‫هﺬﻩ‬ ‫إﺧﺘﺼﺎر‬ ‫ﻣﻦ‬: sin cosP Px y aφ φ φ− = cos sinP Px y aφ φ+ = ‫اﻟﺪاﺋﺮة‬ ‫ﻣﻨﺸﺄ‬ ‫ﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻧﺤﺼﻞ‬ ‫اﻟﻤﻌﺎدﻟﺘﻴﻦ‬ ‫هﺬﻩ‬ ‫ﻣﻦ‬: (cos sin ) (sin cos ) P P x a y a φ φ φ φ φ φ = +⎧ ⎪ ⎨ ⎪ = −⎩ ‫ﻣﺜﻞ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬P‫ﺣﺒﻞ‬ ‫ﺑﺤﻴﺚ‬ ‫ﺳﻠ‬ ‫أو‬‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﺣﻮل‬ ‫ﻳﻠﻒ‬ ‫ﻣﺮن‬ ‫ﻚ‬a‫اﻟﺤﺒﻞ‬ ‫ﻳﻔﺘﺢ‬ ‫ﻣﺴﺤﻮب‬ ‫اﻟﺤﺒﻞ‬ ‫ﻓﻴﻬﺎ‬ ‫ﻳﻜﻮن‬ ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺪاﺋﺮة‬ ‫ﺣﻮل‬ ‫ﻣﻦ‬ ‫اﻟﻘﻮس‬ ‫ﻳﻜﻮن‬ ‫أن‬ ‫هﺬا‬ ‫ﻳﺴﺘﻄﻠﺐ‬AP‫ﻋﻠﻰ‬ ‫اﻟﻤﻤﺎس‬ ‫ﻳﺴﺎوي‬ ‫اﻟﺪاﺋﺮة‬PM‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬P‫أي‬: P P OH x PH y = =
  20. 20. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬20 ‫اﻟﻠﺒﻼﺑﻲ‬ ‫ﻟﻠﻤﻨﺤﻨﻲ‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬Cissoid ‫اﻟﺰاوﻳﻪ‬ ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬ORMcos 2 cos 2 OR OR a a θ θ= ⇒ = ‫اﻟﺰاوﻳﻪ‬ ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬OSM 2 2 cos cos a a OS OS θ θ = ⇒ = OR RS OS+ = ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﺧﺼﺎﺋﺺ‬ ‫ﻣﻦ‬OP RS=‫إذن‬: 2 2 cos cos a RS OP OS OR OP a θ θ = = − ⇒ = − 2 2 sin cos a OP θ θ = ‫اﻟﻨﻘﻄﺔ‬ ‫إﺣﺪاﺛﻴﺎت‬P‫اﻟﺰاوﻳﻪ‬ ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻣﻦ‬OPH cosPx OP θ= sinPy OP θ= ‫هﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﻬﺬا‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬: 2 3 2 sin 2 sin cos P P x a a y θ θ θ ⎧ ⎪ = ⎪⎪ ⎨ ⎪ ⎪ = ⎪⎩ ‫ﻣﺜﻞ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬P‫ﻣﻦ‬ ‫ﻓﺎﺻﻠﺘﻬﺎ‬ ‫ﺑﺤﻴﺚ‬ ‫اﻟﻤﺒﺪأ‬O‫ﻧﻘﻄﺔ‬ ‫ﻓﺎﺻﻠﺔ‬ ‫ﺗﺴﺎوي‬R)‫اﻟﺨﻂ‬ ‫ﺗﻼﻗﻲ‬ ‫ﻣﺤﻞ‬OP‫داﺋﺮة‬ ‫ﻣﻊ‬ ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬a‫ﻣﻦ‬ ‫ﺗﻤﺮ‬O(‫اﻟﻨﻘﻄﺔ‬ ‫اﻟﻰ‬S)‫أﻣﺘﺪاد‬ ‫ﺗﻼﻗﻲ‬ ‫ﻣﺤﻞ‬ OP‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫اﻟﺪاﺋﺮة‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﻤﺎس‬ ‫ﻣﻊ‬M(‫أي‬: OP RS=
  21. 21. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬21 ‫أﻏﻨﻴﺴﻲ‬ ‫اﻟﺴﺎﺣﺮة‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬Witch of Agnesi PAM x= ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻓﻲ‬‫اﻟﺰاوﻳﺔ‬OAMtan( ) 2 tan( ) 2 cot 2 2 2 P P P x x a x a a π π θ θ θ− = ⇒ = − ⇒ = cos( ) 2 sin 2 2 2 2 cos( ) 2 sin OB OBM OB a a a a OAM OA OA π θ θ π θ θ Δ ⇒ − = ⇒ = Δ ⇒ − = ⇒ = 2 2 2 cos 2 sin sin sin a a AB OA OB AB a AB θ θ θ θ = − ⇒ = − ⇒ = ‫اﻟﺰاوﻳﺔ‬ ‫اﻟﻘﺎﺋﻢ‬ ‫اﻟﻤﺜﻠﺚ‬ ‫ﻣﻦ‬BPA‫اﻟﺰاوﻳﺔ‬ ‫إن‬ ‫اﻟﻌﻠﻢ‬ ‫ﻣﻊ‬ ،ABP∠‫ﺗﺴﺎوي‬θ‫أي‬ABP θ∠ =‫إذن‬ ‫اﻟﻤﻮازﻳﺔ‬ ‫اﻟﺨﻄﻮط‬ ‫ﺧﺎﺻﻴﺔ‬: 2 22 cos sin sin 2 cos sin AP a AP AP a AB θ θ θ θ θ = ⇒ = × ⇒ = 2 2 cosAP a θ= ‫ﻟﻠﻨﻘﻄﺔ‬ ‫اﻟﻌﻤﻮدﻳﺔ‬ ‫اﻹﺣﺪاﺛﻴﺎت‬P‫هﻲ‬: 2 2 2 2 2 cos (2 2cos )P P Py a AP y a a y aθ θ= − ⇒ = − ⇒ = − ‫اﻟﻤﺜﻠﺜﺎﺗﻴﺔ‬ ‫اﻟﺘﺤﻮﻳﻼت‬ ‫ﺑﻬﺬﻩ‬ ‫ﻧﺴﺘﻌﻴﻦ‬2 2 2 2 2 2cos cos (1 sin ) 1 (cos sin ) 1 cos2θ θ θ θ θ θ= + − = + − = + (1 cos2 )Py a θ= − ‫هﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﻬﺬا‬ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬: 2 cot (1 cos2 ) P P x a y a θ θ =⎧ ⎪ ⎨ ⎪ = −⎩ ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬a‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫ﻋﻠﻴﻬﺎ‬ ‫ﻣﻤﺎﺳﺎن‬ ‫ﺧﻄﺎن‬ O‫و‬M‫اﻟﻨﻘﻄﺔ‬ ، ‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫آﻤﺎ‬A‫اﻟﻤﺎر‬ ‫اﻟﺨﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻣﻦ‬M‫اﻟﺨﻂ‬ ‫ﻳﻘﻄﻊ‬OA‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺪاﺋﺮة‬B‫اﻟﺨﻂ‬ ، ‫ﻣﻦ‬ ‫اﻟﻤﺎر‬B‫ﻣﻮازي‬ ‫و‬MA‫ﻋﻠﻰ‬ ‫اﻟﻌﻤﻮد‬ ‫اﻟﺨﻂ‬ ‫ﻳﻘﻄﻊ‬ MA‫ﻣﻦ‬A‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬P‫اﻟﻤﺤﻞ‬ ‫هﻮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ، ‫ﻟﻠﻨﻘﻄﺔ‬ ‫اﻟﻬﻨﺪﺳﻲ‬P. ‫ﻣﻦ‬ ‫اﻟﻤﺎر‬ ‫اﻟﺨﻂ‬O‫ﻣﻮازي‬ ‫و‬MA‫ﻣﻘﺎرب‬ ‫هﻮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬.
