Analytical Geometry - Elliptic Cylinder With Axis As Z Axis

 
   
     
       

Analytic Geometry – Elliptic Cylinder

       

Elliptic Cylinder with Axis Along Z-axis

       

          An elliptic cylinder is a type of cylindrical surface formed by extending an ellipse along a straight line — typically the Z-axis. The equation represents all points \((x, y, z)\) such that every cross-section perpendicular to the Z-axis is an ellipse.        

       

          \[           \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1           \]        

       
          Elliptic Cylinder Equation        
     
     
       
         

Key Components:

         
               
  • \(a, b\): Represent the semi-major and semi-minor axes of the elliptical cross-section.
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  • This equation is independent of \(z\), meaning the shape extends infinitely along the Z-axis.
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Key Properties of Elliptic Cylinders:

         
               
  • Each cross-section perpendicular to the Z-axis is an ellipse.
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  • The surface is infinite along the Z-axis.
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  • If \(a = b\), the cylinder becomes a circular cylinder.
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  • The surface is symmetric about the origin and each axis.
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Applications of Elliptic Cylinders:

         
               
  • Used in architecture for vaulted ceilings or tunnels with elliptical arches.
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  • Appears in physics and engineering problems involving elliptical motion or waveguides.
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  • Applied in 3D computer graphics and CAD modeling of elliptical tubes and pipes.
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  • Helpful in solving Laplace's equation in cylindrical coordinates for elliptical domains.
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