There cannot be a third intersection pointR. IndeedR∈ [AB]would forced to have two common pointsP,Rwith[AB]and the side[AB]would be on the lined.
Analogously, one cannot haveR∈ [AC]. On the other hand by Proposition3.4.2.2, havingR∈ [BC]would forceB andC to be in the two different half planes with respect tod, which is again a contradiction since both are ind+. Let us also make more precise the locution “a line enters a triangle”, just by defining the “interior” of a triangle.
Definition 3.5.3 Given a triangleABC in a real affine plane, consider the three half planes:
• ΠA, determined by the linedBC and containingA;
• ΠB, determined by the linedAC and containingB;
• ΠC, determined by the linedAB and containingC.
The interiorIof the triangleABC is the intersection of the three half planesΠA, ΠB,ΠC. This yields a corresponding partition
I∪P∪E
of the plane, whereP is the perimeter of the triangle. The third subsetE is called the exterior of the triangle.
Proposition 3.5.4 Consider a triangleABC in a real affine plane.
1. The interior of the triangle is a convex subset.
2. IfAis interior to the triangle andBis exterior to the triangle, the segment[AB] cuts the perimeter of the triangle at a unique point.
3. A line containing an interior point of a triangle intersects the perimeter at exactly two points.
Proof A real affine plane is a directed plane in the sense of Definition 8.2.1 in [7], Trilogy I (see the proof of Example 8.5.3 in [7], Trilogy I). The result is then
Proposition 8.2.12 in [7], Trilogy I.
3.6 Affine Classification of Real Quadrics
In the real case, our results on quadrics can be made more precise.
First observe that in the real case, the reduced equation of a quadric can further be reduced!
Proposition 3.6.1 LetQ⊆Ebe a quadric in a finite dimensional real affine space.
There exists an affine basis(O;e1, . . . , en)with respect to which the equation of the quadric takes one of the following reduced forms:
k i=1
X2i − l i=k+1
X2i =1 k
i=1
X2i − l i=k+1
X2i =0 k
i=1
X2i − l i=k+1
X2i =Xn
withl < nin the last case.
Proof Under the conditions of Theorem2.24.2, up to a possible renumbering of the coordinates, there is no loss of generality in assuming that in the left hand side of the equation, the strictly positive coefficients come first, then the strictly negative coefficients and finally the zero coefficients. Whenai=0, simply apply the change of basis
ei=
1
|ai|ei.
Otherwise, keepei=ei. With respect to the basis(0;e1, . . . , en)the quadric now has a reduced equation of the form indicated in the statement.
Proposition 3.6.2 Let(E, V )be a finite dimensional real affine space andQ a non-degenerate quadric. Let us make the convention that:
• for equations of type 2, possibly multiplying the equation by −1, the number of strictly positive coefficients is greater than or equal to the number of strictly negative coefficients;
• for equations of type 3, possibly changing the orientation of the last axis and mul- tiplying the equation by−1, we assume again that the number of strictly positive coefficients is greater than or equal to the number of strictly negative coefficients.
Under these conditions, two reduced equations of the non-degenerate quadricQ:
1. are of the same type; and
2. have the same number of strictly positive, strictly negative and zero coefficients.
3.6 Affine Classification of Real Quadrics 131 Proof Assertion 1 is thus condition 1 in Theorem2.25.5. Let us consider two bases
(O;e1, . . . , en),
O;e1, . . . , en with respect to whichQadmits reduced equations. We write
−
→x =M−→x + −→v , M=(mij)1≤i,j≤n for the change of basis formula.
Suppose first that the reduced equations are of type 1:
−
→xtA−→x =1, −→ xtA−→
x =1
thus withAandAdiagonal. The equation ofQwith respect to the first basis can then also be written as
(M−→x + −→v )tA(M−→x + −→v )=1.
Expanding this formula we obtain
−→xtMtAM−→x +2−→vtA−→x + −→vtA−→v =1.
By Proposition2.25.3, this equation has no term of degree 1, thus
−
→vtA=(0, . . . ,0).
Therefore the equation reduces to the form
−
→x tMtAM−→x =1.
By Theorem2.26.9this yields
A=MtAM.
By Sylvester’s theoremG.3.1, the two matricesAandAhave the same number of strictly positive, strictly negative and zero coefficients.
The same argument applies to reduced equations of type 2 to infer a proportion- ality
A=kMtAM, 0=k∈R.
Ifk >0, we are done, again by Sylvester’s theoremG.3.1. The convention in the statement concerning the signs of the coefficients forceskto be positive, unless there are exactly the same number of strictly positive and strictly negative coefficients.
