Question

Sketch the graph of the given cylindrical or spherical equation. $$ \rho=3 \cos \phi $$

Short answer

The graph is a sphere with center (0, 0, 3/2) and radius 3/2.

Step-by-step solution

Understand the Equation in Spherical Coordinates

The given equation is \( \rho = 3 \cos \phi \). Here, \( \rho \) represents the radial distance from the origin to a point in space, \( \phi \) is the angle measured from the positive z-axis, and \( \theta \) is the angle in the xy-plane from the positive x-axis. This equation is given in spherical coordinates.

Relate to Cartesian Coordinates

The spherical coordinate equation \( \rho = 3 \cos \phi \) translates to a surface in three-dimensional space. In spherical coordinates, it suggests that for any given angle \( \phi \), \( \rho \) is determined by the cosine of \( \phi \). This describes a sphere. In Cartesian coordinates, this equation translates as \( x^2 + y^2 + z^2 = 3z \).

Algebraic Manipulation to Identify the Sphere

Transform the equation \( x^2 + y^2 + z^2 = 3z \) by completing the square: \( x^2 + y^2 + (z^2 - 3z) = 0 \). Completing the square for \( z \, \) gives: \( z^2 - 3z = (z - \frac{3}{2})^2 - \frac{9}{4} \). Thus, the equation becomes \( x^2 + y^2 + (z - \frac{3}{2})^2 = \frac{9}{4} \).

Identify the Graph

The final equation \( x^2 + y^2 + (z - \frac{3}{2})^2 = \frac{9}{4} \) describes a sphere. This sphere has a center at \( (0, 0, \frac{3}{2}) \) and a radius of \( \frac{3}{2} \).

Sketch the Graph

To sketch this sphere, plot the center at \( (0,0,\frac{3}{2}) \) in three-dimensional space. From this center, draw a sphere with a radius of \( \frac{3}{2} \). Consider symmetries in the spherical coordinate system that will render a circle in any plane parallel to the xy-plane at different \( z \) values.

Key concepts

Spherical Equations

Spherical equations are a way to represent points in three-dimensional space using a system of coordinates. These coordinates are highly intuitive when dealing with points on or around a sphere. In a spherical coordinate system, each point is defined by three coordinates:

• \( \rho \): the radial distance from the origin to the point.
• \( \phi \): the angle from the positive z-axis.
• \( \theta \): the angle in the xy-plane from the positive x-axis.

In the original exercise, the equation \( \rho = 3 \cos \phi \) describes a relationship where \( \rho \) is dependent on the cosine of the angle \( \phi \). This specific form is often used to represent spheres, cones, or other symmetrical 3D structures. Spherical equations provide a compact and efficient way to describe complex 3D surfaces, like spheres, using fewer constants and simpler relationships than their Cartesian counterparts.

Cartesian Coordinates

Cartesian coordinates are perhaps the most widely used system for graphing in mathematics and are especially useful for defining points in a familiar, grid-like structure. This system uses three mutually perpendicular axes:

• x-axis: which measures horizontal distance.
• y-axis: which measures vertical distance.
• z-axis: which measures depth.

In the context of the spherical equation \( \rho = 3 \cos \phi \), translating it to Cartesian coordinates makes the graphing and visualization of the equation more straightforward for those more familiar with these common coordinates. The transformation converts the spherical expression into the Cartesian form \( x^2 + y^2 + z^2 = 3z \). This showcases how the interplay of variables creates familiar geometries, like spheres, while allowing for deeper exploration into the relationships between the differing coordinate systems.

Graphing in 3D

Graphing in 3D opens up the possibility of visualizing and understanding complex surfaces and figures in mathematics. It involves plotting points (or continuous surfaces) defined by three coordinates into a 3D space. Using either spherical or Cartesian coordinates can alter the complexity of the graph being drawn. With the equation transformed into Cartesian terms \( x^2 + y^2 + (z - \frac{3}{2})^2 = \frac{9}{4} \), we can identify the graph as a sphere centered at \((0, 0, \frac{3}{2})\) with radius \(\frac{3}{2}\). This lens allows for easier sketching and comprehension of how symmetries present in spherical coordinates translate to familiar shapes, such as spheres or ellipsoids, in 3D Cartesian graphs. When visualizing these graphs, it's important to consider how planes slicing through the 3D shape appear in two dimensions, often as circular cross-sections that build intuition about the 3D shape's layout.

Conversion between Coordinate Systems

Converting between coordinate systems, such as from spherical to Cartesian coordinates (or vice versa), involves using mathematical formulas to translate each point defined in one system into the equivalent definition in another system. This transformation is crucial because it allows equations and figures to be examined in the system that is most intuitive or familiar for the specific context being analyzed. To move from spherical to Cartesian we use:

• x = \( \rho \sin \phi \cos \theta \)
• y = \( \rho \sin \phi \sin \theta \)
• z = \( \rho \cos \phi \)

These relationships are derived from the definitions of the spherical coordinates. Conversions like these expand the usability of equations across different forms of analysis and better bridge the gap between theoretical math and practical application in problem-solving and estimation.