Finding the Solution Set for a Rectangular Coordinate System
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In the cylindrical coordinate system, a point P in three-dimensional space is represented by the ordered triple (r, θ, z), where r and θ are polar coordinates of P's projection onto the XY-plane and z is the directed distance from the XY-plane to P.
In three dimensions, two coordinate systems are similar to polar coordinates and give convenient descriptions of some commonly occurring surfaces and solids.
If not familiar, the polar coordinate system gives a convenient description of individual curves and regions. Recall the connection between polar coordinates and cartesian coordinates.
Change From Rectangular to Cylindrical Coordinates and Vice Versa
Remember that in the cylindrical coordinate system, a point P in three-dimensional space is represented by the ordered triple (r, θ, z), where r and θ are polar coordinates of the projection of point P onto the XY-plane while z is the directed distance from the XY-plane to P.
To convert from rectangular to cylindrical coordinates, use the formulas presented below.
r2 = x2 + y2
tan(θ) = y/x
z = z
To convert from cylindrical to rectangular coordinates, use the following equations.
x = r cos(θ)
y = r sin (θ)
z = z
Cylindrical Coordinates in Calculus
Cylindrical coordinates are useful in problems that involve symmetry about an axis, and the z-axis is chosen to coincide with this axis of symmetry. For instance, the circular cylinder axis with Cartesian equation x2 + y2 = c2 is the z-axis. In cylindrical coordinates, the cylinder has the straightforward equation r = c. It is the reason for the name "cylindrical" coordinates.
Example 1: Conversion Between Cylindrical and Rectangular Coordinates
Plot the point with cylindrical coordinates (2, 2π/3, 1) and find its rectangular coordinates.
Solution
The given problem is a conversion from cylindrical coordinates to rectangular coordinates. First, plot the given cylindrical coordinates or the triple points in the 3D-plane as shown in the figure below. Next, substitute the given values in the mentioned formulas for cylindrical to rectangular coordinates.
Recall that the dimensional space is represented by the ordered triple (r, θ, z) whereas r = 2, θ = 2π/3, and z = 1.
x = r cos(θ)
x = (2) cos (2π/3)
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x = (2) (-1/2)
x = -1
y = r sin (θ)
y = (2) sin (2π/3)
y = √3
z = z
z = 1
Answer
The rectangular coordinates of (2, 2π/3, 1) are (-1, √3, 1).
Conversion Between Cylindrical and Rectangular Coordinates
Example 2: Rectangular to Cylindrical Coordinates
Find cylindrical coordinates of the point with rectangular coordinates (3, -3, -7).
Solution
When converting rectangular coordinates to cylindrical coordinates, simply substitute the given ordered triple to the introduced equations above. Recall that the given coordinates can be interpreted as x = 3, y= -3, and r = -7.
r = √x2 + y2
r = √(3)2 + (-3)2
r = 3√2
tan (θ) = y/x
tan (θ) = -3/3
tan (θ) = -1
θ = tan-1 (-1)
θ = (3π/4) + πn
z = z
z = -7
Answer
The converted cylindrical coordinates of the given rectangular coordinates (3, -3, -7) can be expressed in multiple forms such as (3√2, 3π/4, -7) and (3√2, 7π/4, -7), and many more. As with polar coordinates, there are infinitely many solutions.
Rectangular to Cylindrical Coordinates
Example 3: Identifying the Surface Given an Equation
Describe the surface whose equation in cylindrical coordinates is z = r.
Solution
The given equation z = r interprets that the value of z or height of each point on the surface is the same as r, the distance from the point to the z-axis. Since θ does not appear, it can vary. Any horizontal trace in the plane z = k (k > 0) is a circle of radius k. These traces suggest that the surface is a cone. This prediction can be confirmed by converting the equation into rectangular coordinates. From the first equation, we have the following.
z2 = r2
z2 = x2 + y2
Answer
We recognize the equation z2 = x2 + y2 as a circular cone whose axis is the z-axis.
Identifying the Surface Given an Equation
Example 4: Cylindrical Equation for an Ellipsoid
Find an equation in cylindrical coordinates for the ellipsoid 4x2 + 4y2 + z2 = 1.
Solution
Substitute r2 to x2 + y2 and simplify the given equation of ellipsoid in cylindrical coordinates.
4x2 + 4y2 + z2 = 1
z2 = 1 - 4x2 + 4y2
z2 = 1 – 4(x2 + y2)
z2 = 1 – 4r2
Answer
The equation of the ellipsoid in cylindrical coordinates is z2 = 1 – 4r2.
Cylindrical Equation for an Ellipsoid
Example 5: Identifying the Surface Given an Equation
Identify the surface for each of the following equations.
a. r = 10
b. r2 + z2 = 81
Solution
In two-dimensional space, we can infer that the first given equation is a circle of radius 10. Since we are now in three dimensions and there is no z in the equation, it can vary freely.
Therefore, for any given z, we have a circle of radius 5 with its center lying on the z-axis.
For the second equation, substitute r2 = x2 + y2 into the equation r2 + z2 = 81 to express the rectangular form of the equation.
r2 + z2 = 81
x2 + y2 + z2 = 81
Answer
a. The equation is a cylinder of radius 10 centered on the z-axis.
b. The equation is a sphere centered at the origin with radius 9.
Identifying the Surface Given an Equation
Example 6: Converting Cylindrical to Rectangular Coordinates
Plot the point with the cylindrical coordinates (5, 2π/4, -3) and express its location in rectangular coordinates.
Solution
First, plot the coordinates in the three-dimensional space to visualize properly. Then, apply the equations listed above for the conversion from cylindrical coordinates to rectangular coordinates. Given (5, 2π/4, -3), the rectangular coordinates shall be:
x = r cos(θ)
x = (5) cos (2π/4)
x = (5) (0)
x = 0
y = r sin (θ)
y = (5) sin (2π/4)
y = (5) (1)
y = 5
z = z
z = -3
Answer
The point with cylindrical coordinates (5, 2π/4, -3) has rectangular coordinates (0, -1, -3).
Converting Cylindrical to Rectangular Coordinates
Example 7: Rectangular to Cylindrical Coordinates
Convert the rectangular coordinates (1, -2, 6) to cylindrical coordinates.
Solution
To translate from rectangular to cylindrical coordinates, simply substitute to the right equations. Recall that the given coordinates can be interpreted as x = 1, y= -2, and r = 6.
r = √x2 + y2
r = √(1)2 + (-2)2
r = √5
tan (θ) = y/x
tan (θ) = -2/1
tan (θ) = -2
θ = tan-1 (-2)
θ = 2.0344 + πn
z = z
z = 6
Answer
The converted cylindrical coordinates of the given rectangular coordinates (1, -2, 6) can be expressed in multiple forms such as (3√2, 2.034, -7) and many more. As with polar coordinates, there are infinitely many solutions.
Rectangular to Cylindrical Coordinates
Example 8: Identifying the Surfaces in the Cylindrical Coordinate System
Describe the surfaces with the given cylindrical equations.
a. θ = π/2
b. r2 + z2 = 9
Solution
When the angle theta is held constant whole r and z are allowed to vary, the result is a half-plane.
First, substitute r2 = x2 + y2 into the given equation r2 + z2 = 9 in order to arrive to its rectangular form.
r2 = x2 + y2
r2 + z2 = 9
x2 + y2 + z2 = 9
The resulting equation is a sphere centered at the origin with radius 3.
Answer
Therefore, θ = π/2 is a half-plane and r2 + z2 = 9 is a sphere.
Identifying the Surfaces in the Cylindrical Coordinate System
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Finding the Solution Set for a Rectangular Coordinate System
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