Question

Sketch the graph of the given cylindrical or spherical equation. $$ r^{2}+z^{2}=9 $$

Short answer

The equation represents a sphere with center (0,0,0) and radius 3.

Step-by-step solution

Identify the Type of Equation

The given equation is in cylindrical coordinates, expressed as \( r^2 + z^2 = 9 \). This is analogous to the equation of a circle in Cartesian coordinates, \( x^2 + y^2 = 9 \), implying a circular base.

Convert to Cartesian Coordinates

In cylindrical coordinates, \( r \) corresponds to \( \sqrt{x^2 + y^2} \) and \( z \) is the same in both coordinate systems. Thus, the equation \( r^2 + z^2 = 9 \) converts to \( x^2 + y^2 + z^2 = 9 \) in Cartesian coordinates.

Determine the Shape

The equation \( x^2 + y^2 + z^2 = 9 \) represents a sphere in three-dimensional space with its center at the origin \((0, 0, 0)\) and a radius of 3. This is because it follows the standard form of a spherical equation \( x^2 + y^2 + z^2 = R^2 \), where \( R \) is the radius.

Sketch the Sphere

To sketch the graph, draw a sphere centered at the origin with a radius of 3 units. All points that lie at a distance of 3 from the origin in any direction (in three-dimensional space) are included on this sphere.

Key concepts

Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system that allows us to represent points in space with a set of three numbers. These numbers are usually denoted as \(\rho, \theta, \phi\), where \(\rho\) is the distance from the origin to the point, \(\theta\) is the angle from the positive x-axis to the point’s projection in the xy-plane, and \(\phi\) is the angle from the positive z-axis.
Using spherical coordinates can simplify the equation conversion process when dealing with spheres or other spherical objects in three-dimensional geometry. The conversion from Cartesian to spherical coordinates can be carried out using the following relationships:

• \(\rho = \sqrt{x^2 + y^2 + z^2}\)
• \(\theta = \arctan\frac{y}{x}\)
• \(\phi = \arccos\frac{z}{\rho}\)

This system becomes especially handy for sketching figures like spheres, as it aligns perfectly with natural radial symmetry.

Graph Sketching

Sketching a graph in three dimensions is a bit more complex than in two dimensions, but understanding the nature of the equations involved helps. A sphere, such as defined by your equation \(x^2 + y^2 + z^2 = 9\), represents all points in space that are equidistant from a center point.
To sketch such a figure accurately, it can be useful to visualize its shadow or outline in the xy, yz, and zx planes. Imagine looking at a ball; the silhouette you see gives you an idea of the circle when viewed from any of these axes. This is how you would approach sketching in three-dimensional graphing.
Plot the center point, calculate the radius, and then imagine drawing all points equidistant from this center to form the sphere's surface. Tools like projections or interactive graphing software can help depict such three-dimensional objects more vividly.

Equation Conversion

Converting between different coordinate systems is a crucial skill in mathematics and physics, especially in three-dimensional geometry. For example, translating an equation from cylindrical to Cartesian coordinates involves using the definitions of cylindrical coordinates: \(r\) is the radial distance in the xy-plane (equivalent to \(\sqrt{x^2 + y^2}\)), \(\theta\) the angle, and \(z\) is the height.
In our equation, \(r^2 + z^2 = 9\) was translated into \(x^2 + y^2 + z^2 = 9\) by substituting the components related to \(r\) into the Cartesian format.

• \(r = \sqrt{x^2 + y^2}\)
• \(r^2 = x^2 + y^2\)

This translation reveals that the figure is a sphere, something not immediately obvious in the original cylindrical equation format. Such conversion is key to understanding the geometry behind equations.

Three-Dimensional Geometry

Three-dimensional geometry deals with shapes like spheres, cylinders, cones, and prisms. These geometric forms exist in three-dimensional space and often need unique visualization techniques. Understanding how to manipulate and interpret equations in this space expands our ability to model real-world phenomena.
In the case of a sphere, like in your original exercise, it is simply defined by all points that are a fixed distance (radius) from a central point. Spheres stand as the simplest three-dimensional form due to their symmetrical nature.
Visualizing three-dimensional geometry involves projecting shapes onto two-dimensional planes (x-y, y-z, or z-x) or using cross-sectional views to comprehend their structure. This skill becomes especially handy in fields like physics, engineering, and computer graphics, where the representation of objects in space is vital.