Coordinate System
A coordinate system is a mapping of parameters to locations.
Contents
Description
A coordinate system maps a set of coordinate parameters to a unique location.
In 2 dimensions, the most common system is Cartesian coordinates (i.e., (x,y)). Beyond this, there is also polar coordinates (i.e., (r,θ) where r is a radius and θ is an angle).
In 3 dimensions, the most common system is rectangular coordinates (i.e., (x,y,z)). Others include cylindrical coordinates (i.e., (r,θ,z)) and spherical coordinates (i.e., (ρ,θ,φ) where ρ is the radial distance, θ is the polar angle, and φ is the azimuthal angle). Polar angles are measured in comparison to the polar axis, and range from 0 to π (radians). Azimuthal angles are measured as rotation around the polar axis.
Changing System
Polar to Cartesian
x = r cos(θ)
y = r sin(θ)
The Jacobian is given by J = r, as in dA = dxdy = rdrdθ
Cartesian to Polar
tan(θ) = y/x or θ = tan-1(y/x)
r2 = x2 + y2 or r = √(x2 + y2)
J = 1/r, as in dA = drdθ = (1/r)dxdy
Cylindrical to Rectangular
x = r cos(θ)
y = r sin(θ)
z = z
J = r, as in dV = dxdydz = rdrdθdz
Rectangular to Cylindrical
tan(θ) = y/x or θ = tan-1(y/x)
r2 = x2 + y2 or r = √(x2 + y2)
z = z
J = 1/r, as in dV = drdθdz = (1/r)dxdydz
Spherical to Rectangular
Note that in some texts, θ and φ are reversed. Here φ is the polar angle and θ is the azimuthal angle.
x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
J = ρ2 sin(φ), as in dV = dxdydz = ρ2 sin(φ) dρdθdφ
Rectangular to Spherical
ρ2 = x2 + y2 + z2 or ρ = √(x2 + y2 + z2)
tan(θ) = y/x or θ = tan-1(y/x)
cos(φ) = z/ρ = z/√(x2 + y2 + z2) or φ = cos-1(z/√(x2 + y2 + z2))
Cylindrical to Spherical
Note that in some texts, θ and φ are reversed. Here φ is the polar angle and θ is the azimuthal angle.
ρ2 = r2 + z2 or ρ = √(r2 + z2)
θ = θ
cos(φ) = z/ρ = z/√(r2 + z2) or φ = cos-1(z/√(r2 + z2))
Spherical to Cylindrical
r = ρ sin(φ)
θ = θ
z = ρ cos(φ)