- A system of conic sections

- Theorem 1

- Theorem 2

- Basic points and basic conic sections

- One conic section through one point

- Degenerated conic sections in a system

- A conic section through 5 points

- Systems of circles

All conic sections with equation

l Fis called a system of conic sections. The real numbers l and m are homogeneous parameters ( not both = 0 ). All conic sections of the system different from F_{1}(x,y,z) + m F_{2}(x,y,z) = 0

Fwith h = a real non-homogeneous parameter._{1}(x,y,z) + h F_{2}(x,y,z) = 0

Proof:

For FThis equation has degree = 3 and therefore it has always a real root._{1}(x,y,z) + h F_{2}(x,y,z) = 0 | a_{1}+ ha_{2}b_{1}"+ hb_{2}" b_{1}'+ hb_{2}'| DELTA = | b_{1}"+ hb_{2}" a_{1}'+ ha_{2}' b_{1}+ hb_{2}| | b_{1}'+ hb_{2}' b_{1}+ hb_{2}a_{1}"+ ha_{2}"| <=> | ha_{2}+ a_{1}hb_{2}"+ b_{1}" hb_{2}'+ b_{1}'| DELTA = | hb_{2}"+ b_{1}" ha_{2}'+ a_{1}' hb_{2}+ b_{1}| | hb_{2}'+ b_{1}' hb_{2}+ b_{1}ha_{2}"+ a_{1}"| <=> | ha_{2}hb_{2}"+ b_{1}" hb_{2}'+ b_{1}'| |a_{1}hb_{2}"+ b_{1}" hb_{2}'+ b_{1}'| DELTA = | hb_{2}" ha_{2}'+ a_{1}' hb_{2}+ b_{1}|+ |b_{1}" ha_{2}'+ a_{1}' hb_{2}+ b_{1}| | hb_{2}' hb_{2}+ b_{1}ha_{2}"+ a_{1}"| |b_{1}' hb_{2}+ b_{1}ha_{2}"+ a_{1}"| <=> |a_{2}hb_{2}"+ b_{1}" hb_{2}'+ b_{1}'| |a_{1}hb_{2}"+ b_{1}" hb_{2}'+ b_{1}'| DELTA = h |b_{2}" ha_{2}'+ a_{1}' hb_{2}+ b_{1}|+ |b_{1}" ha_{2}'+ a_{1}' hb_{2}+ b_{1}| |b_{2}' hb_{2}+ b_{1}ha_{2}"+ a_{1}"| |b_{1}' hb_{2}+ b_{1}ha_{2}"+ a_{1}"| <=> |a_{2}hb_{2}" hb_{2}'+ b_{1}'| |a_{1}hb_{2}" hb_{2}'+ b_{1}'| DELTA = h |b_{2}" ha_{2}' hb_{2}+ b_{1}|+ |b_{1}" ha_{2}' hb_{2}+ b_{1}| + |b_{2}' hb_{2}ha_{2}"+ a_{1}"| |b_{1}' hb_{2}ha_{2}"+ a_{1}"| |a_{2}b_{1}" hb_{2}'+ b_{1}'| |a_{1}b_{1}" hb_{2}'+ b_{1}'| h |b_{2}" a_{1}' hb_{2}+ b_{1}|+ |b_{1}" a_{1}' hb_{2}+ b_{1}| |b_{2}' b_{1}ha_{2}"+ a_{1}"| |b_{1}' b_{1}ha_{2}"+ a_{1}"| <=> |a_{2}b_{2}" hb_{2}'+ b_{1}'| |a_{1}b_{2}" hb_{2}'+ b_{1}'| DELTA = h^{2}|b_{2}" a_{2}' hb_{2}+ b_{1}|+ h |b_{1}" a_{2}' hb_{2}+ b_{1}| + |b_{2}' b_{2}ha_{2}"+ a_{1}"| |b_{1}' b_{2}ha_{2}"+ a_{1}"| |a_{2}b_{1}" hb_{2}'+ b_{1}'| |a_{1}b_{1}" hb_{2}'+ b_{1}'| h |b_{2}" a_{1}' hb_{2}+ b_{1}|+ |b_{1}" a_{1}' hb_{2}+ b_{1}| |b_{2}' b_{1}ha_{2}"+ a_{1}"| |b_{1}' b_{1}ha_{2}"+ a_{1}"| <=> |a_{2}b_{2}" hb_{2}'| |a_{1}b_{2}" hb_{2}'| DELTA = h^{2}|b_{2}" a_{2}' hb_{2}|+ h |b_{1}" a_{2}' hb_{2}| + |b_{2}' b_{2}ha_{2}"| |b_{1}' b_{2}ha_{2}"| |a_{2}b_{1}" hb_{2}'| |a_{1}b_{1}" hb_{2}'| h |b_{2}" a_{1}' hb_{2}|+ |b_{1}" a_{1}' hb_{2}| + |b_{2}' b_{1}ha_{2}"| |b_{1}' b_{1}ha_{2}"| |a_{2}b_{2}" b_{1}'| |a_{1}b_{2}" b_{1}'| h^{2}|b_{2}" a_{2}' b_{1}|+ h |b_{1}" a_{2}' b_{1}| + |b_{2}' b_{2}a_{1}"| |b_{1}' b_{2}a_{1}"| |a_{2}b_{1}" b_{1}'| |a_{1}b_{1}" b_{1}'| h |b_{2}" a_{1}' b_{1}|+ |b_{1}" a_{1}' b_{1}| |b_{2}' b_{1}a_{1}"| |b_{1}' b_{1}a_{1}"| <=> |a_{2}b_{2}" b_{2}'| |a_{1}b_{2}" hb_{2}'| DELTA = h^{3}|b_{2}" a_{2}' b_{2}|+ h |b_{1}" a_{2}' hb_{2}| + |b_{2}' b_{2}a_{2}"| |b_{1}' b_{2}ha_{2}"| |a_{2}b_{1}" hb_{2}'| |a_{1}b_{1}" hb_{2}'| h |b_{2}" a_{1}' hb_{2}|+ |b_{1}" a_{1}' hb_{2}| + |b_{2}' b_{1}ha_{2}"| |b_{1}' b_{1}ha_{2}"| |a_{2}b_{2}" b_{1}'| |a_{1}b_{2}" b_{1}'| h^{2}|b_{2}" a_{2}' b_{1}|+ h |b_{1}" a_{2}' b_{1}| + |b_{2}' b_{2}a_{1}"| |b_{1}' b_{2}a_{1}"| |a_{2}b_{1}" b_{1}'| |a_{1}b_{1}" b_{1}'| h |b_{2}" a_{1}' b_{1}|+ |b_{1}" a_{1}' b_{1}| |b_{2}' b_{1}a_{1}"| |b_{1}' b_{1}a_{1}"| Since F_{2}(x,y,z) = 0 is not degenerated, |a_{2}b_{2}" b_{2}'| |b_{2}" a_{2}' b_{2}| is not 0. |b_{2}' b_{2}a_{2}"| We have: DELTA = 0 <=> |a_{2}b_{2}" b_{2}'| |a_{1}b_{2}" hb_{2}'| h^{3}|b_{2}" a_{2}' b_{2}|+ h |b_{1}" a_{2}' hb_{2}| + |b_{2}' b_{2}a_{2}"| |b_{1}' b_{2}ha_{2}"| |a_{2}b_{1}" hb_{2}'| |a_{1}b_{1}" hb_{2}'| h |b_{2}" a_{1}' hb_{2}|+ |b_{1}" a_{1}' hb_{2}| + |b_{2}' b_{1}ha_{2}"| |b_{1}' b_{1}ha_{2}"| |a_{2}b_{2}" b_{1}'| |a_{1}b_{2}" b_{1}'| h^{2}|b_{2}" a_{2}' b_{1}|+ h |b_{1}" a_{2}' b_{1}| + |b_{2}' b_{2}a_{1}"| |b_{1}' b_{2}a_{1}"| |a_{2}b_{1}" b_{1}'| |a_{1}b_{1}" b_{1}'| h |b_{2}" a_{1}' b_{1}|+ |b_{1}" a_{1}' b_{1}| = 0 |b_{2}' b_{1}a_{1}"| |b_{1}' b_{1}a_{1}"|

