LECTURE 31

QUADRIC SURFACES

When a standard quadric surface

x2

a2 y

2

b2 z

2

c2= 1

is rotated in space mixed terms xy, xz and yz appear in the equation. It is possible to expressthe resulting quadratic form in matrix form xTAx = 1 where A is a symmetric matrix. Ananalysis of the eigenvectors and eigenvalues of A will reveal both the structure and the principalaxes of the surface.

Consider the curve 2x24xy+5y2 = 54. The presence of the mixed term xy indicatesthat this is a standard object (ellipse or hyperbola) which has been tilted to some degreeso that its major and minor axes no longer point in the x and y directions. To under-stand the curve we need to apply a specific transformation which untilts the curve intostandard form. Our first step is to rewrite the quadratic form in terms of matrices.

Claim: 2x2 4xy + 5y2 = 54 is equivalent to ( x y )(

2 22 5

)(x

y

)= 54

Proof:

This gets us over into the arena of matrices where the theory of eigenvalues andeigenvectors may be brought into play! Observe that by its nature of construction, thematrix A will be symmetric and thus its eigenvectors will be naturally orthogonal to eachother. We now undertake a complete eigen-analysis of A.

1

2

So the eigenvalues of A are 1 and 6 with associated unit eigenvectors

25

15

and

1

5

25

respectively. Observe that the eigenvectors are indeed orthogonal.These di-

rections X =

25

15

and Y =

1

5

25

actually form the principal axes of the

curve. That

is the curve sits properly on the eigenvectors of A. We will prove this formally in the nextlecture. Furthermore in the {X, Y } system the equation of the curve is 1X2 + 6Y 2 = 54.(Note the use of the eigenvalues). We can now identify the curve as an ellipse whoseclosest point to the origin is 3 units in the Y direction. Thus the closest points to the

origin are

3

5

65

in the {x, y} system. The transformation which interrelates the

two coordinate systems is

(x

y

)= P

(X

Y

)where P is the usual matrix of eigenvectors

25

15

15

25

. Sketch:

In summary:The quadratic form xTAx where A is a symmetric matrix has principal axes given by

the orthogonal eigenvectors of A and the associated quadratic curves and quadric surfacesmay be transform into standard objects with the eigenvalues as coefficients.

3

Example 1 Express the equation of the surface

x2 + 2y2 + 2z2 + 4xy 4xz + 6yz = 30

in terms of its principal axes X , Y and Z and hence determine the nature of the surface.

Find an orthogonal matrix P implementing the transformation through

xy

z

= P

XY

Z

.

Deduce the shortest distance from the surface to the origin and the

xy

z

coordinates

of these closest point(s).

The equation may be written as(x y z

) 1 2 22 2 32 3 2

xy

z

= 30

Now the matrix A =

1 2 22 2 32 3 2

has eigenvalues -3, 3 and 5 with associated eigen-

vectors

11

1

, 21

1

and

01

1

. Observe the orthogonality of the eigenvectors!

4

P =

13

26

0

13

16

12

13

16

12

, 3X2 + 3Y 2 + 5Z2 = 30

Hyperboloid of one sheet with closest distance to the origin of6 at

03

3

We will start the next lecture with a formal proof of these algorithms and results.

31You can now do Q 91 and 92

5

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