Spherical To Rectangular Coordinates: A Step-by-Step Guide

Hey everyone! Today, we're diving into the fascinating world of coordinate systems, specifically how to convert from spherical coordinates to rectangular coordinates. This is a crucial skill in various fields, including physics, engineering, and computer graphics, where we often need to represent points in 3D space. So, let's break down the process step-by-step and solve the problem at hand: finding the rectangular coordinates of the point given in spherical coordinates as (3, 3π/4, π/6).

Understanding Spherical Coordinates

Before we jump into the conversion, let's quickly recap what spherical coordinates are. In a spherical coordinate system, a point in 3D space is defined by three parameters: ρ (rho), θ (theta), and φ (phi). Think of it like this: Imagine you're standing at the origin (0, 0, 0) and want to describe the location of a floating ball in front of you.

• ρ (rho): This represents the distance from the origin to the point. It's like the length of an imaginary straight line connecting you to the ball. Rho is always a non-negative value.
• θ (theta): This is the azimuthal angle, measured in the xy-plane from the positive x-axis. Imagine projecting the ball onto the xy-plane; theta is the angle formed between the positive x-axis and this projection. It ranges from 0 to 2π radians (or 0 to 360 degrees).
• φ (phi): This is the polar angle (or zenith angle), measured from the positive z-axis down to the point. It's the angle between the imaginary line connecting you to the ball and the vertical (positive z-axis). Phi ranges from 0 to π radians (or 0 to 180 degrees).

So, in our case, we're given the spherical coordinates (ρ, θ, φ) = (3, 3π/4, π/6). This means the point is 3 units away from the origin, has an azimuthal angle of 3π/4 radians, and a polar angle of π/6 radians. Now, the challenge is to find the equivalent x, y, and z coordinates in the rectangular coordinate system.

The Conversion Formulas: Spherical to Rectangular

The key to converting between spherical coordinates and rectangular coordinates lies in a set of elegant formulas. These formulas establish the relationship between (ρ, θ, φ) and (x, y, z). Here they are:

• x = ρ sin(φ) cos(θ)
• y = ρ sin(φ) sin(θ)
• z = ρ cos(φ)

These formulas might look intimidating at first, but they're actually quite intuitive. Let's break them down and see how they work. The z coordinate is the easiest to understand: It's simply the distance from the origin (ρ) multiplied by the cosine of the polar angle (φ). As φ increases from 0 to π, the z coordinate decreases, which makes sense since we're moving further down from the positive z-axis. The formulas for x and y are a bit more involved but follow a similar logic. They both involve the sine of the polar angle (φ), which gives us the projection of the point onto the xy-plane. Then, we use the cosine and sine of the azimuthal angle (θ) to determine the x and y components of this projection, respectively. These formulas are derived using basic trigonometry and vector projections, and they are the cornerstone of converting between these two coordinate systems. Mastering these formulas will unlock your ability to navigate and manipulate 3D space with ease. Now that we have the formulas, let's apply them to our specific problem.

Applying the Formulas: Step-by-Step Calculation

Now comes the fun part: plugging in the values and calculating the rectangular coordinates! We have (ρ, θ, φ) = (3, 3π/4, π/6), and we want to find (x, y, z). Let's go through each coordinate one by one.

Calculating x

The formula for x is x = ρ sin(φ) cos(θ). Let's substitute the values:

• x = 3 * sin(π/6) * cos(3π/4)

We need to recall the values of sin(π/6) and cos(3π/4). From the unit circle or trigonometric tables, we know that sin(π/6) = 1/2 and cos(3π/4) = -√2/2. Plugging these in:

• x = 3 * (1/2) * (-√2/2)
• x = -3√2/4

So, the x-coordinate is -3√2/4. This negative value makes sense because the azimuthal angle θ = 3π/4 places the point in the second quadrant of the xy-plane, where x-coordinates are negative.

Calculating y

Next, we'll calculate the y-coordinate using the formula y = ρ sin(φ) sin(θ). Substituting the given values:

• y = 3 * sin(π/6) * sin(3π/4)

We already know that sin(π/6) = 1/2. We also know that sin(3π/4) = √2/2 (note the positive sign here). Plugging these in:

• y = 3 * (1/2) * (√2/2)
• y = 3√2/4

The y-coordinate is 3√2/4. This positive value is also consistent with the fact that the point lies in the second quadrant, where y-coordinates are positive.

Calculating z

Finally, let's calculate the z-coordinate using the formula z = ρ cos(φ). Substituting the values:

• z = 3 * cos(π/6)

We know that cos(π/6) = √3/2. Plugging this in:

• z = 3 * (√3/2)
• z = 3√3/2

The z-coordinate is 3√3/2. This positive value indicates that the point is above the xy-plane, which is expected since the polar angle φ = π/6 is less than π/2.

The Final Answer: Rectangular Coordinates

Alright, we've done all the calculations! We found the x, y, and z coordinates. Therefore, the rectangular coordinates of the point with spherical coordinates (3, 3π/4, π/6) are:

• x = -3√2/4
• y = 3√2/4
• z = 3√3/2

So, the point in rectangular coordinates is (-3√2/4, 3√2/4, 3√3/2). Awesome! We've successfully converted from spherical coordinates to rectangular coordinates. This skill is super useful for visualizing and working with points in 3D space, especially in applications where both coordinate systems are commonly used.

Key Takeaways and Applications

Let's recap what we've learned and discuss some real-world applications. We've seen how to convert between spherical coordinates and rectangular coordinates using the following formulas:

• x = ρ sin(φ) cos(θ)
• y = ρ sin(φ) sin(θ)
• z = ρ cos(φ)

These formulas are fundamental tools for anyone working with 3D geometry and spatial data. But where do we actually use these conversions in the real world? Well, there are numerous applications, including:

• Computer Graphics: In 3D modeling and rendering, objects are often represented using spherical coordinates because they can simplify certain calculations, especially when dealing with shapes that have spherical symmetry. Converting to rectangular coordinates is necessary for displaying these objects on a 2D screen.
• Physics: In physics, spherical coordinates are frequently used to describe the motion of objects in space, such as planets orbiting a star or particles interacting with each other. Converting to rectangular coordinates can be helpful for analyzing forces and trajectories.
• Navigation and Mapping: Spherical coordinates are used in GPS systems and mapping applications to represent locations on the Earth's surface. Converting to rectangular coordinates allows for easier distance calculations and map projections.
• Medical Imaging: Techniques like MRI and CT scans generate 3D images of the human body. The data is often acquired in a spherical coordinate system, and converting to rectangular coordinates is necessary for visualization and analysis.
• Robotics: In robotics, spherical coordinates can be used to control the movement of robot arms and other manipulators. Converting to rectangular coordinates is essential for precise positioning and object manipulation.

In summary, understanding the relationship between spherical coordinates and rectangular coordinates is crucial for a wide range of applications. By mastering the conversion formulas and practicing with examples, you'll be well-equipped to tackle problems involving 3D spatial data.

Practice Makes Perfect: Try it Yourself!

Now that we've worked through an example together, the best way to solidify your understanding is to practice! Try converting other points from spherical coordinates to rectangular coordinates. You can find plenty of examples online or in textbooks. Experiment with different values of ρ, θ, and φ, and see how the corresponding x, y, and z coordinates change. You can even try working backwards: converting from rectangular coordinates to spherical coordinates (which involves a slightly different set of formulas). The more you practice, the more comfortable you'll become with these coordinate systems and their conversions. So go ahead, give it a try, and unlock the power of 3D coordinate transformations!