A locus in 3-dimensional space that can be represented in a Cartesian coordinate system by a polynomial equation in x, y, and z of degree two; that is, an equation of the form

where the constants a, b, c, f, g, and h are not all zero. When the equation represents a non-empty locus, it can be reduced by translation and rotation to one of the following canonical forms:

1. (i) Ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.$

2. (ii) Hyperboloid of one sheet: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1.$

3. (iii) Hyperboloid of two sheets: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1.$

4. (iv) Elliptic paraboloid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{2z}{c}.$

5. (v) Hyperbolic paraboloid: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{2z}{c}.$

6. (vi) Quadric cone: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}.$

7. (vii) Elliptic cylinder: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$

8. (viii) Hyperbolic cylinder: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

9. (ix) Parabolic cylinder: $\frac{x^2}{a^2}=\frac{2y}{b}.$

10. (x) Pair of non-parallel planes: $\frac{x^2}{a^2}=\frac{y^2}{b^2}\left(\text{that}\text{is},y=\pm \frac{b}{a}x\right).$

11. (xi) Pair of parallel planes: $\frac{x^2}{a^2}=1\left(\text{that}\text{is},x=\pm a\right).$

12. (xii) Plane: $\frac{x^2}{a^2}=0\left(\text{that}\text{is},x=0\right).$

13. (xiii) Line: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=0\left(\text{that}\text{is},x=y=0\right).$

14. (xiv) Point: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=0\left(\text{that}\text{is},x=y=z=0\right).$

Forms (i)–(v) are the non-degenerate quadrics.