Graph Vector Functions: Explaining Appearance of Graph on Sphere

In summary, the conversation discusses how to graph a curve with parametric equations and explains that it lies on a sphere. The suggestion is made to manipulate the equations to obtain the equation of a sphere and to use cartesian coordinates to graph points.
  • #1
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Homework Statement


Graph the curve with the parametric equations

x=√(1-0.25cos^2(10t)) * cost
y=√(1-0.25cos^2(10t)) * sint
z=0.5cos10t

explain the appearance of the graph by showing that it lies on a sphere


Homework Equations





The Attempt at a Solution


could anyone please just point me in the right direction on how to start this problem? any help would be greatly appreciated! thanks pf!
 
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  • #2
'Graph the curve' seems pretty straightforward. Did you do that?

Did you manipulate the parametric equations for x, y, and z to see if you can obtain the equation of a sphere?

That would be pretty conclusive if you did.
 
  • #3
A sphere is the set of points equidistant from the same point. If a sphere has radius R, what equation in x,y,z describes it? Take the x, y, z you were given, with the assorted sines and cosines and see what happens when you plug those variables into the equation of a sphere.

As far as graphing, I am a lousy artist, but if I had to graph it I'd sketch 3-d coordinates and graph some points based on cartesian x, y and z. This is maybe cheating, but I can't think too well in terms of cos 10t. The 10t will turn out to be irrelevant anyway-- try paragraph 1 and you'll see.
 
  • #4
thank you brmath and steamking! brmath...that's exactly what I am trying now. I am plugging in x, y, and z into the equation of a sphere and seeing if my t's cancel and the equation is satisfied.
 

1. What is a graph vector function?

A graph vector function is a mathematical function that maps a set of input values to a set of output values in a multidimensional space. It is commonly represented as a vector with multiple components, each representing a different variable or dimension.

2. How does a graph vector function appear on a sphere?

A graph vector function can be graphed on a sphere by plotting the output values as points on the surface of the sphere. The input values are typically represented by the angles or coordinates on the sphere, and the resulting graph shows the relationship between the input and output values in a three-dimensional space.

3. What factors affect the appearance of a graph vector function on a sphere?

The appearance of a graph vector function on a sphere can be affected by the type of function, the input values, and the scale of the graph. The orientation and rotation of the sphere can also impact the way the function is represented.

4. How can a graph vector function on a sphere be used in real-world applications?

Graph vector functions on a sphere can be used to model and analyze various physical phenomena, such as weather patterns, ocean currents, or molecular structures. They can also be used in computer graphics and animation to create realistic 3D images.

5. What are some common examples of graph vector functions on a sphere?

Some common examples of graph vector functions on a sphere include spherical harmonics, which are used to represent complex waveforms, and spherical coordinates, which are used to describe the position of objects in three-dimensional space. Other examples include contour maps, wind vector fields, and celestial coordinate systems.

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