Coordinate Systems, Transformations
and their Line-Surface-Volume Elements
Coordinate Systems:
Three most common coordinate systems used in 3-dimensional representations are:
a)
Cartesian coordinates
b) Cylindrical (polar) coordinates
c) Spherical (polar) coordinates
Their basic coordinates and associated unit vectors are shown in Figures 1, 2 and 3.
z
z
unit vectors
in Cartesian
coordinates
P(x,y,z)
&
&
&
x
x
y
Cartesian coordinates
x = constant
y = constant
y
O
y
x
z = constant
Figure 1 Cartesian Coordinate System
z
z
&
ez
unit vectors in
cylindrical
coordinates
&
= constant
er
y
r
&
P(r,,z)
r = constant
z
O
Cylindrical coordinates
z = constant
&
e r : radially outward from the center of the cylinder (Oz), on z = constant plane
&
e : tangent to the cylinder lateral surface in counterclockwise direction, on z = constant plane
&
e z : in +z direction
Figure 2 Cylindrical Coordinate System
2001Spring/Blent E. Platin & Merve Erdal
ME 210 / Coordinate Systems - 1
unit vectors
in spherical
coordinates
&
O
x
= constant
P(r,,)
z
= constant
&
e
y
&
= constant surface
Spherical coordinates
&
e : radially outward from the center of the sphere, i.e. the origin
&
e : tangent to the sphere in counterclockwise direction, on z = constant plane
&
e : tangent to the sphere in clockwise direction on = constant plane
Figure 3 Spherical Coordinate System
In Cartesian coordinates,
x = constant [,+] represents a plane parallel to Oyz plane,
y = constant [,+] represents a plane parallel to Oxz plane,
z = constant [,+] represents a plane parallel to Oxy plane.
In cylindrical coordinates,
r = constant [0,+] represents a cylinder whose axis is Oz and radius is r,
= constant [0,+2] represents a vertical half plane containing Oz,
z = constant [,+] represents a plane parallel to Oxy plane.
In spherical coordinates,
= constant [0,+] represents a sphere whose center is O and radius is ,
= constant [0,+] represents a cone whose axis is Oz and tip is located at O.
= constant [0,+2] represents a vertical half plane containing Oz,
Transformations between Coordinate Systems:
Cylindrical to Cartesian coordinates
x = r cos
y = r sin
z=z
2001Spring/Blent E. Platin & Merve Erdal
Cartesian to cylindrical coordinates
r = x 2 + y2
= tan 1 ( y / x )
z=z
ME 210 / Coordinate Systems - 2
Spherical to Cartesian coordinates
x = sin cos
y = sin sin
z = cos
Cartesian to spherical coordinates
= x 2 + y2 + z2
= tan 1 ( y / x )
= tan 1 x 2 + y 2 / z
Cylindrical to spherical coordinates
Spherical to cylindrical coordinates
r = sin
z = cos
=
= r2 + z2
= tan 1 (r / z )
=
Line elements:
In Cartesian coordinates:
ds = (dx ) 2 + (dy ) 2 + (dz ) 2
In cylindrical coordinates:
ds = (dr ) 2 + r 2 (d) 2 + (dz )2
In spherical coordinates:
ds = (d )2 + 2 (d) 2 + 2 (sin ) 2 (d)2
Surface and volume elements:
In Cartesian coordinates:
dS3
z
dz
dx
dy
dS2
y
dS1
Figure 2 Surface and Volume Elements in Cartesian Coordinates
Surface elements:
dS1 = dy dz
dS2 = dx dz
dS3 = dx dy
Volume element:
dV = dx dy dz
2001Spring/Blent E. Platin & Merve Erdal
ME 210 / Coordinate Systems - 3
In cylindrical coordinates:
z
dS3
dz
rd
dr
y
dS1
dS2
Figure 3 Surface and Volume Elements in Cylindrical Coordinates
Surface elements:
dS1 = dr dz
dS2 = r d dz
dS3 = r dr d
Volume element:
dV = r dr d dz
In spherical coordinates:
z
dS3
dS2
sind
y
dS1
x
Figure 4 Surface and Volume Elements in Spherical Coordinates
Surface elements:
dS1 = d d
dS2 = 2 sin d d
dS3 = sin d d
Volume element:
dV = 2 sin d d d
2001Spring/Blent E. Platin & Merve Erdal
ME 210 / Coordinate Systems - 4
