क्यूबिक्स हिंदी उपयोग और उदाहरण

""फ्रांसीसी कला समीक्षक लुइस वॉक्सेलस ने 1908 में बराक द्वारा बनाये गए एक चित्र को देखने के बाद पहली बार 'क्यूबिज्म' या $बिजारे क्यूबिक्स$ शब्द का प्रयोग किया।

क्यूबिक्स इसके अंग्रेजी अर्थ का उदाहरण

Assume that two cubics and in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field.

A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and .

In that case, every cubic through also passes through the intersection of any two different cubics through , which has at least nine points (over the algebraic closure) on account of Bézout's theorem.

For the Cayley"ndash;Bacharach theorem, it is necessary to have a family of cubics passing through the nine points, rather than a single one.

A special case is Pascal's theorem, in which case the two cubics in question are all degenerate: given six points on a conic (a hexagon), consider the lines obtained by extending opposite sides – this yields two cubics of three lines each, which intersect in 9 points – the 6 points on the conic, and 3 others.

The following eight points are common to both cubics: A, B, C, A+B, -A-B, B+C, -B-C, O.

Thus if the nine points lie on more than one cubic, equivalently on the intersection of two cubics (as ), they are not in general position – they are overdetermined by one dimension – and thus cubics passing through them satisfying one additional constraint, as reflected in the "eight implies nine" property.

These first agree for , which is why the Cayley–Bacharach theorem occurs for cubics, and for higher degree is greater, hence the higher degree generalizations.

9 and 9: nine points determine a cubic, two cubics intersect in nine points,.

For cubics, nine points determine a cubic, but in general they determine a unique cubic – thus having two different cubics pass through them (and thus a pencil) is special – the solution space is one dimension higher than expected, and thus the solutions satisfy an additional constraint, namely the "8 implies 9" property.

Curves of types (ii) and (iii) are the rational cubics and are call nodal and cuspidal respectively.

Curves of type (i) are the nonsingular cubics (elliptic curves).