- #1

gary0000

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- Homework Statement
- This problem began with solving for an 2D ellipse whose axis aligned with the y-axis and the line x=-z. Then the ellipse needs to be translated, rotated, and then revolved to form a spheroid. I've already found the rotated and translated ellipse.

Rotate the following ellipse about its major axis to obtain a prolate spheroid:

(195.010)^2*((x-72.850)*cos(1.423)+(y-490.030)*sin(1.423))^2+(532.419)^2*((x-72.850)*sin(1.423)-(y-490.030)*cos(1.423))^2-(532.419)^2*(195.010)^2=0

[Each x-value corresponds to a negative z-value of equal magnitude as well, since this ellipse is in the plane formed by line x=-z and the y-axis, not the xy plane.]

- Relevant Equations
- General equation of an ellipse: x^2/a^2+y^2/b^2=1

(in my case b>a)

Equation of an ellipse with a rotated axis and translated center:

((x−h)cos(A)+(y−k)sin(A))^2/(a^2)+((x−h)sin(A)−(y−k)cos(A))^2/(b^2)=1

General equation of a prolate spheroid: (x^2+y^2)/a^2+z^2/c^2=1, where c > a

I was able to find the equation of an ellipse where its major axis is shifted and rotated off of the x,y, or z axis. However, I could not find anywhere an equation for a spheroid that does not have its axis or revolution along the x,y, or z axis. How might I go about deriving such an equation?For this spheroid specifically I have found that:

-It's axis of revolution passes through the origin

-Its center its located at the point (72.846,490.034,-72.846)

-It's radius is 195.010 units long

-Its semi-major axis is 532.419 units long

(I attached an image of a plot of the ellipse that needs to be revolved with its axis of revolution represented by a yellow line)

-It's axis of revolution passes through the origin

-Its center its located at the point (72.846,490.034,-72.846)

-It's radius is 195.010 units long

-Its semi-major axis is 532.419 units long

(I attached an image of a plot of the ellipse that needs to be revolved with its axis of revolution represented by a yellow line)