Interactive 3D Surface Grapher. Visualize Paraboloids, Hyperboloids, Torus knots, and Spheres in real-time.

Shape Properties

Select a shape to see its formula.
    This calculation runs locally in your browser. View Source

    How to Use

    Exploring 3D Topology

    Topologies are the shapes of space. While 2D graphs show lines, 3D graphs show surfaces. This tool helps you visualize how changing parameters affects the curvature of space.

    The Shapes

    1. Elliptic Paraboloid: Often called a “Bowl.” It has a single minimum point. Used in satellite dishes to focus signals.
    2. Hyperbolic Paraboloid: The “Saddle” or “Pringle.” It curves up in one direction and down in the other. This is a classic example of a surface with “negative curvature.”
    3. Torus: A Donut shape. It is a surface with a “genus” of 1 (one hole).

    The Math Behind It

    We use Multivariable Calculus to plot these surfaces.

    $$ \text{Paraboloid: } z = \frac{x^2}{a^2} + \frac{y^2}{b^2} \quad | \quad \text{Saddle: } z = \frac{y^2}{b^2} - \frac{x^2}{a^2} $$

    Where:

    • $x, y, z$ are the spatial coordinates.
    • $R$ is the Major Radius (distance from center to the tube).
    • $r$ is the Minor Radius (thickness of the tube).