Hyperbolic Paraboloid Example

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2014-12-16 10:17

This example demonstrates the 3D graph of a hyperbolic paraboloid and its properties. Key math concepts include solid geometry, function graphing, and surface visualization. It uses GeoGebra's Surface command to construct the paraboloid and highlights focal points with Focal_{max} and Focal_{min} objects. Suitable for high school and above students to learn spatial geometry and function graphs, ideal for educational demonstrations and geometric concept visualization.

GeoGebra Commands & Parameters

Surface(u, v, (v^(2) - u^(2)) / 2, u, -10, 10, v, -10, 10) → Surface
Surface(u + u / ((((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)) + sqrt(((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)))^(2) + (1 + u^(2) + v^(2))^(-2))) * (u^(2) + v^(2) + 1)^(0.5))), v - v / ((((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)) + sqrt(((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)))^(2) + (1 + u^(2) + v^(2))^(-2))) * (u^(2) + v^(2) + 1)^(0.5))), (v^(2) - u^(2)) / 2 + 1 / ((((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)) + sqrt(((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)))^(2) + (1 + u^(2) + v^(2))^(-2))) * (u^(2) + v^(2) + 1)^(0.5))), u, -10, 10, v, -10, 10) → Focal_{max}
Surface(u + u / ((((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)) - sqrt(((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)))^(2) + (1 + u^(2) + v^(2))^(-2))) * (u^(2) + v^(2) + 1)^(0.5))), v - v / ((((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)) - sqrt(((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)))^(2) + (1 + u^(2) + v^(2))^(-2))) * (u^(2) + v^(2) + 1)^(0.5))), (v^(2) - u^(2)) / 2 + 1 / ((((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)) - sqrt(((((u^(2) - v^(2)) * 0.5) * (1 + u^(2) + v^(2))^(((-3)) / 2)))^(2) + (1 + u^(2) + v^(2))^(-2))) * (u^(2) + v^(2) + 1)^(0.5))), u, -10, 10, v, -10, 10) → Focal_{min}

AI Drawing Prompt Reference

Create a 3D hyperbolic paraboloid (saddle surface) defined by z=(v^2-u^2)/2 with parameters u and v ranging from -10 to 10. Add two focal surfaces, one for the maximum and one for the minimum focal points, obtained by offsetting the original surface along its normal direction with a distance derived from curvature-related formulas. The result shows the main saddle surface with two auxiliary surfaces that illustrate its geometric properties.

The prompt above was compiled from the original drawing commands and is for reference only.