Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics

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section. If C is a parabola, then L

is called its axis. If C is an ellipse, then L

and L

will be called the major and minor (coordinate) axis of C, respectively. If C is a

hyperbola, then L

and L

will be called the transverse and conjugate (coordinate) axis

of C, respectively. In the case of either an ellipse or hyperbola the point O is called its

center.

Figure 3.21 shows the coordinate systems and coordinate axes for parabolas,

ellipses, and hyperbolas. Ellipses and hyperbolas are called central conic sections

because they have a center. One can show that the conic sections are symmetric about

their axes, meaning that if a point belongs to them, then the reﬂected point about

their axes will belong to them also.

In the case of the plane R

, we can use Theorem 3.6.1 and write out the constraints

on the distances of points from the focus and directrix in terms of an equation. It is

easily seen to be a quadratic equation. Let us look at the equations for some special

well-known cases in their natural coordinate system.

The parabola: y

= 4ax (3.32)

Focus: (a, 0), a > 0

Directrix: x =-a

Eccentricity: e = 1

3.6 Conic Sections 169

Directrix

x + a = 0

F(a,0)

Parabola

x + a/e = 0

x + a/e = 0 x – a/e = 0

x – a/e = 0

(0,b)

F¢(–ae,0)

F(ae,0)

(–a,0)

(0,–b)

(a,0)

Ellipse

directrix directrix

y = –bx/a y = bx/a

Hyperbola

(a)

(b)

(c)

Figure 3.21. The natural coordinate systems for conic sections.

The ellipse:

(3.33)

Focus: (c, 0)

Directrix:

Eccentricity: , where c ≥ 0 and c

= a

- b

The segments [(-a,0),(a,0)] and [(0,-b),(0,b)] are also sometimes called the major

and minor axis of the ellipse, respectively, or simply the principal axes.

The hyperbola:

(3.34)

Focus: (c, 0)

Directrix:

Eccentricity: , where c

= a

+ b

The segments [(-a,0),(a,0)] and [(0,-b),(0,b)] are also sometimes called the trans-

verse and conjugate axis of the hyperbola, respectively. The asymptotes of the

hyperbola are the lines y =±bx/a.

Because of the symmetry present in the case of a hyperbola and ellipse, they actu-

ally have two foci and directrices. The second of the pair is obtained by reﬂecting the

ones given above about the y-axis.

The discussion above shows that conic sections are solutions to quadratic equa-

tions. To connect the two we start from the other direction.

Deﬁnition. An (afﬁne) conic is any subset of R

deﬁned by an equation of the

form

where (a,b,c) π (0,0,0).

Our object is to show that the terms “conic” and “conic section” refer essentially

to the same geometric spaces. To that end it is convenient to rewrite the equation

deﬁning a conic in the form

(3.35)

ax hxy by fx gy c

2220+++++=.

ax bxy cy dx ey f

0+++++=,

10-= >,,

if=>

()

0 c

10+= ≥>,

170 3 Projective Geometry

Although this may look a little artiﬁcial right now, it prevents a factor of 2 from enter-

ing equations at a later stage and so we shall use this form of the equation from now

on. In any case, to understand the solution set of our quadratic equation, the idea will

be to transform it into a simpler one. We shall show that in the nondegenerate cases

a simple change in coordinate systems will change the general quadratic equation

(3.35) into one of the equations (3.32), (3.33), or (3.34). The main problem is ﬁnding

a change in coordinates that will eliminate the xy term.

We start off by describing two straightforward simple-minded but ad hoc

approaches to solve our problem. The second is actually good enough to give us the

formulas that one uses to convert (3.35) to one of the standard forms. However, there

are also some well-deﬁned invariants associated to (3.35), which can tell us right away

what sort of curve one has without actually transforming the equation. To be able to

prove that these invariants work as speciﬁed is what motivates us to present a third,

more elegant approach to analyzing (3.35) using homogeneous coordinates and the

theory of quadratic forms.

