Quadric Surfaces -- Identification

Identifying Quadric Surfaces

an introduction

The goal of this page is to aid in the identification of quadric surfaces. There is no supplement to understanding how the following formulas generate the given shapes. Still, one can see patterns in the formulas (generating patterns in the shapes) that help in identification.

Cylinders

An equation composed with a combination of only 2 variables (such as x and y, or y and z) forms a cylinder. The name of the shape is derived from the shape of the graph in two dimensions. For example, x² = 4 is a parabola in 2 dimensions and forms a parabolic cylinder in 3 dimensions. Following this example, what would the name of the surface formed from the equation x² + y² = 3/4 be called? To check your answer, click here.

Cones, Ellipsoids and Hyperboloids

Next, we consider those equations which can be resolved into
(a second order polynomial containing quadratic terms in x, y, and z) = constant
In such cases, we have one of the following 4 forms:

-- forms a cone. A signal of this quadric surface is the constant equaling zero. In each of the following examples, the constant equals 1.

-- forms an ellipsoid. Notice, the sign of each variable is positive.
-- forms a hyperboloid of one sheet. A signal of this shape is the presence of one minus sign. Can you see the difference between this equation and the equation for the cone given above?
-- forms a hyperboloid of two sheets. A signal of this shape is the presence of two minus signs.

Remember, the order of the variables can change but the name of the surface remains the same. For example, is a hyperboloid of one sheet. We have only changed the order of the variables. So, what is would the equation be called? To check your answer click here.

Elliptic paraboloids, and Hyperbolic paraboloids

Equations for elliptic paraboloids and hyperbolic paraboloids contain two quadratic terms and one linear term. The presence of the linear term aides in the ease of identifying these surfaces.

		-- forms an elliptic paraboloid. In this equation, x and y are quadratic and z is linear. Notice, the quadratic terms have the same sign.
		-- forms a hyperbolic paraboloid. This equation also contains one linear variable and two quadratic terms. Notice, the quadratic terms in this equation differ in sign.

From the above information, which surface would the following equation form? To check your answer click here.

Written by Tim Chartier

		-- forms a cone. A signal of this quadric surface is the constant equaling zero. In each of the following examples, the constant equals 1.
		-- forms an ellipsoid. Notice, the sign of each variable is positive.
		-- forms a hyperboloid of one sheet. A signal of this shape is the presence of one minus sign. Can you see the difference between this equation and the equation for the cone given above?
		-- forms a hyperboloid of two sheets. A signal of this shape is the presence of two minus signs.