Hyperbolic Paraboloid – Equation and 3D Surface :: Danielitte

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Analytical Geometry - Hyperbolic Paraboloid

Analytic Geometry – Hyperbolic Paraboloid

Equation and Properties of Hyperbolic Paraboloid

A hyperbolic paraboloid is a saddle-shaped surface that curves upward in one direction and downward in the other.

Standard Equation:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c} \] where \( a, b, c \) are constants that scale the surface along each axis.

Hyperbolic Paraboloid

Key Properties:

It is a **saddle surface** – curves up along one axis and down along another.
Every cross-section parallel to the xz-plane is a **parabola**.
Every cross-section parallel to the yz-plane is a **parabola** (with opposite curvature).
Cross-sections parallel to the xy-plane are **hyperbolas**.
The surface has a **saddle point** at the origin (0,0,0) if centered there.

Applications:

Used in **architecture** for modern roofs and bridges (e.g., saddle roofs).
Found in **reflector antennas** and solar panels to optimize angle of incidence.
Appears in **optimization problems** and **differential geometry**.
Studied in **economics** in utility and cost functions with saddle point behavior.