The ﬁrst simple-minded approach uses a form of “completing the square.” Prob-

ably the most straightforward way to do this is to rewrite the equation as

where a

satisﬁer a = a

. The substitution

will then produce an equation in x¢ and y¢ that has no x¢y¢ term. Although this sub-

stitution is satisfactory in some applications and produces simple formulas, it has the

disadvantage that the linear transformation of coordinates to which it corresponds is

not a rigid motion and may deform shapes in undesirable ways. Thus, since the xy

term arises from a rotation of the axes of the conic, a second and “better” way to

eliminate this term is via a rotation about the origin. Consider the substitution

which corresponds to rotating the conic about the origin through an angle -q. This

substitution will transform equation (3.35) into an equation in x¢ and y¢ for which the

coefﬁcient of the x¢y¢ term is

Setting this expression to zero and using some simple trigonometric identities gives

the equation

ba h-

()

sin cos cos sin .qq q q

yy y=¢ +¢sin cosqq

xx y=¢ -¢cos sinqq

¢=yy

¢= -xax

ab h

yfxgyc

22 0+

++ +=

3.6 Conic Sections 171

If a = b, then cos 2q=0, which means that q is ±45 degrees. If a π b, then the solution

can be written either as

(3.36a)

(3.36b)

Note that if q

and q

are angles satisfying (3.36b) where we use the + and - sign,

respectively, in the formula, then

which shows that the angles differ by 90 degrees. We shall see that this basically

afﬁrms that the “axes” of the conic are perpendicular. In any case, there is an angle q

which will eliminate the x¢y¢ cross-term in equation (3.35). Thus, we end up with an

equation of the form

(3.37)

The rest of the steps involved in analyzing equation (3.35) are very simple. Equa-

tion (3.37) still has some linear terms that need to be eliminated if the corresponding

quadratic term is present. This is done by completing the square in the standard way.

For example, if a¢π0, then make the substitution

From this one sees that, in the nondegenerate case, equation (3.38) splits into two

cases. If a¢b¢=0, then (3.35) represents a parabola; otherwise, (3.35) represents an

ellipse or hyperbola depending on whether the sign of a¢b¢ is positive or negative.

In summary, we have shown how some simple manipulations of equation (3.35)

enable us to determine the geometry of the solution set. However, we have glossed

over some degenerate cases. For example, a degenerate case of equation (3.35) occurs

when it factors into two linear terms, that is, that it can be written in the form

(3.38)

In this case the equation represents a pair of lines. To be able to detect the special

cases, and all the other cases for that matter, in a nice simple way, we need to make

use of some more powerful tools. In particular, we need to switch to homogeneous

coordinates and projective space.

axbycaxbyc

111222

0++

()

= .

¢= ¢¢+

¢¢+ ¢¢+ ¢¢+ ¢¢+¢=ax by fx gy c

22 0.

tan tan ,qq

1=-

tan .q=

-± -

()

+ba ba h

tan2

ba h-

()

+=sin cos .22 20qq

172 3 Projective Geometry

Deﬁnition. A conic in the projective plane, or a projective conic, is a set of points in

whose homogeneous coordinates (x,y,z) satisfy a quadratic equation of the form

(3.39)

where (a,b,d) π (0,0,0).

First, projective conics are well deﬁned in the sense that if one homogeneous coor-

dinate representative of a point in P

satisﬁes equation (3.39), then they all do. Second,

every conic in the plane deﬁnes a unique projective conic because, switching to homo-

geneous coordinates, equation (3.35) changes into

(3.40)

Third, the condition that at least one of the values a, b, and d be nonzero is simply

the analog of the condition on the coefﬁcients of an afﬁne conic. It is an extremely

artiﬁcial condition at the projective level. The fact is that when someone says “conic”

one thinks of ellipses, hyperbolas, and parabolas. We were trying to exclude grossly

degenerate cases of equation (3.39) such as 0 = 0, which would have all of P

as its

solution. The projective conics we are after are the “nondegenerate” ones that are

deﬁned below. Our condition on the coefﬁcients happens to capture (with foresight)

the equation form of the conic sections. Having said this, the fact that one can give a

complete analysis of the solutions to the general quadratic equation (3.39) is inter-

esting on its own, independent of any relation to conic sections. This equation-solving

mindset is the context in which we will be working right now.

Now equation (3.40) is also the convenient form for projective conics. Consider

the quadratic form

and its matrix

(3.41)

Note that equation (3.40) can be rewritten in matrix form as

(3.42)

Conversely, equation (3.42) deﬁnes a projective conic for any symmetric 3 ¥ 3 matrix

A satisfying (a,b,h) π (0,0,0). The next theorem shows that projective conics are actu-

ally very easy to describe if one chooses the correct coordinate system.

3.6.3. Theorem. We can coordinatize projective space P

in such a way that in the

new coordinate system the equation of the projective conic deﬁned by equation (3.40)

has one of the following forms:

xyzAxyz

,, ,, .

