Help: Perpendicular project is regular surface?

In summary, the problem asks to show that the perpendicular projection of the center of an ellipsoid onto its tangent planes is a regular surface. Using the gradient and normal equations, the tangent plane is found to be xx0/a^2 + yy0/b^2 + zz0/c^2 = 1. To continue, tn is considered and a value of t is found to make tn lie in the plane, which is then used to verify that the desired equation is satisfied. There is some uncertainty about the exact equation as it has yielded different results.
  • #1
Help: Perpendicular project is regular surface??

(edit found a better way of showing this)

Homework Statement



Hi

I have this problem here which is causing me trouble.

Show that the perpendicular projection of the center (0,0,0) of the ellipsoid
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

onto its tangent planes constitutes a regular surface given by

{(x,y,z) \in R^3; (x^2+y^2+z^2)/2 = a^2x^2 + b^2y^2 + c^2z^2}-{(0,0,0)}

What can do here is arrive at tangent plane


The Attempt at a Solution



First I find the tangent

gradF(x0,y0,z0) = <2x0/a^2, 2y0/b^2, 2z0/c^2>

Which gives us a tangentplane

2x0/a^2*(x-x0)+2y0/b^2(y-y0)+2z0/c^2(z-z0) = 0

which by rearrangement gives

xx0/a^2 + yy0/b^2 + zz0/c^2 = 1


the normal is x0/a^2*x = y0/b^2*y = z0/c^2*z


But how do I continue from here?


Best Regards
Alphaboy

Homework Equations

 
Last edited:
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  • #2


You have a normal n=<x0/a^2, y0/b^2, z0/c^2>.

Can you consider tn and determine what value of t makes tn lie in the plane? Isn't that the projection of the origin into the plane?

Then verify that this point satisfies your desired equation:

(x^2+y^2+z^2)/2 = a^2x^2 + b^2y^2 + c^2z^2

which unfortunately I tried three times, getting two different results, neither of which was exactly this. Are you sure it's divided by 2?
 

1. What is a perpendicular project?

A perpendicular project is a geometric concept where a line or plane is drawn at a right angle to another line or plane. This creates a 90-degree angle between the two lines or planes.

2. What is a regular surface?

A regular surface is a mathematical term for a surface that can be described by a continuous function with two independent variables. It is smooth and does not have any sharp edges or corners.

3. How does a perpendicular project create a regular surface?

A perpendicular project creates a regular surface by intersecting a plane at a right angle with another plane or line. This creates a smooth, continuous surface with no sharp edges or corners.

4. What are some real-world applications of a perpendicular project on a regular surface?

A perpendicular project on a regular surface has many real-world applications, such as creating 3D models in computer graphics, designing structures in engineering, and mapping terrains in geography and cartography.

5. Are there any limitations or constraints when working with a perpendicular project on a regular surface?

There may be limitations or constraints when working with a perpendicular project on a regular surface, such as the need for precise measurements and calculations, and the possibility of errors or inaccuracies in the resulting surface. Additionally, the regular surface may not accurately represent the actual shape or structure in certain situations.

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