Feel like "cheating" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.
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 . 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 :
Ax2 + B2 + C2 + 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 :
- Elliptic paraboloids,
- Hyperbolic paraboloid,
- 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.
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).
The equation for a cone is :
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:
 Quadric Surfaces. Retrieved July 31, 2021 from: https://web.cs.wpi.edu/~matt/courses/cs563/talks/renderman/quadric.html
 Quadric Surfaces. Retrieved August 1, 2021 from: https://opentextbc.ca/calculusv3openstax/chapter/quadric-surfaces/
 Quadratic Surfaces. Retrieved August 8, 2021 from: http://www.staff.city.ac.uk/o.castro-alvaredo/teaching/surfaces.pdf
Stephanie Glen. "Quadric Surface: Equations, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/quadric-surface/
Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!