()()

= 0

ahf

hbg

fgc

q x y z ax hxy by fxz gyz cz

22 2

222,,

()

=+ ++ + +

ax hxy by fxz gyz cz

22 2

2220+++++=.

ax by cz dxy exz fyz

222

0+++++=,

3.6 Conic Sections 173

(3.43a)

(3.43b)

(3.43c)

Proof. By Theorem 1.9.11, any symmetric matrix, the matrix A in (3.41) in particu-

lar, is congruent to a diagonal matrix with ±1 or 0 along the diagonal. This clearly

implies the result since A is not the zero matrix. Note that since we are working in

projective space here the change of coordinates transformations that produce equa-

tions (3.43) are projective and not afﬁne transformation in general.

Translating things back to Euclidean space, it is easy to see that equations (3.43a)

and (3.43b) correspond to cases where the solution set to (3.35) is either empty or

consists of lines. Equations (3.43c) is the case where A is nonsingular.

Deﬁnition. The afﬁne conic deﬁned by equation (3.35) or the projective conic

deﬁned by equation (3.40) is said to be nondegenerate if the matrix A in (3.41) is non-

singular; otherwise it is said to be degenerate.

Note. The deﬁnition of a nondegenerate conic has the advantage of simplicity but

has a perhaps undesirable aspect to it, at least at ﬁrst glance. If A is nonsingular, then

one possibility for equation (3.43c) is

This equation has no real nonzero solutions. Therefore, a “nondegenerate” conic

could be the empty set. For that reason, some authors add the condition that a conic

be nonempty before calling it nondegenerate. On the other hand, we would get a non-

empty set if we were to allow complex numbers and we were talking about conics in

the complex plane. See Section 10.2.

We have just seen that matrix A in (3.41) determines one important invariant for

conics, but there is another. Consider the quadratic form that is the homogeneous

part of equation (3.35), namely,

(3.44)

Let

(3.45)

be the matrix associated to q

. By Theorem 1.9.10 there is a change of basis that will

diagonalize B, that is, in the new coordinate system q

will have the form

q x y ax hxy by

2,.

()

=+ +

xyz orxy

222 22

010++= ++=

()

using Cartesian coordinates .

xyz

222

0+±=

0±=

174 3 Projective Geometry

The numbers a¢ and b¢ are just the eigenvalues of the transformation associated to B

and are the roots of the characteristic polynomial for B. Since the change corresponds

to a linear change of variables, in this new coordinate system equation (3.35) will

have been transformed into an equation of the form (3.37). We now have all the pieces

of the puzzle.

3.6.4. Theorem. Deﬁne numbers D, D, and I for equation (3.35) by

(1) The quantities D, D, and I are invariant under a change of coordinates via a

rigid motion (translation or rotation).

(2) If Dπ0, then equation (3.35) deﬁnes a nondegenerate conic. More precisely,

(a) D > 0: We have an ellipse if ID<0 and the empty set otherwise.

(Note that since a and b have the same sign in this case, the sign

of ID is the same as the sign of bD or aD.)

(b) D < 0: We have a hyperbola.

(3) If D=0, then equation (3.35) factors into two factors of degree one (the con-

verse is also true) and deﬁnes the empty set, a point, or a pair of lines. The

pair of lines may be parallel, intersecting, or coincident. More precisely,

(a) D > 0: We get a single point.

(b) D < 0: We get two intersecting lines.

D= = = - = +

ahf

hbg

fgc

ab h and I a b,,.

qxy ax by

¢¢

()

=¢¢+¢¢,.

3.6 Conic Sections 175

b π 0: There are three cases depending on E = g

- bc.

E > 0: We get two parallel lines.

E < 0: We get the empty set.

E = 0: We get a single line.

b = 0: Then h = 0. There are three cases depending on F = f

- ac.

F > 0: We get two parallel lines.

F < 0: We get the empty set.

F = 0: We get a single line.

Proof. Part (1) follows from properties of the determinant and the trace function (I

is the trace of the matrix B). The main ideas behind the proof of (2) have been sketched

above. The condition on the product ID in (2)(a) is equivalent to saying that I and D

have opposite signs. The effect that the signs of I and D have on the conic is best seen

by looking at the matrices after they have been diagonalized. The details of the rest

of the proof are lengthy and messy but not hard. They basically only involve rewrit-

ing equation (3.35) and solving for various conditions. See [Eise39].

