Question

Sketch the graph of the given cylindrical or spherical equation. $$ \phi=\pi / 6 $$

Short answer

The graph is a double cone centered at the origin, with an apex angle of \( \pi/6 \).

Step-by-step solution

Understanding the Spherical Coordinates

In spherical coordinates, a point is described by three parameters: \( (\rho, \theta, \phi) \). Here, \( \rho \) is the radial distance from the origin, \( \theta \) is the azimuthal angle in the \( xy \)-plane from the positive \( x \)-axis, and \( \phi \) is the polar angle from the positive \( z \)-axis.

Identifying the Surface

The equation given is \( \phi=\pi/6 \). This specifies that the polar angle \( \phi \) is constant, which describes a conical surface. The surface consists of a cone centered around the \( z \)-axis with the angle at the apex being \( \pi/6 \) radians.

Visualizing the Shape

A cone with an apex angle of \( \pi/6 \) has its apex at the origin and opens upward and downward around the \( z \)-axis. The cross-section in any plane containing the \( z \)-axis will show lines meeting at the origin, forming an angle of \( \pi/6 \) with the \( z \)-axis.

Drawing the Graph

To sketch the graph, begin by drawing the \( z \)-axis. Then, draw straight lines representing the conical surface. These lines should form an angle of \( \pi/6 \) with the \( z \)-axis, extending symmetrically around it to create a double cone (one part above and one part below the \( xy \)-plane).

Completing the Sketch

Ensure the completed drawing shows two cones meeting at the origin. The cones should be symmetrical and smooth, accurately reflecting the \( \pi/6 \) angle to the \( z \)-axis. This completes the graph of the spherical equation.

Key concepts

Spherical Coordinates

Spherical coordinates offer a unique way to describe points in three-dimensional space. Unlike Cartesian coordinates which use x, y, and z axes, spherical coordinates utilize three parameters:

• \( \rho \): This is the radial distance from the origin to the point.
• \( \theta \): The azimuthal angle, measured in the xy-plane from the positive x-axis.
• \( \phi \): Known as the polar angle, it is the angle measured from the positive z-axis.

Understanding these parameters is crucial in translating complex 3D forms into simpler representations for analysis, modeling, or visualization. The main advantage is their efficiency in representing shapes like spheres and cones.
For example, a sphere centered at the origin has a simple equation in spherical coordinates: \( \rho = R \), where R is the radius. Compared to Cartesian coordinates, spherical coordinates can simplify many geometrical problems.

Cone Surface

When a spherical equation specifies a constant polar angle, such as \( \phi=\pi/6 \), this defines a cone surface. This concept can be visualized as a 3D cone, with the following characteristics:

• It is centered around the z-axis. The z-axis acts like the spine of the cone, going through its center.
• The apex of the cone is located at the origin of the spherical coordinate system.
• The apex angle, which is \( \phi = \pi/6 \), determines how "wide" or "narrow" the cone is. A smaller \( \phi \) means a narrower cone.

The conical surface extends both above and below the xy-plane, forming a double cone. Such a structure is essential in fields like physics, where cone geometries occur naturally in problems involving jets, beams, and other axial systems.

Polar Angle

The polar angle, \( \phi \), is a core component of spherical coordinates with pivotal roles in geometry representation.

• It measures the angle from the positive z-axis to the point of interest.
• A polar angle of \( 0 \) means the point is directly above the z-axis, at the top of the axis (like the North Pole on a globe), while a polar angle of \( \pi \) places it directly below.

In the example equation \( \phi = \pi/6 \), the constant polar angle describes a consistent spread of points from the z-axis. This defines the slope of the cone's surface.
Each point on this conical surface maintains this \( \phi \) angle from the z-axis, making the understanding of polar angles essential to sketching or conceptualizing spherical shapes, particularly cones.