Proof:

- If at least one conic section is degenerated in line d1 and d2, then d1 cuts the other conic section in 2 points, and d2 cuts the other conic section in 2 points.
- If both conic sections are not degenerated:

The common points of both conic sections are the solutions of/ F

We choose h such that F_{1}(x,y,z) = 0 \ F_{2}(x,y,z) = 0 <=> / F_{1}(x,y,z) = 0 \ F_{1}(x,y,z) + h F_{2}(x,y,z) = 0_{1}(x,y,z) + h F_{2}(x,y,z) = 0 is degenerated. The system has 4 solutions and the conic sections have 4 common points.

Each conic sections of the system

l Fgoes through the 4 common points of F_{1}(x,y,z) + m F_{2}(x,y,z) = 0

Two arbitrary conic sections of the system go through the four basic points. These two conic sections can be chosen as basic conic sections of the system.

The proof is left as an exercise.

If there is at least one non-degenerated conic section in a system, then there are at least one and at most three degenerated conic sections in that system.

Proof:

Take a system with basic conic sections F_{1}(x,y,z) = 0 and F_{2}(x,y,z) = 0.
Say F_{2}x,y,z) = 0 is not degenerated. An element of the system
different from F_{2} has equation

FFrom above we know that the DELTA of that conic section can be written as a polynomial in h with degree = 3._{1}(x,y,z) + h F_{2}(x,y,z) = 0

Thus, there are at least one and at most 3 real values of h, such that DELTA = 0.

Through 4 of this points there is a system of conics. From this system, there is just one element going through the fifth point.

The proof is left as an exercise.

The tutorial address is http://home.scarlet.be/math/

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