z
Cylindrical Coordinates
But, first, let’s go back to 2D
y
x
Cartesian Coordinates – 2D
y
x
(x,y)
y
x
x= distance from +y axis
y= distance from +x axis
y
Polar Coordinates
r
θ
(r, θ)
r= distance from origin
θ = angle from + x axis
x
y
Relationship between Polar and
Cartesian Coordinates
x
r
θ
From Polar to Cartesian
cos θ = x/r  x = r cos θ
sin θ = y/r

y = r sin θ
y
x
x
From Cartesian to Polar
By Pythagorean Theorem
tan θ = y/x

x  y r
2
2
2
y
Example: Plot the point (2,7π/6)
and convert it into rectangular
coordinates
7π/6
x
2
2
(x,y)
x = r cos θ
y = r sin θ


x = 2cos(7π/6)
y = 2sin(7π/6)


x 3
x  1
y
Example: Convert the point (-1,2)
into polar coordinates
(-1,2)
r
θ
x
-63o
r  x  y
y
tan(  ) 
x
2
2
2


r  (  1)  ( 2 )
2
tan(  ) 
1
2
2
2


r
5
tan    2

 tan
1
(2) ?
   63 ?
o
No! (wrong quadrant)
   63  180  117
o
o
z
Cylindrical Coordinates are
Polar Coordinates in 3D.
(x,y,z)
Imagine the projection of the
point (x,y,z) onto the xy plane..
y
x
y
x
z
Cylindrical Coordinates are
Polar Coordinates in 3D.
(r, θ, z)
Now, imagine converting the x
& y coordinates into polar:
z
y
r = distance in the xy plane
θ = angle in xy plane
(from the positive x axis)
z = vertical height
x
θ
r
z
It’s very important to recognize
where certain angles lie on the
xy plane in 3D coordinates:
π
5π/4
3π/4
3π/2
π/2
0
7π/4
x
2π
π/4
y
z
Now, let's do an example.
Plot the point (3,π/4,6)
First, draw the radius r along
the x axis
Then estimate where the
angle θ would be and
redraw the same radius r
along that angle
Final point = (3,π/4,6)
y
Then put the z coordinate
on the edges of the angle
And finally, redraw the
radius and angle on top
x
z
Conversion:
Rectangular to Cylindrical
x2+y2=r2
tan(θ)=y/x
Z always = Z
y
x
θ
y
x
z
Conversion:
Cylindrical to Rectangular
x=r*cos(θ)
y=r*sin(θ)
Z always = Z
y
x
r
θ
y
x