  22. 22. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬22 ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻣﻌﺎدﻟﺔ‬Catenary ‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻘﻮى‬ ‫ﺗﻮازن‬1 P0 cos sin T T sg T θ μ θ = = ‫راﺑﻄﺘﻲ‬ ‫ﺗﻘﺴﻴﻢ‬‫ﺑﻌﻀﻬﻤﺎ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻘﻮى‬ ‫ﺗﻮازن‬ 0 tan sg T μ θ= ‫اﻟﻘﻮى‬ ‫ﺗﻮازن‬ ‫راﺑﻄﺘﻲ‬ ‫ﻣﻦ‬ ‫آﻞ‬ ‫ﺗﺮﺑﻴﻊ‬ ‫ﻣﺠﻤﻮع‬2 2 2 0( ) ( )T sg Tμ+ = ‫اﻟﺜﺎﺑﺖ‬ ‫اﻟﻌﺪد‬ ‫ﺗﺴﺎوي‬ ‫اﻟﻨﺴﺒﺔ‬ ‫هﺬﻩ‬ ‫ﻧﻔﺮض‬a0T a gμ = 0 0 T a a g T g μ μ = ⇒ = 2 2 2 2 2 0( ) ( )T sg T T g a sμ μ+ = ⇒ = + ‫اﻟﻤﺤﻮر‬ ‫و‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ﺗﻠﻚ‬ ‫ﻓﻲ‬ ‫اﻟﺪاﻟﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﻤﺎس‬ ‫زاوﻳﺔ‬ ‫ﻇﻞ‬ ‫ﻳﺴﺎوي‬ ‫ﻣﻌﻴﻨﺔ‬ ‫ﻧﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫داﻟﺔ‬ ‫ﺗﻔﺎﺿﻞ‬ ‫اﻹﺷﺘﻘﺎق‬ ‫و‬ ‫اﻟﺘﻔﺎﺿﻞ‬ ‫ﻣﻔﻬﻮم‬ ‫ﻓﻲ‬ ‫اﻷﻓﻘﻲ‬)‫اﻟﺴﺎﻋﺔ‬ ‫ﻋﻘﺎرب‬ ‫ﺧﻼف‬ ‫اﻟﻤﻮﺟﺒﻪ‬ ‫اﻟﺠﻬﺔ‬(‫إذن‬: 2 2 1 tan dy dy s d y ds dx dx a dx a dx θ = ⇒ = ⇒ = ‫ﻳﺴﺎوي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻦ‬ ‫ﺟﺰء‬ ‫ﻃﻮل‬ ‫اﻟﺘﻔﺎﺿﻠﻴﻪ‬ ‫اﻟﻬﻨﺪﺳﻪ‬ ‫ﻓﻲ‬2 1 ( ) ds dy dx dx = + ‫اﻟﺜﺎﻧﻴﺔ‬ ‫اﻟﺪرﺟﺔ‬ ‫ﻣﻦ‬ ‫اﻟﺘﻔﺎﺿﻠﻴﻪ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ 2 2 2 1 1 ( ) d y dy dx a dx = + ‫اﻟﻤﻌﺎدﻟ‬ ‫ﺟﻮاب‬‫اﻟﻤﻌﻠﻘﺔ‬ ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﺷﻜﻞ‬ ‫هﻮ‬ ‫اﻟﺘﻔﺎﺿﻠﻴﻪ‬ ‫ﺔ‬cosh x y a c a = + ‫اﻟﺤﺪﻳﺔ‬ ‫ﻟﻠﺸﺮاﺋﻂ‬ ‫اﻟﺘﻔﺎﺿﻠﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫هﺬﻩ‬ ‫ﺟﻮاب‬(0) 0y ′ =‫و‬(0) 0y = (0) 0y c a= ⇒ = − ‫اﻟﺴﻠﺴﻠ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻟﺸﻜﻞ‬ ‫اﻟﻨﻬﺎﺋﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬‫اﻟﺮاﺑﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫ﺔ‬0T a gμ =cosh x y a a a = − ‫ﻣﻌﺎدﻟﺔ‬ ‫ﻧﻘﻄﺘﻴﻦ‬ ‫ﺑﻴﻦ‬ ‫ُﺜﺒﺖ‬‫ﻣ‬ ‫ﻣﺮن‬ ‫ﺳﻠﻚ‬ ‫أو‬ ‫ﺣﺒﻞ‬ ‫أو‬ ‫ﺳﻠﺴﻠﻪ‬ ‫هﻮ‬ ‫وزﻧﻬﺎ‬ ‫ﻧﺘﻴﺠﺔ‬ ‫اﻷﺳﻔﻞ‬ ‫ﻧﺤﻮ‬ ‫اﻟﻤﻨﺤﺪرة‬ ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﺜﺎﻧﻴﺔ‬ ‫اﻟﺪرﺟﺔ‬ ‫ﻣﻦ‬ ‫ﺗﻔﺎﺿﻠﻴﺔ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫ﺟﻮاب‬. T‫ا‬ ‫ﺷﺪة‬‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫ﻟﺴﺤﺐ‬P 0T‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻋﻠﻰ‬ ‫اﻷﻓﻘﻲ‬ ‫اﻟﺴﺤﺐ‬ ‫ﺷﺪة‬P μ‫ﻟﻠﺴﻠﺴﻠﺔ‬ ‫اﻟﻄﻮﻟﻴﺔ‬ ‫اﻟﻜﺜﺎﻓﺔ‬ s‫و‬ ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻃﻮل‬ds‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻣﻦ‬ ‫ﺟﺰء‬ ‫ﻃﻮل‬ g‫اﻷرض‬ ‫ﺟﺎذﺑﻴﺔ‬ ‫ﺛﺎﺑﺖ‬ ‫أو‬ ‫ﻟﻸرض‬ ‫اﻟﺘﻌﺠﻴﻞ‬ ‫ﺛﺎﺑﺖ‬ θ‫اﻷﻓﻘﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫و‬ ‫اﻟﺴﺤﺐ‬ ‫ﺷﺪة‬ ‫ﺑﻴﻦ‬ ‫اﻟﺰاوﻳﺔ‬ 1-‫ﺗﺴﺎوي‬ ‫و‬ ‫ﺛﺎﺑﺘﺔ‬ ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫آﻞ‬ ‫ﻓﻲ‬ ‫اﻷﻓﻘﻴﻪ‬ ‫اﻟﺴﺤﺐ‬ ‫ﺷﺪة‬0T
  23. 23. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬23
  24. 24. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬24 Bicorn ‫اﻟﻬﻼﻟﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ Astroid ‫اﻟﻘﺮن‬ ‫رﺑﺎﻋﻲ‬ ‫ﺗﺤﺘﻲ‬ ‫دوﻳﺮي‬‫اﻟﻨﺠﻤﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬/‫ﺳﺘﺮوﺋﻴﺪ‬ ‫إ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 3 32 2 23 x y a+ = ‫اﻟﻜﺎرﺗﻴﺰﻳﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ 3 3 cos ( ) sin ( ) x a t y a t = = ⎧⎪ ⎨ ⎪⎩ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻻت‬ ‫اﻟﻤﺴﺎﺣﺔ‬ 2 3 8 aπ ‫اﻟﻄﻮل‬6a ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻨﺤﻨﻲ‬‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬‫إﻧﺰﻻق‬ ‫دون‬ ‫ﺗﺘﺪﺣﺮج‬ ‫أآ‬ ‫أﺧﺮى‬ ‫داﺋﺮة‬ ‫داﺧﻞ‬‫ﻣﻨﻬﺎ‬ ‫ﺒﺮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 2 ( ) ( 2 )y a x x ay a− = + − 2 2 sin( ) ((2 cos( )) cos ( )) /(3 sin ( )) x a t y a t t t = = + + ⎧ ⎨ ⎩ ‫اﻟﻤﺴﺎﺣﺔ‬:‫آﺎن‬ ‫إذا‬1a =‫اﻟﻤﺴﺎﺣﺔ‬ 1 3 (16 3 27)A π= − ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﺴﺎﺣﺔ‬:‫آﺎن‬ ‫إذا‬0 1a< <‫اﻟﺤ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬‫ﺎﻟﺔ‬ 1 3 4 3 6 3(4 3 7)A aπ= − + −⎡ ⎤⎣ ⎦ ‫اﻟﺮاﺑﻌﺔ‬ ‫اﻟﺪرﺟﺔ‬ ‫ﻣﻦ‬ ‫ﻣﻌﺎدﻟﺘﻪ‬ ‫آﺬﻟﻚ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫أﺳﻢ‬Cocked hat‫اﻟﻘﺒﻌﺎت‬ ‫أﻧﻮاع‬ ‫ﻣﻦ‬ ‫ﻧﻮع‬
