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Mathematics-Online problems:

Interactive Problem 70: Euclidian Normal Form of a Quadric

 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Problem Interactive Problem

in .

Bring the quadric into matrix form

(with , and symmetric!):

, , .

Find the eigenvalues of and give them in descending order into the diagonal of the following matrix :

 0 0 0 0 0 0
.

Find (using eigenvectors of ) an orthogonal matrix with . The first row of shall consist of nonnegative entries. Bring the entries of to the common denominator . Give the numerator values of :

.

The next intention is to find the Euclidean normal form of . First you have to find a representation of with respect to a coordinate system adapted to the symmetry (principle axes transformation), i.e. make the transformation . In the new coordinates is specified as:

.

Eliminate the linear terms by translation: The transformation

, ,

.

The total transformation is:

.

The quadric is the following geometric figure:

 n/a ellipsoid one-sheeted hyperboloid hyperbolic paraboloid zeppelin cone

The signature of the quadric is:

 n/a equal to the determinant of

The length of the principle axes of a quadric determines:

 n/a the intersection of the quadric and the coordinate axes the intersection of the quadric and the unit sphere the expansion of the quadric

Find two intersecting lines and , which are contained in and are orthogonal to each other. Thereby the point shall be element of . The intersection of and is:

, , .

Complete the following two normalized direction vectors of the lines and :

, .

(Authors: Hertweck/Höfert)