أشهر المنحنيات والمجسمات

1. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬1

2. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬2

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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫ﻤ‬‫اﻟ‬ ‫أﺷﻬﺮ‬‫ﻨﺤﻨﻴﺎ‬‫ت‬‫ا‬ ‫و‬ُ‫ﻤ‬‫ﻟ‬‫ﺠﺴﻤﺎت‬‫اﻟﻬﻨﺪﺳﻴﺔ‬‫ﻋﺒﺪ‬ ‫اﻟﺤﺎج‬ ‫ﺟﻼل‬23

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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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. ُ‫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