General 3d Surface Equations

Ellipsoid (Graph)

x²/a² + y²/b² + z²/c² = 1

all traces are ellipses

if a = b = c, this is a sphere

Cone (Graph)

x²/a² +y²/b² = z²/c²

horizontal traces are ellipses

vertical traces in the planes

x − k and y − k are hyperbolas

if k =/= 0 but are pairs of lines if k = 0

Cylinder (Graph)

x² + y² = r²

Elliptic Paraboloid

x²/a² + y²/b² = z/c

horizontal traces are ellipses

vertical traces are parabolas

the variable raised to the first power indicates the axis of the paraboloid

Hyperbolic Paraboloid

x²/a² - y²/b² = z/c

horizontal traces are hyperbolas

vertical traces are parabolas

Hyperboloid of One Sheet

x²/a² + y²/b² - z²/c² = 1

horizontal traces are ellipses

vertical traces are hyperbolas

the axis of symmetry corresponds to the variable whose coefficient is negative

Hyperboloid of Two Sheets

-x²/a² - y²/b² + z²/c² = 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