However, in the latter case, the sign ofkno longer matters.
In the case of reduced equations of type 3, let us write A=(aij)1≤i,j≤n−1, A=
aij
1≤i,j≤n−1, M=(mij)1≤i,j≤n−1
so that the two reduced equations are
(X1, . . . , Xn−1)A
⎛
⎜⎝ X1
... Xn−1
⎞
⎟⎠=Xn,
X1, . . . , Xn− 1 A
⎛
⎜⎝ X1
... Xn−1
⎞
⎟⎠=Xn
withAandAdiagonal. Again we can write the equation of the quadric with respect to the first basis by expressing the fact thatM−→
X + −→v satisfies the second equation.
By Theorem2.26.9, this new equation is proportional to the first one and therefore reduces to the form
(X1, . . . , Xn−1)MtAM
⎛
⎜⎝ X1
... Xn−1
⎞
⎟⎠=mnnXn.
Let us switch back to the non-degenerate quadricQofRn−1
n−1
i=1
aiiX2i =1
already considered in the proof of Theorem2.26.9. The equation of this quadric with respect to the first basis can thus also be written as:
(X1, . . . , Xn−1)MtAM
⎛
⎜⎝ X1
... Xn−1
⎞
⎟⎠=mnn.
In particularmnn=0 and dividing by this quantity, the two equations ofQcoincide term by term, by Theorem2.26.9. Again the Sylvester’s theoremG.3.1and the con- vention in the statement forceAandAto have the same number of strictly positive,
strictly negative and zero coefficients.
Finally, let us consider the following relation on quadrics, which is trivially an equivalence relation.
Definition 3.6.3 Let (E, V ) be a finite dimensional affine space. Two given quadricsQ andQ are said to be affinely equivalent when there exists an affine isomorphism(f,−→
f ):(E, V )−→(E, V )such thatf (Q)=Q. We now present the classification theorem of real affine quadrics:
Theorem 3.6.4 Let (E, V ) be a finite dimensional real affine space. With the convention of Proposition 3.6.2 on the signs of coefficients, two non-degenerate quadricsQandQare affinely equivalent if and only if:
3.6 Affine Classification of Real Quadrics 133 1. their reduced equations are of the same type (see Theorem2.24.2); and
2. their reduced equations contain the same number of strictly positive, strictly neg- ative and zero coefficients.
Proof Consider an affine basis
(O;e1, . . . , en)
whereQadmits a reduced equation and an affine isomorphism(f,−→
f )transforming QintoQ. The equation off (Q)with respect to the basis
f (0);−→
f (e1), . . . ,−→ f (en)
is identical to the equation ofQwith respect to the original basis. Butf (Q)=Q, so thatQadmits in the second basis the same reduced equation as Qin the first basis.
Conversely, consider a quadricQhaving a reduced equation in a first basis and a quadricQhaving a reduced equation in a second basis; assume that these reduced equations satisfy the conditions of the statement. Up to renumbering the coordinates, there is no loss of generality in assuming, with the notation of Theorem2.24.2, that in both reduced equations:
• the firstrcoefficients are strictly negative;
• the nextscoefficients are zero;
• the remaining coefficients are strictly positive.
Let us write
(a1, . . . , an), (b1, . . . , bn)
for these two lists of coefficients, withan=1=bnin the case of equations of type 3.
Consider the linear isomorphism(f,−→
f )sending the first origin to the second origin and having diagonal matrix
⎧⎪
⎨
⎪⎩
mii=baii ifi≤r;
mii=1 ifr+1≤i≤r+s; mii=baii ifr+s+1≤i≤n;
with respect to the two bases. This isomorphism transforms the first quadric into the
second one.
Corollary 3.6.5 In a real affine plane a conic is:
1. an ellipse if and only if it admits a reduced equation of the form x2+y2=1;
2. a hyperbola if and only if it admits a reduced equation of the form x2−y2=1;
3. a parabola if and only if it admits a reduced equation of the form x2=y.
Proof In the case ofR2and its canonical basis, the three equations are indeed those of an ellipse, a hyperbola and a parabola (see Sect.1.10). Since an affine isomor- phism transforms a line into a line, a quadric equivalent to an ellipse, a hyperbola or a parabola can never be empty, a point, a line or the union of two lines; thus it must be an ellipse, a hyperbola or a parabola. Since moreover the ellipse, the hy- perbola and the parabola admit equations of different types, the result follows by
Theorem3.6.4.