Quadric Surfaces

ellipsoid

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
all traces are ellipses
if a=b=c, the ellipsoid is a sphere

cone

z^2/c^2 = x^2/a^2 + y^2/c^2
horizontal traces are ellipses
vertical traces are hyperbolas but pairs of lines if k=0

elliptic paraboloid

z/c = x^2/a^2 + y^2/b^2
horizontal traces are ellipses
vertical traces are parabolas
the variable raised to the first power indicates the axis of the paraboloid

hyperboloid of one sheet

x^2/a^2 + y^2/b^2 - z^2/c^2 = 1
horizontal traces are ellipses
vertical traces are hyperbolas
the axis of symmetry corresponds to the variable whose coefficient is negative

hyperbolic paraboloid

z/c = x^2/a^2 - y^2/b^2
horizontal traces are hyperbolas
vertical traces are parabolas

hyperboloid of two sheets

-x^2/a^2 - y^2/b^2 + z^2/c^2 = 1
horizontal traces in z=k are ellipses if k>c or k<-c
vertical traces are hyperbolas
the two minus signs indicate two sheets