Elliptic Paraboloid Equation – 3D Analytic Geometry :: Danielitte

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Analytical Geometry - Elliptic Paraboloid

Analytic Geometry – Elliptic Paraboloid

Equation and Properties of Elliptic Paraboloid

An elliptic paraboloid is a three-dimensional surface shaped like a paraboloid with elliptical cross-sections. It opens along the Z-axis and has parabolic curves in the xz- and yz-planes.

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

Elliptic Paraboloid

Key Components:

\(a, b\): Control the curvature in x- and y-directions respectively.
\(c\): Determines the rate of change of \(z\) with respect to the radial distance from the origin.
Opens upward along the z-axis if \(c > 0\), downward if \(c < 0\).

Key Properties of Elliptic Paraboloids:

Each horizontal cross-section (\(z = \text{constant}\)) is an ellipse.
Each vertical cross-section is a parabola.
The surface is smooth and continuous, with a single vertex (minimum or maximum point).
It is symmetric with respect to the z-axis.

Applications of Elliptic Paraboloids:

Design of satellite dishes and telescopes (reflector shape).
Used in architectural structures like parabolic domes and bridges.
Modeling projectile motion in physics and engineering simulations.
Optimization problems in mathematics and economics.