A **quadric surface** is a 3D extension of a conic (ellipsis, hyperbola, or parabola). In 2D space, they are defined by quadratic equations in 2D space [1]. *Quadric *refers to the degree of the equation describing the surface: variables in these equations are raised to the 2nd power. They can be described, in general, as a graph of an equation expressed in the form [2]:

Ax^{2} + B^{2} + C^{2} + Dxy + Exz + Fyz + Hx + Iy + Jz + K = 0.

Where A, B, C, D, E, F, H, I, J, K are fixed constants and x, y, z, are variables.

Different types of quadric surfaces can be obtained from simplifying this equation by rotations and translations of the x, y, z coordinate axes.

## Types of Quadric Surfaces

There are six basic types of quadric surface [2]:

- Ellipsoid,
- Elliptic paraboloids,
- Hyperbolic paraboloid,
- Cones,
- Hyperboloids of one sheet,
- Hyperboloids of two sheets.

Other examples of quadratic surfaces include the cylinder, elliptic cone, elliptic cylinder, elliptic hyperboloid, hyperbolic cylinder, paraboloid, sphere, and spheroid.

**1. Ellipsoid**

The general equation for an ellipsoid is:

If A = B = C, the shape is a sphere.

**2. Elliptic paraboloids**

The equation for an elliptic paraboloid is:

Elliptic paraboloids have ellipses as cross sections. If A = B, the cross section is a circle.

**3. Hyperbolic paraboloid**

The equation for a hyperbolic paraboloid is:

Hyperbolic paraboloids are saddle-shaped, like a Pringle. The sign of c determines whether the graph opens up or down. A positive value for c has the saddle right side up (as if it were being placed on a horse).

**4. Cones**

The equation for a cone is [3]:

**5. Hyperboloids of one sheet**

The equation for the hyperboloid of one sheet is:

**6. Hyperboloids of two sheets**

The equation for the hyperboloid of two sheets is:

## References

[1] Quadric Surfaces. Retrieved July 31, 2021 from: https://web.cs.wpi.edu/~matt/courses/cs563/talks/renderman/quadric.html

[2] Quadric Surfaces. Retrieved August 1, 2021 from: https://opentextbc.ca/calculusv3openstax/chapter/quadric-surfaces/

[3] Quadratic Surfaces. Retrieved August 8, 2021 from: http://www.staff.city.ac.uk/o.castro-alvaredo/teaching/surfaces.pdf

**CITE THIS AS:**

**Stephanie Glen**. "Quadric Surface: Equations, Examples" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/quadric-surface/

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