Section 12.12

Cylindrical Coordinates

A 3D coordinate system where points are defined by a radius, an angle, and a height.

Introduction

Cylindrical Coordinates are essentially Polar Coordinates with a vertical -axis added.
- : Distance from the z-axis.
- : Angle in the xy-plane (from positive x-axis).
- : Height above the xy-plane.

Interactive: The Cylindrical Frame

Drag the sliders to move point P. Notice how it moves on the surface of a cylinder when you change or , but changes cylinders when you change .

Key Formulas

Cylindrical to Cartesian

Cartesian to Cylindrical

Common Surfaces

: Cylinder.
: Vertical Half-Plane.
: Horizontal Plane.
: Cone.

Worked Examples

Example 1: Converting Points (Level 1)

Convert to Cartesian.

.
.
.

Result: .

Example 2: Identifying Surfaces (Level 2)

Identify the surface given by .

Substitute :
.

This is the standard equation for a Sphere centered at origin with radius 10.

Note: would be a cylinder.

Example 3: Converting Equations (Level 3)

Convert to Cartesian and identify it.

Square both sides (to handle r easily):
.