In mathematics, a twisted cubic is a smooth,

rational 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 ...

''C'' of

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

. 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 genera ...

(''the'' twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

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 Eucl ...

, 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 sho ...

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

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...

of a Veronese map of degree three on the

projective line In mathematics, 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 resultant elimination of special cases; ...

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 In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...

:T /math> the value

: \nu: :T \mapsto^3:S^2T:ST^2:T^3 In one coordinate patch of projective space, the map is simply the

moment curve In geometry, the moment curve is an algebraic curve in ''d''-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form :\left( x, x^2, x^3, \dots, x^d \right). In the Euclidean plane, the moment curve is a parabol ...

\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. A ...

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 (quadric hypersurface in higher dimensions), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...

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 to ...

s vanish identically when using the explicit parameterization above; that is, substitute ''x''³ for ''X'', and so on. More strongly, the homogeneous ideal 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 the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

and

secant line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or recip ...

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_ ...

) 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 ...

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 set of solutions of a system of polynomial equations over the real or complex numbers ...

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 In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...

References

*. {{Algebraic curves navbox Algebraic curves