Analytical Geometry - Hyperboloid Of One Sheets

 
   
     
       

Analytic Geometry – Hyperboloid of One Sheet

       

Equation and Properties of Hyperboloid of One Sheet

       

          A hyperboloid of one sheet is a type of quadric surface that appears curved along all axes but remains connected. It looks like an hourglass or a cooling tower.        

     
     
       
         
Standard Equation:
         

            \[             \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1             \]             where \(a\), \(b\), and \(c\) are real, positive constants defining the axis scaling.          

         
            Hyperboloid of One Sheet          
         

Key Properties:

         
               
  • It is a **ruled surface**, meaning it can be generated by straight lines.
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  • The cross-sections parallel to the x-y plane are **ellipses**.
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  • Cross-sections parallel to the x-z or y-z plane are **hyperbolas**.
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  • It is continuous and extends infinitely in all directions.
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  • The surface is **doubly ruled**, useful in structural design.
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Applications:

         
               
  • Used in **architecture** (e.g., cooling towers, hyperboloid buildings).
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  • Seen in **physics** as potential surfaces or wavefronts.
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  • Important in **structural mechanics** for tension-compression balance.
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  • Used in **art and design** due to its symmetrical and graceful shape.
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