Theorem 3.6.4 tells us the conic that equation (3.35) represents but does not

directly tell us the transformation that transforms it to our standard equations. The

basic steps in ﬁnding this transformation were sketched earlier. We summarize the

results along with some additional details below. They are divided into three cases

that subdivide into the subcases in Theorem 3.6.4. Again see [Eise39]. For a slightly

different approach, see [RogA90].

Let C be the conic deﬁned by equation (3.35). Our object is to ﬁnd a rigid motion M

so hat M(C) is deﬁned by the standard equation for a conic. The transformation M will

have the form RT, where T is a translation and R is a rotation about the origin.

Case 1: D π 0 (the central conics if Dπ0)

The “center” of the conic is at

(3.46)

Let T be the translation that sends (x

) to the origin. The conic C¢=T(C) will have

its center at the origin. It follows that replacing x by x + x

and y by y + y

will elim-

inate the linear terms in (3.35) and give us an equation for C¢ of the form

(3.47)

Let q be an angle deﬁned by equations (3.36) and let R be the rotation about the origin

through the angle -q. This will rotate an axis of the conic into the x-axis. The equa-

tion for C≤=R(C¢) is obtained by replacing x by x cosq-y sin q and y by x sin q+y

cosq in equation (3.47). We will get

(3.48)

In fact, a¢ and b¢ are the roots of

(3.49)

that is,

(3.50)

Equation (3.49) is actually the characteristic equation for the matrix B in (3.45)

(Theorem C.4.8(3)), so that a¢ and b¢ are the eigenvalues of B. In other words, instead of

dealingwith angles directly, ﬁnd an orthonormal pair u

and u

of eigenvectors for B. Let

(3.51)

xy x y

()

=¢¢

()

¢=

++ -

()

¢=

+- -

()

ab ab h

and b

ab ab h

xIxD

0-+=,

¢¢+ ¢¢+ =ax by

ax hxy by

20+++ =constant .

hg bf

hf ag

176 3 Projective Geometry

and substitute the x and y that one gets from equation (3.51) into equation (3.47).

This will also produce equation (3.48).

Case 2: D = 0 and Dπ0 (parabola)

One can show that the vertex of the parabola is at (x

), where x

and y

are the

solution to the equations

(3.52)

and

(3.53)

Furthermore,

(3.54)

is the equation of the axis of the parabola. Therefore, translate the origin to the vertex

of the parabola and then rotate about that point through an angle -q, where

(3.55)

These transformations will change equation (3.35) into

(3.56a)

where

(3.56b)

Case 3: D = 0 and D=0 (degenerate cases)

If b π 0, then equation (3.35) reduces to

(3.57)

If b = 0 (and also h = 0), then we get

(3.58)

ffac

-± -

hx g g bc

()

±-

ag hf

Ia h

¢= ¢ypx

tan .q=- =-

a b hx by fh gb+

()

++ =0

fb hg

ag hf

yc+

ax hy

af hg

0++

3.6 Conic Sections 177

Alternatively, rotate about the origin through the angle -q, where q is deﬁned by

equation (3.55). This changes equation (3.35) into

(3.59)

If there are real roots to this equation, then we get a factorization

which corresponds to one or two straight lines.

3.6.5. Example. To transform

(3.60)

into standard form.

Solution. We have

Therefore, I = 125, D=det (A) =-250000, and D = det (B) = 2500 and we fall into Case

1. In fact, D > 0, Dπ0, and ID<0 means that we have an ellipse. After translating the

center of the ellipse, deﬁned by equation (3.46), to the origin, we can ﬁnish the reduc-

tion of equation (3.60) in two ways: we can rotate through the angle speciﬁed by equa-

tions (3.36), or we could use the eigenvector approach indicated by equation (3.51).

Of course, we could also just simply use equations (3.48) and (3.50), but, although

this may seem simpler, the advantage of the other two methods is that they also give

us the coordinate transformation that transforms the standard coordinate system

into the one in which the curve has the standard form. One often needs to know this

transformation.

We start with the angle approach. Using equation (3.46), the center (x

) of the

curve turns out to be (2,3). Substituting x + 2 and y + 3 for x and y in equation (3.60) gives

(3.61)

We now want to rotate the coordinate axes through an angle -q, where, using formula

(3.36b),

We arbitrarily choose 3/4, so that

tan .q=-

52 72 73 100 0

xxyy-+-=.

A and B=

52 36 4

36 73 147

4 147 333

52 36

36 73

52 72 73 8 294 333 0

xxyyxy-++-+=

¢-

()

¢-

()

=yyyy

af hg

yc¢+

¢+ =

20.

178 3 Projective Geometry