Z x y Cylindrical Coordinates But, first, let’s go back to 2D.

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