  25. 25. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬25 Cartesian Oval ‫ﺑﻴﻀ‬‫ﺎ‬‫دﻳﻜﺎرت‬ ‫وي‬ Cardioid ‫اﻟﻘﻠﺒﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 2 ( ) ( )x y ax a x y+ + = + ‫آﺎرﺗﻴﺰﻳﺔ‬ (1 cos( ))r a θ= − ‫ﻗﻄﺒﻴﺔ‬ (1 cos( ))cos( ) (1 cos( ))sin( ) x a t t y a t t = −⎧ ⎨ = −⎩ ‫اﻟﻤﺴﺎﺣﺔ‬23 2 A aπ=‫اﻟﻄﻮل‬8L a= ‫ﺗﻮﺿﻴﺢ‬:‫ﺗﺘﺪﺣﺮج‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫أﺧﺮى‬ ‫داﺋﺮة‬ ‫ﻋﻠﻰ‬‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬ ‫ﻓﻲ‬ ‫ﻟﻬﺎ‬ ‫ﻣﺴﺎوﻳﺔ‬ ‫ااﻟﻘﺮﻧﺔ‬ ‫ﻧﻘﻄﺔ‬cusp‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﻬﺬا‬4a ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 ( ) ( )m x a y n x a y k− + + + + = ‫اﻟﻜﺎرﺗﻴﺰﻳﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻳﻀﻞ‬ ‫ﻋﻨﺪﻣﺎ‬ ‫ﻣﺜﻠﺚ‬ ‫ﻟﺮأس‬ ‫اﻟﻬﻨﺪﺳﻲ‬ ‫اﻟﻤﺤﻞ‬‫ﻣﺠﻤﻮع‬ً‫ﺎ‬‫ﺛﺎﺑﺘ‬ ‫ﻟﻠﺮأس‬ ‫اﻟﻤﺠﺎورﻳﻦ‬ ‫اﻟﻀﻠﻌﻴﻦ‬.‫ﻟﻠﻨﻘﻄﺔ‬ ‫اﻟﻬﻨﺪﺳﻲ‬ ‫اﻟﻤﺤﻞ‬p‫ﺑﺤﻴﺚ‬ ‫ﺗﺴﺎوي‬ ‫و‬ ‫ﺛﺎﺑﺘﺔ‬ ، ‫ﺛﺎﺑﺘﺘﻴﻦ‬ ‫ﻧﻘﻄﺘﻴﻦ‬ ‫ﻣﻦ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﺎﺻﻠﺔ‬ ‫ﻣﺠﻤﻮع‬k‫أي‬mr nr k′± = ‫ﻋﺎم‬ ‫دﻳﻜﺎرت‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻃﺎﻟﻊ‬ ‫ﻣﻦ‬ ‫أول‬1637‫ﻧﻴﻮﺗﻦ‬ ‫آﺬﻟﻚ‬ 1 m n Ellipse m n Hyperpola m Limacon = → = − → = →
  26. 26. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬26 Catenary ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ Cassinian Ovals ‫آﺎﺳﻴﻨﻲ‬ ‫ﺑﻴﻀﻮﻳﺎت‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 4 ( ) ( )x a y x a y c⎡ ⎤ ⎡ ⎤− + + + =⎣ ⎦ ⎣ ⎦ ‫اﻟﻜﺎرﺗﻴﺰﻳﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ [ ]4 4 2 2 4 2 1 cos(2 )r a a r cθ+ − + = ‫اﻟﻘﻄﺒﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:ً‫ﺎ‬‫ﺛﺎﺑﺘ‬ ‫ﻟﻠﺮأس‬ ‫اﻟﻤﺠﺎورﻳﻦ‬ ‫اﻟﻀﻠﻌﻴﻦ‬ ‫ﺟﺪاء‬ ‫ﻳﻀﻞ‬ ‫ﻋﻨﺪﻣﺎ‬ ‫ﻣﺜﻠﺚ‬ ‫ﻟﺮأس‬ ‫اﻟﻬﻨﺪﺳﻲ‬ ‫اﻟﻤﺤﻞ‬.‫ﻟﻠﻨﻘﻄﺔ‬ ‫اﻟﻬﻨﺪﺳﻲ‬ ‫اﻟﻤﺤﻞ‬p‫ﺑﺤﻴﺚ‬ ‫ﺣﺎﺻﻞ‬‫ﺑﻔﺎﺻﻠﺔ‬ ‫و‬ ‫ﺛﺎﺑﺘﺘﻴﻦ‬ ‫ﻧﻘﻄﺘﻴﻦ‬ ‫ﻣﻦ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫هﺬﻩ‬ ‫ﻓﺎﺻﻠﺔ‬ ‫ﺿﺮب‬a2‫ﺑﻌﻀﻬﻤﺎ‬ ‫ﻣﻦ‬1r‫و‬2r‫ﻳﺴﺎوي‬ ‫ﺛﺎﺑﺖ‬ ‫ﻣﻘﺪار‬2 c‫أي‬ 2 c=1r*2r ‫اﻷرض‬ ‫و‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﺮآﺔ‬ ‫ﻣﻄﺎﻟﻌﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫أﺳﺘﻌﻤﻞ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ cosh( ) x y a a = ‫اﻟﻜﺎرﺗﻴﺰﻳﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ( ) ( ) cosh( ) x t t t y t a a =⎧ ⎪ ⎨ =⎪⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﺛﺎﺑﺘﺘﻴﻦ‬ ‫ﻧﻘﻄﺘﻴﻦ‬ ‫ﺑﻴﻦ‬ ‫ﺑﺤﺮﻳﺔ‬ ‫ﻣﻌﻠﻘﺔ‬ ‫ﻣﺮﻧﺔ‬ ‫ﺛﻘﻴﻠﺔ‬ ‫ﺳﻠﺴﻠﺔ‬ ‫أو‬ ‫آﺒﻞ‬ ‫أو‬ ‫ﺣﺒﻞ‬ ‫ﻳﺸﻜﻠﻪ‬ ‫اﻟﺬي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﻮ‬‫اﻟﺜﺎﺑﺖ‬ ،a‫اﻟﻤﺴﺎﺋﻞ‬ ‫ﻓﻲ‬ ‫اﻟﺴﻠﺴﻠﺔ‬ ‫أو‬ ‫اﻟﻜﺒﻞ‬ ‫ﺑﻮزن‬ ‫ﻳﺮﺗﺒﻂ‬ ‫اﻟﻬﻨﺪﺳﻴﺔ‬.
  27. 27. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬27 Circle ‫ا‬‫ﻟﺪاﺋﺮة‬ Cayley’s sextic ‫آﺎﻳﻠﻲ‬ ‫ُﺪاﺳﻴﺔ‬‫ﺳ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 3 2 2 2 2 4( ) 27 ( )x y ax a x y+ − = + 3 4 cos ( ) 3 r a θ = 3 3 3 3 4 cos ( / )cos 4 cos ( / )sin x a t t y a t t ⎧ =⎪ ⎨ =⎪⎩ ‫اﻟﻤﺴﺎﺣﺔ‬:‫اﻟﺨﺎرﺟﻴﺔ‬2 23.50219A a=‫اﻟﺪاﺧﻠﻴﺔ‬2 0.05975299A a= ‫اﻟﻄﻮل‬6L aπ= ‫ﺗﻮﺿﻴﺢ‬: ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 ( ) ( )x x y y R° °− + − = r R= cos sin x x R t y y R t ° ° = +⎧ ⎨ = +⎩ ‫اﻟﻤﺴﺎﺣﺔ‬2 A Rπ= ‫اﻟﻄﻮل‬2L Rπ= ‫ﺗﻮﺿﻴﺢ‬:R‫و‬ ‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬( , )x y° °‫اﻟﺪاﺋﺮة‬ ‫ﻣﺮآﺰ‬ ‫إﺣﺪاﺛﻴﺎت‬
  28. 28. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬28 Cochleoid ‫ﻣﻨﺤﻨﻲ‬‫ﺣﻠﺰوﻧﻲ‬‫اﻟﺸﻜﻞ‬ Cissoid ‫ا‬‫اﻟﻠﺒﻼﺑﻲ‬ ‫ﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 3 2 2 x y a x = − 2 tan( )sin( )r a θ θ= 2 3 2 sin (2 sin )/ cos x a t y a t t ⎧ =⎪ ⎨ =⎪⎩ ‫اﻟﻤﺴﺎﺣﺔ‬3A aπ= ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻣﺤﻮر‬ ‫ﺑﻴﻦ‬ ‫اﻟﻔﺎﺻﻠﺔ‬y‫ﺗﺴﺎوي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻘﺎرب‬ ‫و‬a2 ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 ( ) tan( / )x y Arc y x ay+ = sina r θ θ = 2 ( sin cos )/ ( sin )/ x a t t t y a t t =⎧ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﺣﻠﺰوﻧﻲ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫أﺳﻢ‬ ‫ﻳﻌﻨﻲ‬snail-form‫اﻟﺤﻠﺰون‬ ‫ﺷﻜﻞ‬ ‫أي‬
  29. 29. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬29 Conchoid of de Siuze ‫ﺳﻴﻮز‬ ‫دي‬ ‫ﺻﺪﻓﻴﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ Conchoid ‫ﺻﺪﻓﻲ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 ( ) ( )x a x y b x− + = secr b a θ= + ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:a‫اﻟﻤﻘﺎرب‬ ‫و‬ ‫اﻟﻘﻄﺐ‬ ‫ﺑﻴﻦ‬ ‫اﻟﻔﺎﺻﻠﺔ‬b‫اﻟﺜﺎﺑﺘﺔ‬ ‫اﻟﻘﻄﻌﺔ‬ ‫ﻃﻮل‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 ( 1)( )x x y ax− + = sec cosr aθ θ= + (sec cos )cos (sec cos )sin x t a t t y t a t t = +⎧ ⎨ = +⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ 1 (2 ) 1 (4 ) sec 2 loopA a a a a Arc a⎡ ⎤= − − − + + −⎣ ⎦ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:
