Finding the Interval for Theta in Parametric Representation of a Sphere

In summary, to find the parametric representation for the surface x^2 + y^2 + z^2 = 16 between the planes z = -2 and z = 2, one would use spherical coordinates x = 4sin(phi)cos(theta), y = 4sin(phi)sin(theta), and z = 4cos(phi). The interval for phi can be found, but the interval for theta depends on the angle in a plane parallel to the xy plane, which can be determined by setting phi to a constant and observing what curve results on the sphere's surface. This implies that there are restrictions on theta, as the sphere is cut by a plane for z between -2 and 2.
  • #1
eunhye732
11
0
Find the parametric representation for the surface:
The part of the sphere x^2 + y^2 + z^2 = 16 that lies between the planes z = -2 and z = 2.

okay, i know that i have to use spherical coordinates which is
x = 4sin(phi)cos(theta)
y = 4sin(phi)sin(phi)
z = 4cos(phi)

i know how to find the interval for phi, but how do you find the interval for theta? this is probably a stupid question, but i don't get it.
thanks!
 
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  • #2
eunhye732 said:
...
y = 4sin(phi)sin(phi)
...
That should be y=4sin(phi)sin(theta).
eunhye732 said:
...
i know how to find the interval for phi, but how do you find the interval for theta?
...
What is theta on the sphere ? If you set phi to some constant, what curve results on the sphere's surface ? What does this imply about the restriction on theta ?
 
  • #3
Remember that [itex]\theta[/itex] measures the angle in a plane parallel to the xy plane. Imagine the sphere cut by such a plane for z between -2 and 2. What restrictions are there on [itex]\theta[/itex]?
 

FAQ: Finding the Interval for Theta in Parametric Representation of a Sphere

What is parametric representation?

Parametric representation is a method of describing a mathematical object, such as a curve or surface, by using a set of parameters. These parameters are typically variables that represent the coordinates of points on the object, and their values determine the position of the object in space.

How is parametric representation different from Cartesian representation?

In Cartesian representation, objects are described using equations in terms of x, y, and z coordinates. In parametric representation, objects are described using equations in terms of parameters, which can be any variables. This allows for more flexibility in describing complex shapes and curves.

What are some advantages of using parametric representation?

One advantage of using parametric representation is that it allows for easier manipulation and calculation of complex shapes and curves. It also allows for more efficient and accurate rendering in computer graphics and animation, as the equations can be easily translated into code.

Can any object be represented parametrically?

Yes, any object can be represented parametrically as long as it can be described using equations. However, some objects may be more difficult to represent parametrically than others, and may require more complex equations.

What are some real-world applications of parametric representation?

Parametric representation is widely used in fields such as engineering, computer graphics, and physics. It is often used to describe and model complex objects and systems, such as the movement of particles in a fluid or the shape of a car body. It is also used in computer-aided design (CAD) software to create and manipulate 3D objects and surfaces.

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