mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a twisted cubic is a smooth, rational curve ''C'' of degree three in projective 3-space P³. It is a fundamental example of a

skew curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

. It is essentially unique, up to

projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...

(''the'' twisted cubic, therefore). In

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the twisted cubic is a simple example of a

projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...

that is not linear or a

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

, in fact not a

complete intersection In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there s ...

. It is the three-dimensional case of the rational normal curve, and is the

image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

of a Veronese map of degree three on the

projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...

Definition

The twisted cubic is most easily given parametrically as the image of the map :

\nu:\mathbf^1\to\mathbf^3

which assigns to the homogeneous coordinate

:T /math> the value

: \nu: :T \mapsto^3:S^2T:ST^2:T^3 In one

coordinate patch In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies th ...

of projective space, the map is simply the moment curve :

\nu:x \mapsto (x,x^2,x^3)

That is, it is the closure by a single

point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...

of the affine curve

(x,x^2,x^3)

. The twisted cubic is a

, defined as the intersection of three

quadric In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hype ...

s. In homogeneous coordinates

:Y:Z:W /math> on P

³, the twisted cubic is the closed

subscheme This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

defined by the vanishing of the three

homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...

s :

F_0 = XZ - Y^2

F_1 = YW - Z^2

F_2 = XW - YZ.

It may be checked that these three

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

s vanish identically when using the explicit parameterization above; that is, substitute ''x''³ for ''X'', and so on. More strongly, the

homogeneous ideal In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, but ...

of the twisted cubic ''C'' is generated by these three homogeneous polynomials of degree 2.

Properties

The twisted cubic has the following properties: * It is the set-theoretic complete intersection of

XZ - Y^2

and

Z(YW-Z^2)-W(XW-YZ)

, but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed. (An attempt to use only two polynomials make the resulting ideal not radical, since

(YW-Z^2)^2

is in it, but

YW-Z^2

is not). * Any four points on ''C'' span P³. * Given six points in P³ with no four coplanar, there is a unique twisted cubic passing through them. * The union of the

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

and

secant line In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant inter ...

s (the

secant variety In algebraic geometry, the secant variety \operatorname(V), or the variety of chords, of a projective variety V \subset \mathbb^r is the Zariski closure of the union of all secant lines (chords) to ''V'' in \mathbb^r: :\operatorname(V) = \bigcup_ \o ...

) of a twisted cubic ''C'' fill up P³ and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the

and secant lines of any non-planar smooth

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

is three-dimensional. Further, any smooth

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...

with the property that every length four subscheme spans P³ has the property that the tangent and secant lines are pairwise disjoint, except at points of the variety itself. * The projection of ''C'' onto a plane from a point on a tangent line of ''C'' yields a cuspidal cubic. * The projection from a point on a secant line of ''C'' yields a nodal cubic. * The projection from a point on ''C'' yields a

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

References

*. {{Algebraic curves navbox Algebraic curves