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 axis y= distance from +x axis

y x Polar Coordinates θ r (r, θ) r= distance from origin θ = angle from + x axis

y x Relationship between Polar and Cartesian Coordinates θ r x y From Polar to Cartesian cos θ = x = r cos θ sin θ = y = r sin θ From Cartesian to Polar By Pythagorean Theorem 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 2 2 (x,y) x = r cos θ y = r sin θ x = 2cos(7π/6) y = 2sin(7π/6)

y x Example: Convert the point (-1,2) into polar coordinates (-1,2) θ r No! (wrong quadrant) -63 o

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) r θ θ = angle in xy plane (from the positive x axis) r = distance in the xy plane z = vertical height z

z x y 0 π/4 π/2 3π/4 π 5π/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 x 2 +y 2 =r 2 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