z x y cylindrical coordinates but, first, let’s go back to 2d
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z
x
y
Cylindrical Coordinates
But, first, let’s go back to 2D
y
x
Cartesian Coordinates – 2D
(x,y)x
y
x= distance from +y axisy= distance from +x axis
y
x
Polar Coordinates
θ
r (r, θ)
r= distance from originθ = angle from + x axis
y
x
Relationship between Polar and Cartesian Coordinates
θ
rx
y
From Polar to Cartesian
cos θ = x = r cos θ
sin θ = y = r sin θ
From Cartesian to Polar
By Pythagorean Theorem 222 ryx tan θ = y/x
x/r
y/r
x
y
x
Example: Plot the point (2,7π/6) and convert it into rectangular coordinates
7π/6
22
(x,y)
x = r cos θ
y = r sin θ
x = 2cos(7π/6) 3xy = 2sin(7π/6) 1x
y
x
Example: Convert the point (-1,2) into polar coordinates
5r
2tan
(-1,2)
θ
r
222 yxr 222 )2()1( r
x
y)tan(
1
2)tan(
?)2(tan 1
?63oNo! (wrong quadrant)
oo 11718063
-63o
z
x
y
Cylindrical Coordinates are Polar Coordinates in 3D.
Imagine the projection of the point (x,y,z) onto the xy plane..
(x,y,z)
x
y
z
x
y
Cylindrical Coordinates are Polar Coordinates in 3D.
Now, imagine converting the x & y coordinates into polar:
(r, θ, z)
rθ
θ = angle in xy plane (from the positive x axis)
r = distance in the xy plane
z = vertical height
z
z
x
y
0π/4
π/2
3π/4
π5π/4
3π/2
7π/4 2π
It’s very important to recognize where certain angles lie on the xy plane in 3D coordinates:
z
x
y
Now, let's do an example.
Plot the point (3,π/4,6)
Then estimate where the angle θ would be and redraw the same radius r along that angle
First, draw the radius r along the x axis
Then put the z coordinate on the edges of the angle
And finally, redraw the radius and angle on top
Final point = (3,π/4,6)
z
x
y
Conversion:
Rectangular to Cylindrical
θ
y
x
x2+y2=r2
tan(θ)=y/x
Z always = Z
z
x
y
Conversion:
Cylindrical to Rectangular
θ
y
x
x=r*cos(θ)
y=r*sin(θ)
Z always = Z
r