  30. 30. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬30 Devil’s Curve ‫اﻟﺸﻴﻄﺎن‬ ‫ﻣﻨﺤﻨﻲ‬ Cycloid ‫دوﻳﺮي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 cos(1 ( / ) 2x aArc y a ay y= − − − ( sin ) (1 cos ) x a t t y a t = −⎧ ⎨ = −⎩ ‫اﻟﻤﺴﺎﺣﺔ‬2 3A aπ= ‫اﻟﻄﻮل‬8L a= ‫واﺣﺪة‬ ‫ﻟﺪورة‬ ‫اﻟﻄﻮل‬ ‫و‬ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﺧﻂ‬ ‫ﻋﻠﻰ‬ ‫أﻧﺰﻻق‬ ‫دون‬ ‫ﺗﺘﺪﺣﺮج‬ ‫ﻋﻨﺪﻣﺎ‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫ﺗﺮﺳﻤﻪ‬ ‫اﻟﺬي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﻮ‬‫اﻟﺪاﺋﺮة‬ ‫ﻗﻄﺮ‬ ‫ﻧﺼﻒ‬a ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 4 2 2 4 2 2 y a y x b x− = − 2 2 2 2 2 2 2 (sin cos ) sin cosr a bθ θ θ θ− = − 2 2 2 2 2 2 cos ( sin cos )/(sin cos ) sin x E t E a t b t t t y E t =⎧ = − − ⇒ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:
  31. 31. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬31 Durer’s shell curve ‫دوورﻳﺮ‬ ‫ﺻﺪﻓﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ Double of Folium ‫ﻣﻨﺤﻨﻲ‬‫ﻣﺰدو‬ ‫ﻓﻮﻟﻴﻮم‬‫ج‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 ( ) 4x y axy+ = 2 4 cos sinr a θ θ= ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫آﻠﻤﺔ‬ ‫ﺗﻌﻨﻲ‬folium‫اﻟﺸﺠﺮ‬ ‫ورﻗﺔ‬ ‫ﺷﻜﻞ‬ ‫ﻣﻌﺎدﻟﺔ‬‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ 2 2 2 2 2 2 ( ) ( )( )x xy ax b b x x y a+ + − = − − + ‫اﺧﺮى‬ ‫ﻋﺪدﻳﺔ‬ ‫ﻣﻘﺎدﻳﺮ‬ ‫و‬ ‫اﻟﻄﻮل‬ ‫و‬ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫ﺗﻮﺿﻴﺢ‬:
  32. 32. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬32 Ellipse ‫ﻧﺎﻗﺺ‬ ‫ﻗﻄﻊ‬/‫إهﻠﻴﻠﻴﺞ‬/‫ﺑﻴﻀﻮي‬ Eight Curve ‫اﻟﺜﻤﺎﻧﻴﺔ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 4 2 2 2 ( )x a x y= − 2 2 4 cos2 secr a θ θ= sin sin cos x a t y a t t =⎧ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬24 3 A a= ‫اﻟﻄﻮل‬6.09722L a= ‫ﺗﻮﺿﻴﺢ‬: ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 ( ) ( ) 1 x x y y a b ° °− − + = 2 2 2 2 2 0ax bxy cy dx fy g+ + + + + = cos sin x a t y b t =⎧ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬A abπ= ‫اﻟﻄﻮل‬2 4 61 1 1 ( )(1 )& 4 64 256 a b L a b h h h h a b π − = + + + + +⋅⋅⋅ = + ‫ﺗﻮﺿﻴﺢ‬:‫اﻷﻋﻈﻢ‬ ‫اﻟﻤﺤﻮر‬ ‫ﻃﻮل‬ ‫ﻧﺼﻒ‬a‫اﻷﻗﺼﺮ‬ ‫اﻟﻤﺤﻮر‬ ‫ﻃﻮل‬ ‫ﻧﺼﻒ‬ ‫و‬b‫و‬( , )x y° °‫اﻹهﻠﻴﻠﺞ‬ ‫ﻣﺮآﺰ‬ ‫إﺣﺪاﺛﻴﺎت‬
  33. 33. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬33 Epitrochoid ‫إﺑﻴﺘﺮوآﻮﺋﻴﺪ‬ Epicycloid ‫ﺧﺎرﺟﻲ‬ ‫دﺣﺮوج‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ ( )cos cos[( )/ ] ( )sin sin[( )/ ] x R r t r R r r t y R r t r R r r t = + − +⎧ ⎨ = + − +⎩ ‫اﻟﻤﺴﺎ‬‫ﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻧﻘﻄﺔ‬ ‫ﺗﺮﺳﻤﻪ‬ ‫اﻟﺬي‬ ‫اﻟﻤﻨﺤﻨﻲ‬P‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬r ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫ﺛﺎﺑﺘﺔ‬ ‫أﺧﺮى‬ ‫داﺋﺮة‬ ‫ﺣﻮل‬ ‫اﻟﺨﺎرج‬ ‫ﻣﻦ‬ ‫اﻟﺪاﺋﺮة‬ ‫هﺬﻩ‬ ‫ﺗﺘﺪﺣﺮج‬ ‫ﻋﻨﺪﻣﺎ‬R ‫اﻟﻤﻨ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬‫ﺤﻨﻲ‬ ( )cos cos[( )/ ] ( )sin sin[( )/ ] x R r t d R r r t y R r t d R r r t = + − +⎧ ⎨ = + − +⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬P‫ﺑﻔﺎﺻﻠﺔ‬d‫داﺋﺮة‬ ‫ﻣﺮآﺰ‬ ‫ﻋﻦ‬ ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬r‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﺧﺎرج‬ ‫ﺗﺘﺪﺣﺮج‬R
  34. 34. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬34 Fermat’s Spiral ‫ﺣﻠﺰون‬‫ﻓﻴﺮﻣﺎ‬ Equiangular Spiral ‫اﻟﺰواﻳﺎ‬ ‫ﻣﺘﺴﺎوي‬ ‫ﺣﻠﺰون‬ ‫اﻟﻤﻨ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬‫ﺤﻨﻲ‬ cota r aeθ = ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻠﻮﻏﺎرﻳﺜﻤﻲ‬ ‫اﻟﺤﻠﺰون‬ ‫أﺟﻞ‬ ‫ﺁﺧﺮﻣﻦ‬ ‫ﻣﺼﻄﻠﺢ‬ ‫دﻳﻜﺎرت‬ ‫اﻟﺤﻠﺰون‬ ‫ﻣﻦ‬ ‫اﻟﻨﻮع‬ ‫هﺬا‬ ‫أآﺘﺸﻒ‬ ‫ﻣﻦ‬ ‫أول‬1638 ‫ﻋﻨﺪ‬ 2 a π =‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬‫داﺋﺮة‬ ‫ﻋﻦ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 r a θ= ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻋﺎم‬ ‫ﻓﻴﺮﻣﺎ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻧﺎﻗﺶ‬1636 ‫أرﺧﻤﻴﺪس‬ ‫ﺣﻠﺰون‬ ‫أﻧﻮاع‬ ‫أﺣﺪ‬ ‫هﻮ‬ ‫و‬ ‫اﻟﻨﺎﻗﺼﻲ‬ ‫ﺑﺎﻟﺤﻠﺰون‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻳﻌﺮف‬ ‫آﺬﻟﻚ‬ ‫اﻟﺨﻂ‬ ‫ﺣﻮل‬ ‫ﻣﺘﻨﺎﻇﺮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬y x= −
