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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