Parametric Equations
Polar, Cylindrical and Spherical Coordinates
Parametric Equations Polar, Cylindrical and Spherical Coordina
References
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George B. Thomas, Maurice D. Weir and Joel Hass, Thomas’
Calculus, Addison Wesley, 12th Edition, 2009
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James Stewart, Calculus, 7th edition, 2011
Parametric Equations Polar, Cylindrical and Spherical Coordina
Parametric Equations
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Usually functions are described using cartesian coordinates,
also referred to as rectangular coordinates, and cartesian
equations in the form y = f (x).
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Sometimes it is more convenient and concise to describe
functions in a different way, where both x and y are described
in terms of another, third variable, e.g. t, x = f (t) and
y = f (t).
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In this case t is called a parameter and x(t) and y (t) are
parametric equations. In general t represents an arbitrary
parameter, and not time (although it can be time).
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Other commonly used parameters are s, u, v .
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If parameters are angles θ (theta) φ or ϕ (both phi), ψ (psi)
or α (alpha), β (beta), γ (gamma) might be used.
Parametric Equations Polar, Cylindrical and Spherical Coordina
Parametric Equation Examples
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Example: line y = 2x + 3
x(t) = t
(1)
y (t) = 2t + 3
(2)
or more succinctly
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x
= t
(3)
y
= 2t + 3
(4)
Example: circle x 2 + y 2 = r 2
x
= r cos t
(5)
y
= r sin t
(6)
x
= r cos θ
(7)
y
= r sin θ
(8)
or using θ as the parameter
Parametric Equations Polar, Cylindrical and Spherical Coordina
Cartesian Coordinates
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The cartesian coordinate system specifies points in the plane
by ordered pairs (x, y ) which are the distances from two
perpendicular axes.
3
2
(1, 2)
1
0
-3
-2
-1
-1
0
1
2
3
-2
-3
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Similarly in 3D ordered triples (x, y , z) are used
Parametric Equations Polar, Cylindrical and Spherical Coordina
Polar Coordinates
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The cartesian coordinate system specifies points in the plane
by ordered pairs (x, y ) which are the distances from two
perpendicular axes.
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Polar coordinates are another way to describe points in the
plane, using (r , θ) ordered pairs.
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The origin O in polar coordinates is called the pole and a ray
starting at O is called the polar axis. The distance r is the
length OP and the angle θ is measured counter clockwise
from the polar axis.
P(r, θ)
r
O
θ
polar axis
Parametric Equations Polar, Cylindrical and Spherical Coordina
Polar Coordinates (cont)
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Example: (2, π/6)
(2, π/6)
0
1
2
3
Parametric Equations Polar, Cylindrical and Spherical Coordina
Polar Coordinates (cont)
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In polar coordinates functions are specified as r = f (θ).
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Example: r = a is a circle of radius a centred at the pole.
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Example: θ = θ0 is a straight line which makes an angle θ0
with the polar axis.
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r = cos θ is a unit circle centred at (0.5, 0).
Parametric Equations Polar, Cylindrical and Spherical Coordina
Polar Coordinates (cont)
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Polar coordinates can be converted to cartesian coordinates
x
= r cos θ
(9)
y
= r sin θ
(10)
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These are basically parametric equations, where x and y are
given in terms of parameters r and θ.
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This assumes the pole is at the origin and the polar axis is
aligned with the x axis.
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And rectangular coordinates can be converted to polar
coordinates
p
x2 + y2
r =
x
θ = tan−1
y
(11)
(12)
Parametric Equations Polar, Cylindrical and Spherical Coordina
Cylindrical Coordinates
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Cylindrical polar coordinates, or just cylindrical coordinates,
are 3D coordinates formed by adding to polar coordinates a
third coordinate z along the longitudinal axis to give (ρ, θ, z)
z
(r, θ, z)
z
θ
r
y
x
To convert from cylindrical to cartesian coordinates
x
= r cos θ
(13)
y
= r sin θ
(14)
z
= z
(15)
Parametric Equations Polar, Cylindrical and Spherical Coordina
Spherical Coordinates
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Spherical polar coordinates, or just spherical coordinates, are
also 3D coordinates given by ordered triples (ρ, θ, φ).
z
(r, θ, ϕ)
ϕ
θ
r
z
y
x
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Again there is a pole or origin O. The radial distance of OP is
given by r (or ρ). There are two angles, the azimuthal angle θ
which measures the angle of the projection of OP onto the xy
plane from the x axis and the polar angle or inclination angle
φ which measures the angle of OP from the z axis or zenith
direction.
Parametric Equations Polar, Cylindrical and Spherical Coordina
Spherical Coordinates (cont)
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To convert from spherical coordinates to cartesian/rectangular
coordinates
x
= ρ sin φ cos θ
(19)
y
= ρ sin φ sin θ
(20)
z
= ρ cos φ
(21)
Again these are essentially parametric equations. Instead of
using θ and φ, more general parameters e.g. u and v might be
used, and instead of ρ use r , which gives
x
= r sin v cos u
(22)
y
= r sin v sin u
(23)
z
= r cos v
(24)
Parametric Equations Polar, Cylindrical and Spherical Coordina
Torus
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A torus can be similarly be specified using parametric
equations with two angles and two radii. Using u and v for
the angles, instead of θ and φ (which could equally be used)
and R and r for the major and minor radii:
x
= (R + r cos v ) cos u
(25)
y
= (R + r cos v ) sin u
(26)
z
= r sin v
(27)
z
R
r
x
u
y
v
Parametric Equations Polar, Cylindrical and Spherical Coordina
Torus Tangents and Normals
A normal vector N on a 2D surface can be calculated as the cross
product of two tangent vectors. Tangent vectors may be calculated
using partial derivatives in u and v :
Pu = ∂P/∂u = ((R + rcosv )sinu, (R + rcosv )cosu, 0)
|Pu | = ((R + rcosv )2 sin2 u + (R + rcosv )2 cos 2 u)1/2
= (R + rcosv )
Pu /|Pu | = (−sinu, cosu, 0)
Similarly
Pv /|Pv | = (−cos u sin v , −sin u cos v , −sin v )
And
N = Pu /|Pu | × Pv /|Pv | = (cos u cos v , sin u cos v , sin v )
Parametric Equations Polar, Cylindrical and Spherical Coordina