# Homogeneous deformation

1. Aug 24, 2009

### sara_87

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

homogeneous deformations are motions of the form:

xi=ci + AiRXR

where ci and AiR are constants or functions of time.

3. 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. Im confused about this.

2. Aug 24, 2009

### HallsofIvy

Staff Emeritus
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.

Last edited: Aug 24, 2009
3. Aug 24, 2009

### sara_87

Thanks, but wouldnt 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?

4. Aug 24, 2009

### HallsofIvy

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