2020/11/9
Cylindrical coordinates
Dynamics
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Cylindrical coordinates
#rvy
Warning!
The cylindrical coordinate system extends polar coordinates into 3D by using the standard vertical
coordinate z. This gives coordinates (r, θ, z) consisting of:
coordinate
r
θ
z
name
radius
azimuth
height
range
0 ≤ r < ∞
−π < θ ≤ π
−∞ < z < ∞
We normally write r ⃗ for the position vector
of a point, but if we are using cylindrical
coordinates r, θ, z then this is dangerous.
This is because r might mean the
magnitude of r ⃗ or the radial coordinate,
which are different. To avoid this
confusion we use ρ ⃗ for the position vector
and r for the radial coordinate.
definition
distance from the z-axis
angle from the x-axis in the x–y plane
vertical height
The diagram below shows the cylindrical coordinates of a point P . By changing the display options, we
can see that the basis vectors are tangent to the corresponding coordinate lines. Changing θ moves P
^θ , and similarly for the other coordinates.
along the θ coordinate line in the direction e
reset
Show:
labels
B
coordinates
basis vectors
Coordinate lines:
r
,
, ,
θ
#rvy‑ir
B
z
,
Radius:
r =
4
Azimuth:
θ =
45°
Elevation:
z =
4
Cylindrical coordinates are defined with respect to a set of Cartesian coordinates, and can be converted to
and from these coordinates using the atan2 function as follows.
Conversion between cylindrical and Cartesian coordinates
#rvy‑ec
2
2
x = r cos θ
r = √x
y = r sin θ
θ = atan2(y, x)
z = z
z = z
+ y
atan2(y,x) = arctan(y/x)
Derivation #rvy‑ec‑d
+
The basis vectors are tangent to the coordinate lines and form a right-handed orthonormal basis
^
^r , e
^θ , e
^z that depends on the current position P ⃗ as follows. We can write either e
^z or k
e
for the vertical
basis vector.
Cylindrical basis vectors
#rvy‑eb
e
^r = cos θ ^
ı + sin θ ^
ȷ
e
^θ = − sin θ ^
ı + cos θ ^
ȷ
^
e
^z = k
ı = cos θ e
^r − sin θ e
^θ
^
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2020/11/9
Cylindrical coordinates
r
θ
^
ȷ = sin θ e
^r + cos θ e
^θ
^
^z
k = e
Derivation #rvy‑eb‑d
+
If the cylindrical coordinates change with time then this causes the cylindrical basis vectors to rotate
with the following angular velocity.
Angular velocity of the cylindrical basis
#rvy‑ew
˙ ^
ω⃗ = θ e
z
Derivation #rvy‑ew‑d
+
The rotation of the basis vectors caused by changing coordinates gives the time derivatives below.
Time derivatives of cylindrical basis vectors
#rvy‑et
˙
˙
e
^r = θ e
^θ
˙
˙
e
^θ = − θ e
^r
˙
e
^z = 0
Derivation #rvy‑et‑d
A point P at a time-varying position (r, θ, z) has position vector ρ ⃗, velocity v⃗
˙
= ρ⃗
, and acceleration a⃗
+
¨
= ρ⃗
given by the following expressions in cylindrical components.
Position, velocity, and acceleration in cylindrical components
#rvy‑ep
ρ⃗ = r e
^r + z e
^z
˙
v⃗ = ṙ e
^r + r θ e
^θ + ż e
^z
2
˙
¨
˙ ^
^r + (rθ
^z
a⃗ = (r̈ − rθ ) e
+ 2ṙ θ ) e
θ + z̈ e
Derivation #rvy‑ep‑d
+
Copyright (C) 2012-2015 Matthew West
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