Eliminate Parameters to Find Ellipsoid Cartesian Equation

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In summary, the conversation discusses eliminating parameters u and v to obtain a Cartesian equation for a given vector equation representing a portion of the surface named Ellipsoid. The approach involves using the cross product and identifying identities to combine the parameters.
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andmcg
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Homework Statement


Eliminate the parameters u and v to obtain a Cartesian equation, thus showing that the given vector equation represents a portion of the surface named. Also, compute the fundamental vector product dr/du x dr/dv in terms of u and v.

Homework Equations

Ellipsoid
r(u,v) = asinucosvi + bsinusinvj + ccosuk

The Attempt at a Solution

I can find the cross product easy enough(abcsinu((sinucosv/a)i+(sinusinv/b)j+(cosu/c)k), but how should I go about getting rid of the parameters?
 
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  • #2
Let x = asinu*cosv, y = bsinu*sinv, and z = ccosu.

You can combine x and y by using the identity sin^2(t) + cos^2(t) = 1. That's where I would start.
 

1. What is an ellipsoid cartesian equation?

An ellipsoid cartesian equation is a mathematical representation of an ellipsoid shape in three-dimensional space. It consists of three variables (x, y, and z) and three parameters (a, b, and c) that determine the size and shape of the ellipsoid.

2. Why do we need to eliminate parameters in the ellipsoid cartesian equation?

Eliminating parameters in the ellipsoid cartesian equation allows us to express the shape in a simpler, more standard form. This makes it easier to analyze and manipulate mathematically, as well as visually represent the ellipsoid.

3. How do we eliminate parameters in the ellipsoid cartesian equation?

To eliminate parameters, we use algebraic techniques such as substitution or elimination. This involves replacing one or more of the parameters with expressions involving the other variables, until all parameters have been eliminated and the equation is in terms of x, y, and z only.

4. What are the benefits of eliminating parameters in the ellipsoid cartesian equation?

Eliminating parameters allows us to easily determine properties of the ellipsoid, such as its center, axes, and symmetry. It also makes it easier to compare and contrast different ellipsoids and their characteristics.

5. Are there any real-world applications of the ellipsoid cartesian equation?

Yes, the ellipsoid cartesian equation is used in many fields, including physics, engineering, and geodesy. It can be used to model the shape of planets and celestial bodies, as well as to describe the shape of objects in engineering and architecture.

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