Coordinate Surfaces

Coordinate surfaces are formed by setting one variable to be constant and by placing no restriction on the other two variables. Below are the the possible coordinate surfaces along with how to describe the surfaces algebraically.

Surface Rectangular Coordinates Cylindrical Coordinates Spherical Coordinates
Constant x x = 1 r cos θ = 1 ρ sin φ cos θ = 1

Constant y y = 1 r sin θ = 1 ρ sin φ sin θ = 1

Constant z z = 1 z = 1 ρ cos φ = 1

Constant r x² + y² = 1 r=1 ρ sin φ = 1

Constant Theta x = y, x nonnegative θ = π/4 θ = π/4

Constant rho x² + y² + z² = 1 r² + z² = 1 ρ = π/4

Constant phi z = sqrt(x² + y²) z = r φ = π/4

Rectangular Coordinates	Cylindrical Coordinates	Spherical Coordinates
x = 1	r cos θ = 1	ρ sin φ cos θ = 1
y = 1	r sin θ = 1	ρ sin φ sin θ = 1
z = 1	z = 1	ρ cos φ = 1
x² + y² = 1	r=1	ρ sin φ = 1
x = y, x nonnegative	θ = π/4	θ = π/4
x² + y² + z² = 1	r² + z² = 1	ρ = π/4
z = sqrt(x² + y²)	z = r	φ = π/4

Back to Describing Surfaces Using Different Coordinate Systems.