  35. 35. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬35 Folium of Descartes ‫دﻳﻜﺎرت‬ ‫ﻣﻨﺤﻨﻲ‬ Folium ‫اﻟﺸﺠﺮ‬ ‫ورق‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 ( )[ ( )] 4x y y x x b axy+ + + = 2 cos 4 cos sinr b aθ θ θ= − + ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟ‬ ‫هﺬﻩ‬ ‫ﻣﻦ‬ ‫أﻧﻮاع‬ ‫ﺛﻼﺛﺔ‬ ‫ﻳﻮﺟﺪ‬‫ﺛﻨﺎﺋﻲ‬ ‫ﻓﻮﻟﻴﻮم‬ ‫و‬ ، ‫ﺑﺴﻴﻂ‬ ‫ﻓﻮﻟﻴﻮم‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫هﻲ‬ ‫و‬ ‫ﻤﻨﺤﻨﻴﺎت‬)‫ﻣﺰدوج‬(‫و‬ ‫ﺛﻼﺛﻲ‬ ‫ﻓﻮﻟﻴﻮم‬ ‫و‬ ، ‫اﻟﻰ‬ ‫ﻳﺮﺟﻊ‬ ‫هﺬا‬4b a=‫و‬0b =‫و‬b a= ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 3 3 3x y axy+ = 3 (3 sec tan )/(1 tan )r a θ θ θ= + 3 2 3 (3 )/(1 ) (3 )/(1 ) x at t y at t ⎧ = +⎪ ⎨ = +⎪⎩ 23 2 A a= ‫اﻟﻤﺴﺎﺣﺔ‬‫اﻟﻤﻐﻠﻘﺔ‬ ‫اﻟﻨﺎﺣﻴﺔ‬ ‫اﻟﻄﻮل‬‫اﻟﻤﻐﻠﻘﺔ‬ ‫اﻟﻨﺎﺣﻴﺔ‬4.91748L a= ‫ﺗﻮﺿﻴﺢ‬:‫ﻣﻦ‬ ‫أول‬‫ﻋﺎم‬ ‫دﻳﻜﺎرت‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻧﺎﻗﺶ‬1638 ‫ﻣﻌﺎدﻟﺘﻪ‬ ‫ﺧﻂ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻣﻘﺎرب‬0x y a+ + =
  36. 36. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬36 Frequency Curve ‫ﻣﻨﺤ‬‫اﻟﺘﺮدد‬ ‫ﻨﻲ‬ Freeth’s Nephroid ‫ُﻠﻮي‬‫آ‬ ‫ﻣﻨﺤﻨﻲ‬‫ﻓﺮﻳﺚ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ [1 2sin( / 2)]r a θ= + ‫اﻟﻤﺴﺎﺣﺔ‬2 (8 3 )A a π= +‫ا‬‫اﻟﺪاﺧﻠﻴﺔ‬ ‫ﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬21.203405L a=‫اﻟﻄﻮل‬ ‫آﻞ‬ ‫ﺗﻮﺿﻴﺢ‬:‫ا‬ ‫ﻣﻊ‬ ‫اﻟﺪاﺋﺮة‬ ‫ﺳﺘﺮوﻓﻮﺋﻴﺪ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬‫ﻟ‬‫ﻘﻄﺐ‬O‫اﻟﺜﺎﺑﺘﺔ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫و‬ ‫اﻟﻤﺮآﺰ‬ ‫ﻓﻲ‬P‫اﻟﺪاﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻞ‬ ‫ﻣﻦ‬ ‫ﺧﻂ‬ ‫رﺳﻤﻨﺎ‬ ‫إذا‬P‫اﻟﻤﺤﻮر‬ ‫ﻳﻮازي‬y‫اﻟﻨﻘﻄﺔ‬ ‫ﻗﻲ‬ ‫اﻟﻨﻴﻔﺮوﺋﻴﺪ‬ ‫ﺳﻴﻘﻄﻊ‬A‫اﻟﺰاوﻳﺔ‬AOP‫ﺗﺴﺎوي‬ 2 7 π ‫ﻳﻤﻜﻦ‬ ‫ﻣﻨﺘﻈﻢ‬ ‫أﺿﻼع‬ ‫ﺳﺒﺎﻋﻲ‬ ‫ﻟﺮﺳﻢ‬ ‫اﻟﺰاوﻳﺔ‬ ‫ﺑﻬﺬﻩ‬ ‫اﻷﺳﺘﻌﺎﻧﺔ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 22 x y eπ − = ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻷﺣﺼﺎء‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺑﻬﺬا‬ ‫ﻳﺴﺘﻌﺎن‬ ‫اﻟﻨﺎﻇﻤﻲ‬ ‫اﻟﺨﻄﺄ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫هﻮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬
  37. 37. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬37 Hyperbolic Spiral ‫هﺬﻟﻮﻟﻲ‬ ‫أو‬ ‫زاﺋﺪي‬ ‫ﺣﻠﺰون‬ Hyperbola ‫زاﺋﺪ‬ ‫ﻗﻄﻊ‬/‫ُﺬﻟﻮﻟﻲ‬‫ه‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 1 x y a b − = 2 [ ( 1)]/(1 cos )r a e e θ= − − ‫ﻣﺮآﺰﻳﺔ‬ ‫ﻻ‬ e ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻘﺎرب‬ ‫ﻣﻌﺎدﻟﺔ‬ b y x a = ± ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬cosh & sinhx a t y b t= =‫آﺬﻟﻚ‬ ‫و‬sec & tanx a t y b t= = ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ a r θ = cos sin t x a t t y a t ⎧ =⎪⎪ ⎨ ⎪ = ⎪⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻋﺎم‬ ‫ﺑﺮﻧﻮﻟﻲ‬ ‫ﻳﻮهﺎن‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻧﺎﻗﺶ‬1710
  38. 38. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬38 Hypotrochoid ‫هﺎﻳﺒﻮﺗﺮوآﻮﺋﻴﺪ‬/‫دﺣﺮوج‬ ‫ﻣﻨﺤﻨﻲ‬ Hypocycloid ‫دﺣﺮوج‬‫داﺧﻠﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ ( )cos cos ( )sin sin a b x a b b b a b y a b b b θ θ θ θ −⎧ = − −⎪⎪ ⎨ −⎪ = − + ⎪⎩ ‫اﻟﻤﺴﺎﺣﺔ‬2 2 [( 1)( 2) / ]nA n n n aπ= − − ‫اﻟﻄﻮ‬‫ل‬8 ( 1)/nL a n n= − /n a b= ‫ﺗﻮﺿﻴﺢ‬:‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫ﺗﺮﺳﻤﻪ‬ ‫اﻟﺬي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻗﻄﺮهﺎ‬b‫ﺛﺎﺑﺘﺔ‬ ‫أﺧﺮى‬ ‫داﺋﺮة‬ ‫داﺧﻞ‬ ‫اﻟﺪاﺋﺮة‬ ‫هﺬﻩ‬ ‫ﺗﺘﺪﺣﺮج‬ ‫ﻋﻨﺪﻣﺎ‬ ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬a ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ ( )cos cos( ) ( )sin sin( ) R r x R r t d t r R r y R r t d t r −⎧ = − +⎪⎪ ⎨ −⎪ = − − ⎪⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬p‫ﺑﻔﺎﺻﻠﺔ‬d‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻣﺮآﺰ‬ ‫ﻋﻦ‬r ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫داﺧﻞ‬ ‫أﻧﺰﻻق‬ ‫دون‬ ‫ﺗﺘﺪﺣﺮج‬R
  39. 39. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬39 Kamplyle of Eudoxus ‫أودوآﺴﻴﻮس‬ ‫ﻣﻨﺤﻨﻲ‬ Involute of Circle ‫اﻟﺪاﺋﺮة‬ ‫ﻣﻦ‬ ‫اﻟﻤﻨﺸﺄ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ (cos sin ) (sin cos ) x a t t t y a t t t = +⎧ ⎨ = −⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬21 ( ) 2 L t at= ‫ﺗﻮﺿﻴﺢ‬:‫داﺋﺮة‬ ‫ﺣﻮل‬ ‫ﻳﻠﻒ‬ ‫ﻋﻨﺪﻣﺎ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻋﻠﻰ‬ ‫ﺛﺎﺑﺘﺔ‬ ‫ﻧﻘﻄﺔ‬ ‫ﻣﻮﺿﻊ‬ ‫ﻣﻦ‬ ‫اﻟﻨﺎﺷﺊ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﻮ‬)‫اﻟﺪاﺋﺮة‬ ‫ﻋﻠﻰ‬ ‫ﻣﻤﺎس‬ ‫اﻟﻤﺴﺘﻘﻴﻢ‬( ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 4 2 2 2 ( )x a x y= + 2 secr a θ= sec tan sec x a t y a t t =⎧ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻜﻌﺐ‬ ‫ﺗﻀﻌﻴﻒ‬ ‫ﻣﺴﺌﻠﺔ‬ ‫ﻣﻄﺎﻟﻌﺔ‬ ‫و‬ ‫ﺣﻞ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫أﺳﺘﻌﻤﻞ‬
