Cylindrical Coordinates

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

2. y Cartesian Coordinates – 2D x (x,y) y x x= distance from +y axis y= distance from +x axis

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

4. y Relationship between Polar and Cartesian Coordinates x r y θ x x From Cartesian to Polar From Polar to Cartesian x/r x = r cos θ cos θ = By Pythagorean Theorem sin θ = y = r sin θ y/r y/x tan θ =

5. y Example: Plot the point (2,7π/6) and convert it into rectangular coordinates 7π/6 x 2 2 (x,y) x = 2cos(7π/6) x = r cos θ y = r sin θ y = 2sin(7π/6)

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

7. 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

8. z Cylindrical Coordinates are Polar Coordinates in 3D. (r, θ, z) Now, imagine converting the x & y coordinates into polar: z y r r = distance in the xy plane θ θ = angle in xy plane (from the positive x axis) z = vertical height x

9. 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 y π/2 0 7π/4 2π π/4 x

10. z Now, let's do an example. Plot the point (3,π/4,6) First, draw the radius r along the x axis Final point = (3,π/4,6) Then estimate where the angle θ would be and redraw the same radius r along that angle y Then put the z coordinate on the edges of the angle And finally, redraw the radius and angle on top x

11. z Conversion: Rectangular to Cylindrical x2+y2=r2 tan(θ)=y/x Z always = Z y x θ y x

12. z Conversion: Cylindrical to Rectangular x=r*cos(θ) y=r*sin(θ) Z always = Z y r x θ y x