Translation and rotation of quadric surface

[songoku]

Homework Statement

This is not homework. I try to study calculus by myself using James Stewart book and below is part of text that I want to ask about

Relevant Equations

Not sure

I want to ask how ##Ax^2+By^2+Cz^2+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0## can be brought to ##Ax^2+By^2+Cz^2+J=0## or ##Ax^2+By^2+Iz=0## using translation and rotation. There is no explanation in the book. What kind of translation and rotation are needed?

Thanks

 

[pasmith]

(1) The constants in the various forms are not the same.

(2) The variables int he various forms are not the same.

The general form can be written [tex]
\begin{pmatrix} x & y & z \end{pmatrix} \begin{pmatrix} A & \frac12D & \frac12 F \\ \frac12 D & B & \frac12 E \\ \frac12 F & \frac12 E & C \end{pmatrix} \begin{pmatrix} x \\y \\ z \end{pmatrix} + \begin{pmatrix} G & H & I \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} + J = \mathbf{x}^TM\mathbf{x} + \mathbf{b}^T \mathbf{x} + J = 0.[/tex] Now [itex]M[/itex] is symmetric, so its eigenvalues are real and it has a basis of orthogonal eigenvectors; thus there exists a rotation (ie. a matrix [itex]R[/itex] with determinant 1 such that [itex]R^{-1} = R^T[/itex]) such that [itex]R^{-1}MR[/itex] is diagonal. What happens if you now set [itex]\mathbf{x} = R\mathbf{X}[/itex] in the general form and complete the squares in each variable for which the corresponding diagonal entry of [itex]R^{-1}MR[/itex] is non-zero?

 

[e_jane]

I try to study calculus by myself using James Stewart book...

I want to ask how ##Ax^2+By^2+Cz^2+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0## can be brought to ##Ax^2+By^2+Cz^2+J=0## or ##Ax^2+By^2+Iz=0## using translation and rotation. There is no explanation in the book.

They took out that whole chapter? Nice! I remember *hating* that chapter.

 

[songoku]

(1) The constants in the various forms are not the same.

(2) The variables int he various forms are not the same.

The general form can be written [tex]
\begin{pmatrix} x & y & z \end{pmatrix} \begin{pmatrix} A & \frac12D & \frac12 F \\ \frac12 D & B & \frac12 E \\ \frac12 F & \frac12 E & C \end{pmatrix} \begin{pmatrix} x \\y \\ z \end{pmatrix} + \begin{pmatrix} G & H & I \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} + J = \mathbf{x}^TM\mathbf{x} + \mathbf{b}^T \mathbf{x} + J = 0.[/tex] Now [itex]M[/itex] is symmetric, so its eigenvalues are real and it has a basis of orthogonal eigenvectors; thus there exists a rotation (ie. a matrix [itex]R[/itex] with determinant 1 such that [itex]R^{-1} = R^T[/itex]) such that [itex]R^{-1}MR[/itex] is diagonal. What happens if you now set [itex]\mathbf{x} = R\mathbf{X}[/itex] in the general form and complete the squares in each variable for which the corresponding diagonal entry of [itex]R^{-1}MR[/itex] is non-zero?

I will try first and update what I have done.

They took out that whole chapter? Nice! I remember *hating* that chapter.

Can you maybe tell me the name of the chapter or anything related to the transformation of quadric surface so I can learn the basic?

Thanks

 

[e_jane]

They took out that whole chapter? Nice! I remember *hating* that chapter.

Can you maybe tell me the name of the chapter or anything related to the transformation of quadric surface so I can learn the basic?

I sold that textbook back to the university's bookstore about thirty years ago. Sorry.