Proving Homogeneous Deformation: From Spheres to Ellipsoids

[sara_87]

Homework Statement

Prove that in the homogeneous deformation, particles which after the deformation lie on the surface of a shere of radius b originally lay on the surface of an ellipsoid.

Homework Equations

homogeneous deformations are motions of the form:

x_i=c_i + A_iRX_R

where c_i and A_iR are constants or functions of time.

The Attempt at a Solution

I don't know how to prove this, i think i first need to know the equation of motion of a sphere then relate this to the equation above. I am confused about this.

 

[HallsofIvy]

Homework Statement

Prove that in the homogeneous deformation, particles which after the deformation lie on the surface of a shere of radius b originally lay on the surface of an ellipsoid.

Homework Equations

homogeneous deformations are motions of the form:

x_i=c_i + A_iRX_R

where c_i and A_iR are constants or functions of time.

The Attempt at a Solution

I don't know how to prove this, i think i first need to know the equation of motion of a sphere then relate this to the equation above. I am confused about this.

So you have <x, y, z> which satisify $x^2+ y^2+ z^2= R^2$ and your deformation if of the form
$$\begin{bmatrix}x' \\ y' \\ z'\end{bmatrix}= \begin{bmatrix}u \\ v\\ w\end{bmatrix}+ \begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}$$

Go ahead and do the calculation for x', y', z' in terms of x, y, and z and use the equation for the sphere to show that x', y', z' satisfy the equation for an ellipse.

 

[sara_87]

Thanks, but wouldn't the x', y', and z' be in terms of x, y, z, and u, v, w, and all the a's after the matrix multiplication?

 

[HallsofIvy]

Well, yes. I didn't mention the components of A since I assumed that was a constant.