Cylinders and Quadratic Surfaces

y=x² in R²

Parabola. May also include a translation. Defined in R²

(x-h)² / (a²) - (y-k)² / (b²) = 1 *vertical

Or

(y-k)² / (b²) - (x-h)² / (a²) = 1 *horizontal

hyperbola in R²

y+x+z=c

Plane. May also include translations

y=x

Line, may include translations

x²+y²=r² in R²

Circle.

x²+y²+z²=r² in R³

Sphere

x²+y²=1 in R³

Circular cylinder; when only two variables are described in R³

z=y²+1 in R³

Parabolic cylinder; only two variables defined in R³ and mirrors formula for parabola

x²/a² + y²/b² + z²/c² = constant

Ellipse, all coefficients are positive and their sums are a constant

x²/a² + y²/b² = z²/c²

Cone; no constant, all are squared

x²/a² + y²/b² = (±) z/c

Elliptic paraboloid; where c can be positive or negative to change orientation of graph

x²/a² + y²/b² - z²/c² = constant

Or

x²/a² + y²/b² = z²/c² + constant

One sheeted Hyperboloid; constant is positive

-x²/a² - y²/b² - z²/c² = constant

Or

x²/a² + y²/b² = z²/c² - constant

Two sheeted hyperboloid; constant is negative

x²/a² - y²/b² = z/c

Hyperbolic Paraboloid; differing signs