  40. 40. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬40 Lame Curves ‫ﻻﻣﻪ‬ ‫ﻣﻨﺤﻨﻴﺎت‬ Kappa Curve ‫آﺎﺑﺎ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 ( )y x y a x+ = tanr a θ= cos cot cos x a t t y a t =⎧ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿ‬‫ﻴﺢ‬:‫ﻋﺎم‬ ‫آﺎﻧﺖ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﺬهﺎ‬ ‫ﻣﻄﺎﻟﻌﻪ‬ ‫أول‬1662 ‫ﺑﺮﻧﻮﻟﻲ‬ ‫و‬ ‫ﻧﻴﻮﺗﻦ‬ ‫ﻣﻦ‬ ‫آﻞ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻃﺎﻟﻊ‬ ‫آﺬﻟﻚ‬ ‫آﺎﺑﺎ‬ ‫اﻟﻴﻮﻧﺎﻧﻲ‬ ‫اﻟﺤﺮف‬ ‫ﻣﻦ‬ ‫ﻣﺸﺘﻖ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫أﺳﻢ‬κ. ً‫ﺎ‬‫داﺋﻤ‬ ‫آﺎﺑﺎ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻓﻲ‬OP CD=‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫آﻤﺎ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ ( ) ( ) 1n nx y a b + = ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻟﺤﺎﻟﺔ‬ ‫هﻮ‬ ‫أﻋﻼﻩ‬ ‫اﻟﺼﻮرة‬ ‫ﻓﻲ‬ ‫اﻟﻤﺮﺳﻮم‬ ‫اﻟﻤﻨﺤﻨﻲ‬4n = ‫ﺗﺄﺧﺬ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻟﺮﺳﻢ‬n‫ﻓﻘﻂ‬ ‫اﻟﺼﺤﻴﺤﺔ‬ ‫اﻷﻋﺪاد‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﺘﺼﺮ‬ ‫ﻻ‬ ‫و‬ ‫اﻷﻋﺪاد‬ ‫ﺟﻤﻴﻊ‬ ‫آﺎﻧﺖ‬ ‫إذا‬n‫ﺟﺒﺮي‬ ‫ﻣﻨﺤﻨﻲ‬ ‫هﻮ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ُﻨﻄﻖ‬‫ﻣ‬ ‫ﻋﺪد‬ ‫ﻏﻴ‬ ‫آﺎﻧﺖ‬ ‫إذا‬‫ﻣﺘﺴﺎم‬ ‫ﻣﻨﺤﻨﻲ‬ ‫هﻮ‬ ‫ﻓﺎﻟﻤﻨﺤﻨﻲ‬ ‫أﺻﻢ‬ ‫أي‬ ‫ُﻨﻄﻖ‬‫ﻣ‬ ‫ﺮ‬ 2/3n = ⇒ astroid 5 / 2n = ⇒ super ellipse 2n = ⇒ ellipse 3n = ⇒ witch of agnesi 4n = ⇒ rectangular ellipse
  41. 41. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬41 Limacon of Pascal ‫َﻓﺔ‬‫ﺪ‬َ‫ﺻ‬ ‫ﻣﻨﺤﻨﻲ‬‫ﺑﺎﺳﻜﺎل‬ Lemniscate Bernolli ‫اﻟﻌﺮوﺗﻴﻦ‬ ‫ذو‬ ‫ﺑﺮﻧﻮﻟﻲ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 2 ( ) 2 ( )x y a x y+ = − 2 2 2 cos2r a θ= 2 2 ( cos ) /(1 sin ) ( sin cos ) /(1 sin ) x a t t y a t t t ⎧ = +⎪ ⎨ = +⎪⎩ ‫ا‬‫ﻟﻤﺴﺎﺣﺔ‬2 A a= ‫اﻟﻄﻮل‬5.24411L a= ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻃﺎﻟﻊ‬‫ﻋﺎم‬ ‫ﺑﺮﻧﻮﻟﻲ‬1694 ‫ﻧﻘﻄﺘﻴﻦ‬ ‫ﻣﻦ‬ ‫ﻓﺎﺻﻠﺘﻬﺎ‬ ‫ﺿﺮب‬ ‫ﺣﺎﺻﻞ‬ ‫ﻧﻘﻂ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬)‫ﻓﺎﺻﻠﺘﻬﻤﺎ‬ ‫ﺛﺎﺑﺘﺘﻴﻦ‬2a(‫ﺗﺴﺎوي‬2 a ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺑﺆرﺗﻲ‬ ‫هﻤﺎ‬ ‫اﻟﺜﺎﺑﺘﺘﻴﻦ‬ ‫اﻟﻨﻘﻄﺘﻴﻦ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 2 ( ) ( )x y ax b x y+ − = + cosr b a θ= + ( cos )cos ( cos )sin x b a t t y b a t t = +⎧ ⎨ = +⎩ ‫اﻟﻤﺴﺎﺣﺔ‬‫اﻟﺤﻠﻘﺘﻴﻦ‬ ‫ﺑﻴﻦ‬ 2 2 2 2 1 3 ( 2 )sin ( ) b A b a b a b a − = − + + ‫ﺗﻮﺿﻴﺢ‬:‫ﺗﻌﻨﻲ‬ ‫ﻻﺗﻴﻨﻴﺔ‬ ‫آﻠﻤﺔ‬ ‫ﻟﻴﻤﺎآﻮن‬snail‫ﺣﻠﺰون‬ ‫أي‬ ‫ﻓﺎﺻﻠﺘﻬﺎ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬b‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻣﺮآﺰ‬ ‫ﻣﻦ‬ ‫ﻗﻄﺮهﺎ‬a‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫أﺧﺮى‬ ‫داﺋﺮة‬ ‫ﺧﺎرج‬ ‫ﺣﻮل‬ ‫أﻧﺰﻻق‬ ‫دون‬ ‫ﺗﺘﺪﺣﺮج‬a.
  42. 42. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬42 Lituus ‫ﺑﻮﻗﻲ‬ ‫ﻣﻨﺤﻨﻲ‬ Lissajous Curve ‫ﻟﻴﺴﺎﺟﻮ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ sin( ) sin x a t y b t ω δ= −⎧ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻟﻬﺬ‬‫ا‬‫اﻟﻔﻠﻚ‬ ‫و‬ ‫اﻟﻔﻴﺰﻳﺎء‬ ‫ﻓﻲ‬ ‫أﺳﺘﻌﻤﺎﻻت‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻋﺎم‬ ‫ﻟﻴﺴﺎﻳﻮس‬ ‫ﻃﺎﻟﻌﻪ‬1857 ‫اﻟﻤﺮآﺒﺔ‬ ‫اﻟﺘﻮاﻓﻴﻘﻴﺔ‬ ‫اﻟﺤﺮآﺔ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻳﺒﻴﻦ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 a r θ = ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﺗﻌﻨﻲ‬ ‫ﻻﺗﻴﻨﻴﺔ‬ ‫آﻠﻤﺔ‬ ‫ﻟﻴﺘﻮس‬crook‫اﻟ‬ ‫أي‬‫اﻷﻧﻌﻘﺎف‬ ‫أو‬ ‫اﻟﺮاﻋﻲ‬ ‫ﻋﺼﻰ‬ ‫أو‬ ‫ﻤﺤﺘﺎل‬ ‫إﻟﻴﻪ‬ ‫ﻳﺼﻞ‬ ‫أن‬ ‫دون‬ ‫اﻹﺣﺪاﺛﻲ‬ ‫ﻣﺒﺪأ‬ ‫ﺣﻮل‬ ‫ﻳﻠﺘﻒ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫و‬ ‫اﻷﻓﻘﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ُﻘﺎرب‬‫ﻣ‬ ‫واﺣﺪ‬ ‫إﺣﺪاﺛﻲ‬ ‫ﻋﻠﻰ‬ ‫ﺑﻮﻗﻴﻴﻦ‬ ‫ﻣﻨﺤﻨﻴﻴﻦ‬ ‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬
  43. 43. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬43 Nephroid ُ‫ﻜ‬‫اﻟ‬ ‫اﻟﻤﻨﺤﻨﻲ‬‫ﻠﻮي‬ Neile’s Semi-Cubical Parabola ‫ﻗﻄﻊ‬‫ﺗﻜﻌﻴﺒﻲ‬ ‫اﻟﺸﺒﻪ‬ ‫ﻧﻴﻞ‬ ‫ﻣﻜﺎﻓﺊ‬/‫ﺗﻜﻌﻴﺒﻲ‬ ‫ﺷﺒﻪ‬ ‫ﻣﻜﺎﻓﺊ‬ ‫ﻗﻄﻊ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 3 2 y ax= ± 2 (tan sec ) /r aθ θ= 2 3 x t y at ⎧ =⎪ ⎨ =⎪⎩ ‫اﻟﻄﻮل‬2 31 8 ( ) (4 9 ) 27 27 L t t= + − ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻨﺸﺄ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬)involute(‫اﻟﻤﻜﺎﻓﺊ‬ ‫ﻟﻠﻘﻄﻊ‬ ‫ا‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬‫ﻟﻤﻨﺤﻨﻲ‬ 2 2 2 3 4 2 ( 4 ) 108x y a a y+ − = 2 2 0.5 (5 3cos2 )r a θ= − (3cos cos3 ) (3sin sin3 ) x a t t y a t t = −⎧ ⎨ = −⎩ ‫اﻟﻤﺴﺎﺣﺔ‬2 12A aπ= ‫اﻟﻄﻮل‬24L a= ‫ﺗﻮﺿﻴﺢ‬:‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮة‬ ‫ﻣﺤﻴﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫هﻨﺪس‬ ‫ﻣﺤﻞ‬0.5a ‫ﻗﻄﺮهﺎ‬ ‫ﻧﺼﻒ‬ ‫أﺧﺮى‬ ‫داﺋﺮة‬ ‫ﺧﺎرج‬ ‫أﻧﺰﻻق‬ ‫دون‬ ‫ﺗﺘﺪﺣﺮج‬a ‫آﻠﻤﺔ‬ ‫ﻣﻌﻨﻰ‬nephroid‫أي‬kidney shaped‫اﻟﻜﻠﻴﻮي‬ ‫اﻟﺸﻜﻞ‬
