Translation and rotation of quadric surface

songoku · Sep 21, 2023

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 · Sep 21, 2023

(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 <br /> \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. Now M is symmetric, so its eigenvalues are real and it has a basis of orthogonal eigenvectors; thus there exists a rotation (ie. a matrix R with determinant 1 such that R^{-1} = R^T) such that R^{-1}MR is diagonal. What happens if you now set \mathbf{x} = R\mathbf{X} in the general form and complete the squares in each variable for which the corresponding diagonal entry of R^{-1}MR is non-zero?

e_jane · Sep 21, 2023

songoku said:

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. :oldruck:

songoku · Sep 23, 2023

pasmith said:

(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 <br /> \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. Now M is symmetric, so its eigenvalues are real and it has a basis of orthogonal eigenvectors; thus there exists a rotation (ie. a matrix R with determinant 1 such that R^{-1} = R^T) such that R^{-1}MR is diagonal. What happens if you now set \mathbf{x} = R\mathbf{X} in the general form and complete the squares in each variable for which the corresponding diagonal entry of R^{-1}MR is non-zero?

I will try first and update what I have done.

e_jane said:

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 · Sep 24, 2023

e_jane said:

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

songoku said:

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.

Translation and rotation of quadric surface

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