  44. 44. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬44 Parabola ‫اﻟﻤﻜﺎﻓﺊ‬ ‫اﻟﻘﻄﻊ‬/ُ‫ﺸ‬‫اﻟ‬‫ﻠﺠﻤﻲ‬ Newton’s Diverging Parabolas ‫ُﻠﺠﻤﻴﺎت‬‫ﺷ‬‫اﻟﻤﺘﺒﺎﻳﻨﺔ‬ ‫ﻧﻴﻮﺗﻦ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 ( 2 )a y x x bx c= − + ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺑﺠﺬور‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫أﻧﻮاع‬ ‫ﺗﺮﺗﺒﻂ‬2 ( 2 ) 0x x bx c− + = ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 y ax bx c= + + 2 /(1 cos )r a θ= + 2 2 x at y at ⎧ = ⎨ =⎩ ‫اﻟﻤﺴ‬‫ﺎﺣﺔ‬ ‫اﻟﻄﻮل‬2 1 ( ) 1 sinhL t at t t− = + + ‫اﻟﺘﻮﺿﻴﺢ‬:
  45. 45. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬45 Pear-Shaped Quaric ‫اﻟﻜﻤﺜﺮي‬ ‫اﻟﺸﻜﻞ‬ ‫ﻣﻨﺤﻨﻲ‬ Pearls of Sluze ‫ﺳﻠﻮزا‬ ‫ﻵﻟﻲ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ ( )n p m y k a x x= − ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻌﺎدﻟﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬m‫و‬n‫و‬p‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻃﺎﻟﻊ‬ ، ‫ﺻﺤﻴﺤﺔ‬ ‫أﻋﺪاد‬de Sluze‫ﻋﺎم‬1657 ‫ﻟﻸﻋﺪاد‬ ‫اﻟﻤﺮﺳﻮم‬ ‫اﻟﺸﻜﻞ‬4, 2, 3, 4, 2n m p a k= = = = = ‫اﻟﻤ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬‫ﻨﺤﻨﻲ‬ 2 2 3 ( )b y x a x= − 2 3 (1 )y x x= − ‫اﻟﻜﺎرﺗﻴﺰﻳﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫هﺬﻩ‬ ‫آﺬﻟﻚ‬ 1 sin (1 sin )cos x t y t t = +⎧ ⎨ = +⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻟﻌﺎم‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻣﻄﺎﻟﻌﺔ‬ ‫ﺗﺎرﻳﺦ‬ ‫ﻳﺮﺟﻊ‬1886 ‫اﻟﺮاﺑﻌﺔ‬ ‫اﻟﺪرﺟﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﻣﻦ‬
  46. 46. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬46 Pursuit Curve ‫اﻟﻤﻄﺎردة‬ ‫ﻣﻨﺤﻨﻲ‬ Plateau Curves ‫ﺗﻴﻮ‬ ‫ﺑﻼ‬ ‫ﻣﻨﺤﻨﻴﺎت‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ ( ) [ sin( ) ]/[sin( ) ] ( ) [2 sin( )sin( )]/[sin( ) ] x t a m n t m n t y t a mt nt m n t = + −⎧ ⎨ = −⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬2m n=‫داﺋﺮة‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 logy cx x= − ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻟﻌﺎم‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻣﻄﺎﻟﻌﺔ‬ ‫ﺗﺮﺟﻊ‬1732 ‫اﻟﻨﻘﻄﺔ‬ ‫آﺎﻧﺖ‬ ‫إذا‬A‫اﻟﻨﻘﻄﺔ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ، ‫ﻣﻌﻠﻮم‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺘﺤﺮك‬P‫اﻟﻤﻄﺎردة‬ ‫ﻣﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫ﻧﺤﻮ‬ ‫ﻣﺘﺠﻬﺔ‬ ً‫ﺎ‬‫داﺋﻤ‬ ‫آﺎﻧﺖ‬ ‫إذا‬A‫ا‬ ‫ﻧﻔﺲ‬ ‫ﻓﻲ‬ ‫و‬ ،‫ﻟﻮﻗﺖ‬A‫و‬P‫ﺛﺎﺑﺘﺔ‬ ‫ﺑﺴﺮﻋﺔ‬ ‫ﻳﺘﺤﺮآﺎن‬.
  47. 47. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬47 Rhodonea Curve ‫اﻟﻮرود‬ ‫ﻣﻨﺤﻨﻲ‬ Quadratrix of Hippias ‫هﻴﺒﻴﺎس‬ ‫ﺗﺮﺑﻴﻌﻴﺔ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ cot( / 2 )y x x aπ= (2 ) /( sin )r aθ π θ= (2 / ) (2 / )cot x at y at t π π =⎧ ⎨ =⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﺰاوﻳﺔ‬ ‫ﺗﺜﻠﻴﺚ‬ ‫ﻣﺴﺌﻠﺔ‬ ‫ﻣﻄﺎﻟﻌﺔ‬ ‫و‬ ‫ﻟﺤﻞ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫أﺳﺘﻌﻤﻞ‬ ‫ر‬ ‫ﻣﻌﺎدﻟﺔ‬‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺳﻢ‬ sin( )r a kθ= ‫اﻟﻤﺴﺎﺣﺔ‬‫آﺎﻧﺖ‬ ‫إذا‬k‫زوج‬ ‫ﻋﺪد‬ 2 2 a A π =‫آﺎﻧﺖ‬ ‫إذا‬ ،k‫ﻓﺮدي‬ ‫ﻋﺪد‬ 2 4 a A π = ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻟﻌﺎم‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻣﻄﺎﻟﻌﺔ‬ ‫ﺗﺮﺟﻊ‬1723 ‫ﺑﺎﻟﻤﺘﻐﻴﺮ‬ ‫ﻳﺮﺗﺒﻂ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻓﻲ‬ ‫اﻟﺒﺘﻼت‬ ‫ﻋﺪد‬k‫آﺎ‬ ‫إذا‬ ،‫ن‬k‫ﻳﺴﺎوي‬ ‫اﻟﺒﺘﻼت‬ ‫ﻋﺪد‬ ‫زوﺟﻲ‬k‫اﻟﺒﺘﻼت‬ ‫ﻋﺪد‬ ‫ﻓﺮدي‬ ‫آﺎن‬ ‫إذا‬ ‫و‬ ‫ﻳﺴﺎوي‬k2.‫آﺎن‬ ‫إذا‬k‫ﻧﻬﺎﺋﻲ‬ ‫ﻻ‬ ‫اﻟﺒﺘﻼت‬ ‫ﻋﺪد‬ ‫أﺻﻢ‬ ‫أي‬ ‫ُﻨﻄﻖ‬‫ﻣ‬ ‫ﻏﻴﺮ‬ ‫ﻋﺪد‬ 1k =، ‫داﺋﺮة‬ ‫اﻟﻤﻨﺤﻨﻲ‬2k =، ‫أرﺑﻌﺔ‬ ‫اﻟﺒﺘﻼت‬ ‫ﻋﺪد‬5k =‫ﺧﻤﺴ‬ ‫اﻟﺒﺘﻼت‬ ‫ﻋﺪد‬‫ﺔ‬
  48. 48. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬48 Serpentine ‫ُﻠﺘﻒ‬‫ﻤ‬‫اﻟ‬ ‫ﻣﻨﺤﻨﻲ‬/‫اﻷﻓﻌﻮاﻧﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ Right Strophoid ‫ﺳﺘﺮوﻓﻮﺋﻴﺪ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 [( ) ]/( )y a x x a x= − + cos2 secr a θ θ= 2 2 2 2 2 2 2 2 ( ) /( ) ( ) /( ) x a t t a y t a t t a ⎧ = − +⎪ ⎨ = − +⎪⎩ ‫اﻟﻤﺴﺎﺣﺔ‬‫ﻳﺴﺎوﻳﺎن‬ ‫آﻼهﻤﺎ‬ ‫و‬ ‫اﻟﻤﻐﻠﻘﺔ‬ ‫اﻟﻨﺎﺣﻴﺔ‬ ‫ﻣﺴﺎﺣﺔ‬ ‫ﺗﺴﺎوي‬ ‫اﻟﻤﻘﺎرب‬ ‫و‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺑﻴﻦ‬ 2 0.5 (4 )A a π= − ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻣﺜﻞ‬ ‫ﻧﻘﻄﺘﻴﻦ‬ ‫هﻨﺪﺳﻲ‬ ‫ﻣﺤﻞ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬ ‫اﻟﺴﺘﺮوﻓﻮﺋﻴﺪ‬1P‫و‬2P‫اﻟﺨﻂ‬ ‫ﺑﺤﻴﺚ‬ L‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻳﻘﻄﻊ‬C‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬K‫اﻟﺜﺎﺑﺘﺔ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﺎﺻﻠﺔ‬ ،A‫ﻣﻦ‬K‫ﻓﺎﺻﻠﺘﻬﺎ‬ ‫ﺗﺴﺎوي‬ ‫ﻣﻦ‬1P‫و‬2P‫أي‬1 2AK KP KP= =‫اﻟﻨﻘﻄﺔ‬O‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻗﻄﺐ‬.‫اﻟﻤﻨﺤﻨﻲ‬ ‫آﺎن‬ ‫إذا‬ C‫و‬ ‫ﺧﻂ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬OA‫اﻟﻤﻨﺤﻨﻲ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫هﺬﻩ‬ ‫ﻓﻲ‬ ‫ﻋﻠﻴﻪ‬ ‫ﻋﻤﻮد‬Right Striphoid ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 0x y aby a x+ − = 0ab > ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻋﺎم‬ ‫ﻟﻨﻴﻮﺗﻦ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﺗﺴﻤﻴﺔ‬ ‫ﺗﺮﺟﻊ‬1701‫اﻟﻤﻠﺘﻮﻳﺔ‬ ‫ّﺔ‬‫ﻴ‬‫ﺑﺎﻟﺤ‬ ‫ﺷﺒﻴﻪ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺷﻜﻞ‬ ،. ‫ﻣﻦ‬ ‫ﻟﻤﺠﻤﻮﻋﺔ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻳﺮﺟﻊ‬‫ﻣﻌﺎدﻟﺘﻬﺎ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬2 2 3 2 x y ey ax bx cx d+ = + + +
  49. 49. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬49 Spiral of Archimedes ‫أرﺧﻤﻴﺪس‬ ‫ﺣﻠﺰون‬/‫أرﺧﻤﻴﺪس‬ ‫ﻟﻮﻟﺐ‬ Sinusoidal Spirals ‫ﺟﻴﺒﻴﺔ‬ ‫ﻟﻮاﻟﺐ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ cos( )n n r a nθ= ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:n‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫اﻟﻤﻌﺎدﻻت‬ ‫أﻧﻮاع‬ ‫ﺗﺸﻜﻴﻞ‬ ‫ﻳﻤﻜﻦ‬ ، ‫ُﻨﻄﻖ‬‫ﻣ‬ ‫ﻋﺪد‬ 1n = −، ‫ﺧﻂ‬ ‫اﻟﻤﻨﺤﻨﻲ‬1n =، ‫داﺋﺮة‬ ‫اﻟﻤﻨﺤﻨﻲ‬ 1 2 n =‫آﺎردﻳﻮﺋﻴﺪ‬ ‫اﻟﻤﻨﺤﻨﻲ‬cardioid، 1 2 n = −‫ﻗﻄﻊ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ، ‫ﻣﻜﺎﻓﺊ‬2n = −، ‫هﺬﻟﻮﻟﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬2n =‫ﻟﻴﻤﻨﺴﻜﺎت‬ ‫اﻟﻤﻨﺤﻨﻲ‬. ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ r aθ= ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬2 21 ( ) [ 1 ln( 1 )] 2 L aθ θ θ θ θ= + + + + ‫ﺗﻮﺿﻴﺢ‬:‫أرﺧﻤﻴﺪس‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻃﺎﻟﻊ‬225‫ﻗﺒﻞ‬‫اﻟﻤﻴﻼد‬
  50. 50. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬50 Straight line ‫اﻟﻤﺴﺘﻘﻴﻢ‬ ‫اﻟﺨﻂ‬ Spiric Sections ‫ُﺴﺘﺪﻗﺔ‬‫ﻣ‬ ‫َﻘﺎﻃﻊ‬‫ﻣ‬ ‫ﻣﻌ‬‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﺎدﻟﺔ‬ 2 2 2 2 2 2 2 2 2 ( ) 4 ( )r a c x y r x c− + + + = + ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫ﻃﺎرة‬ ‫ﻣﻊ‬ ‫ﺻﻔﺤﺔ‬ ‫ﺗﻘﺎﻃﻊ‬ ‫ﻣﻦ‬ ‫اﻟﻨﺎﺗﺞ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﻮ‬)torus(‫اﻟﺼﻔﺤﺔ‬ ‫هﺬﻩ‬ ‫ﺑﺤﻴﺚ‬ ‫اﻟﻄﺎرة‬ ‫ﻣﺮآﺰ‬ ‫ﻣﻦ‬ ‫اﻟﻤﺎر‬ ‫اﻟﺨﻂ‬ ‫ﻣﻊ‬ ‫ﻣﻮازﻳﺔ‬.‫اﻟﻰ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻣﻄﺎﻟﻌﺔ‬ ‫ﺗﺮﺟﻊ‬150‫م‬ ‫ق‬ a‫اﻟﻄﺎرة‬ ‫ﻣﻘﻄﻊ‬ ‫ﻗﻄﺮ‬ ‫ﻧﺼﻒ‬ r‫اﻟﺪوران‬ ‫ﻗﻄﺮ‬ ‫ﻧﺼﻒ‬‫اﻟﻄﺎرة‬ ‫ﻣﺤﻮر‬ ‫ﺣﻮل‬ c‫اﻟﻄﺎرة‬ ‫ﻣﺮآﺰ‬ ‫و‬ ‫اﻟﺼﻔﺤﺔ‬ ‫ﺑﻴﻦ‬ ‫اﻟﻔﺎﺻﻠﺔ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ y mx b= + x at b y ct d = +⎧ ⎨ = +⎩ ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:
  51. 51. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬51 Tractrix ‫اﻟﻤﻤﺎﺳﺎت‬ ‫ﻣﺘﺴﺎوي‬ ‫ﻣﻨﺤﻨﻲ‬ Talbot’s curve ‫ﺗﺎﻟﺒﻮت‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﻣﻌﺎدﻟﺔ‬ 2 2 2 2 2 2 (1 sin )cos (1 2 sin )sin 1 x a e t t a e e t t y e ⎧ = + ⎪ ⎡ ⎤− +⎨ ⎣ ⎦=⎪ −⎩ ، 2 2 1 b e a = − ‫اﻟﻤﺴﺎﺣﺔ‬ ‫اﻟﻄﻮل‬ ‫ﺗﻮﺿﻴﺢ‬:‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫ﻓﻲ‬ 1 1 2 e< <‫هﻮ‬ ‫و‬‫اﻟﺪواﺳﻪ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫اﻟﻘﺪﻣﻲ‬ ‫ﻣﻨﺤﻨﻲ‬ ‫أو‬)pedal curve(‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻌ‬‫اﻟﻤﻨﺤﻨﻲ‬ ‫رﺳﻢ‬ ‫ﺎدﻟﺔ‬ 2 2 2 2 ( ln[( )/ ] )y a a a x x a x= ± + − − − ‫اﻟﻜﺎرﺗﻴﺰﻳﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ 1 2 2 sec ( / )y a h x a a x− = − − ‫اﻟﻜﺎرﺗﻴﺰﻳﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ( ) 1/ cosh ( ) tanh x t t y t t t =⎧ ⎨ = −⎩ ‫اﻟﻮﺳﻴﻄﻴﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫اﻟﻤﺴﺎﺣﺔ‬‫ﺗﺴﺎوي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﺗﺤﺖ‬ 2 2 a A π = ‫اﻟﻄﻮل‬‫ﻧﻘﻄﺘﻴﻦ‬ ‫ﺑﻴﻦ‬1 2 1 2( ) ln( / )L x x a x x→ =‫أو‬( ) ln coshL t a t= ‫ﺗﻮﺿﻴﺢ‬:‫اﻟ‬ ‫هﺬا‬ ‫ﻃﺎﻟﻊ‬ ‫ﻣﻦ‬ ‫أول‬‫ﻋﺎم‬ ‫هﻮﻳﻐﻨﺲ‬ ‫ﻤﻨﺤﻨﻲ‬1692 ‫ﺗﻌﺘﺒﺮ‬ ، ‫اﻟﺴﻠﺴﻠﺔ‬ ‫ﻟﻤﻨﺤﻨﻲ‬ ‫ﻣﻨﺸﺄ‬ ‫هﻮ‬a‫ﻣﺤﻮر‬ ‫ﺣﻮل‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﺬا‬ ‫دوران‬ ‫ﻣﻦ‬ ‫اﻟﻨﺎﺗﺞ‬ ‫اﻟﺸﻜﻞ‬ ‫و‬ ‫ﻗﻄﺮﻩ‬ ‫ﻧﺼﻒ‬y‫اﻟﻜﺎذﺑﺔ‬ ‫ﺑﺎﻟﻜﺮة‬ ‫ﻳﻌﺮف‬ ‫آﺮة‬ ‫ﺷﺒﻪ‬ ‫أو‬pseudo-sphere ‫ﻳﺴﺎوي‬ ‫و‬ ‫ﺛﺎﺑﺖ‬ ‫ﻣﻤﺎﺳﻪ‬ ‫ﻃﻮل‬ ‫اﻟﺬي‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫هﻮ‬a.‫ﻣﻘﺎرﺑﻪ‬ ‫اﻟﻰ‬ ‫اﻟﻤﻨﺤﻨﻲ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫اﻟﻤﻤﺎس‬ ‫ﻃﻮل‬)‫ﻣﺤﻮر‬y(‫ﻳﺴ‬‫ﺎوي‬a ‫إﻗﻠﻴﺪﻳﺔ‬ ‫ﻻ‬ ‫هﻨﺪﺳﺔ‬ ‫ﻟﺒﻨﺎء‬ ‫اﻟﻤﻬﻤﺔ‬ ‫اﻟﻤﻨﺤﻨﻴﺎت‬ ‫ﻣﻦ‬

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