Editing Quadric (section)

==Rational parametrization==
Given a non-singular point {{mvar|A}} of a quadric, a line passing through {{mvar|A}} is either tangent to the quadric, or intersects the quadric in exactly one other point (as usual, a line contained in the quadric is considered as a tangent, since it is contained in the [[tangent space|tangent hyperplane]]). This means that the lines passing through {{mvar|A}} and not tangent to the quadric are in [[one to one correspondence]] with the points of the quadric that do not belong to the tangent hyperplane at {{mvar|A}}. Expressing the points of the quadric in terms of the direction of the corresponding line provides [[parametric equation]]s of the following forms.

In the case of conic sections (quadric curves), this parametrization establishes a [[bijection]] between a projective conic section and a [[projective line]]; this bijection is an [[isomorphism]] of [[algebraic curve]]s. In higher dimensions, the parametrization defines a [[birational map]], which is a bijection between [[dense set|dense]] [[open set|open]] subsets of the quadric and a projective space of the same dimension (the topology that is considered is the usual one in the case of a real or complex quadric, or the [[Zariski topology]] in all cases). The points of the quadric that are not in the image of this bijection are the points of intersection of the quadric and its tangent hyperplane at {{mvar|A}}.

In the affine case, the parametrization is a [[rational parametrization]] of the form
:<math>x_i=\frac{f_i(t_1,\ldots, t_{n-1})}{f_0(t_1,\ldots, t_{n-1})}\quad\text{for }i=1, \ldots, n,</math>
where <math>x_1, \ldots, x_n</math> are the coordinates of a point of the quadric, <math>t_1,\ldots,t_{n-1}</math> are parameters, and <math>f_0, f_1, \ldots, f_n</math> are polynomials of degree at most two.

In the projective case, the parametrization has the form 
:<math>X_i=F_i(T_1,\ldots, T_n)\quad\text{for }i=0, \ldots, n,</math>
where <math>X_0, \ldots, X_n</math> are the projective coordinates of a point of the quadric, <math>T_1,\ldots,T_n</math> are parameters, and <math>F_0, \ldots, F_n</math> are homogeneous polynomials of degree two.

One passes from one parametrization to the other by putting <math>x_i=X_i/X_0,</math> and <math>t_i=T_i/T_n\,:</math> 
:<math>F_i(T_1,\ldots, T_n)=T_n^2 \,f_i\!{\left(\frac{T_1}{T_n},\ldots,\frac{T_{n-1}}{T_n}\right)}.</math>

For computing the parametrization and proving that the degrees are as asserted, one may proceed as follows in the affine case. One can proceed similarly in the projective case.

Let {{mvar|q}} be the quadratic polynomial that defines the quadric, and <math>\mathbf a=(a_1,\ldots a_n)</math> be the [[coordinate vector]] of the given point of the quadric (so, <math>q(\mathbf a)=0).</math> Let <math>\mathbf x=(x_1,\ldots x_n)</math> be the coordinate vector of the point of the quadric to be parametrized, and <math>\mathbf t=(t_1,\ldots, t_{n-1},1)</math> be a vector defining the direction used for the parametrization (directions whose last coordinate is zero are not taken into account here; this means that some points of the affine quadric are not parametrized; one says often that they are parametrized by [[points at infinity]] in the space of parameters) . The points of the intersection of the quadric and the line of direction <math>\mathbf t</math> passing through <math>\mathbf a</math> are the points <math>\mathbf x=\mathbf a +\lambda \mathbf t</math> such that
:<math>q(\mathbf a +\lambda \mathbf t)=0</math>
for some value of the scalar <math>\lambda.</math> This is an equation of degree two in <math>\lambda,</math> except for the values of <math>\mathbf t</math> such that the line is tangent to the quadric (in this case, the degree is one if the line is not included in the quadric, or the equation becomes <math>0=0</math> otherwise). The coefficients of <math>\lambda</math> and <math>\lambda^2</math> are respectively of degree at most one and two in <math>\mathbf t.</math> As the constant coefficient is <math>q(\mathbf a)=0,</math> the equation becomes linear by dividing by <math>\lambda,</math> and its unique solution is the quotient of a polynomial of degree at most one by a polynomial of degree at most two. Substituting this solution into the expression of <math>\mathbf x,</math> one obtains the desired parametrization as fractions of polynomials of degree at most two.

===Example: circle and spheres===

Let consider the quadric of equation
:<math>x_1^2+ x_2^2+\cdots x_n^2 -1=0.</math>
For <math>n=2,</math> this is the [[unit circle]]; for <math>n=3</math> this is the [[unit sphere]]; in higher dimensions, this is the [[unit hypersphere]].

The point <math>\mathbf a=(0, \ldots, 0, -1)</math> belongs to the quadric (the choice of this point among other similar points is only a question of convenience). So, the equation <math>q(\mathbf a +\lambda \mathbf t)=0</math> of the preceding section becomes
:<math>(\lambda t_1^2)+\cdots +(\lambda t_{n-1})^2+ (1-\lambda)^2-1=0.</math>
By expanding the squares, simplifying the constant terms, dividing by <math>\lambda,</math> and solving in <math>\lambda,</math> one obtains
:<math>\lambda = \frac{2}{1+ t_1^2+ \cdots +t_{n-1}^2}.</math>
Substituting this into <math>\mathbf x=\mathbf a +\lambda \mathbf t</math> and simplifying the expression of the last coordinate, one obtains the parametric equation
:<math>\begin{cases}
x_1=\frac{2t_1}{1+ t_1^2+ \cdots +t_{n-1}^2}\\
\vdots\\
x_{n-1}=\frac{2 t_{n-1}}{1+ t_1^2+ \cdots +t_{n-1}^2}\\
x_n =\frac{1- t_1^2- \cdots -t_{n-1}^2}{1+ t_1^2+ \cdots +t_{n-1}^2}.
\end{cases}</math>

By homogenizing, one obtains the projective parametrization
:<math>\begin{cases}
X_0=T_1^2+ \cdots +T_n^2\\
X_1=2T_1 T_n\\
\vdots\\
X_{n-1}=2T_{n-1}T_n\\
X_n =T_n^2- T_1^2- \cdots -T_{n-1}^2.
\end{cases}</math>

A straightforward verification shows that this induces a bijection between the points of the quadric such that <math>X_n\neq -X_0</math> and the points such that <math>T_n\neq 0</math> in the projective space of the parameters. On the other hand, all values of <math>(T_1,\ldots, T_n)</math> such that <math>T_n=0</math> and <math>T_1^2+ \cdots +T_{n-1}^2\neq 0</math> give the point <math>A.</math>

In the case of conic sections (<math>n=2</math>), there is exactly one point with <math>T_n=0.</math> and one has a bijection between the circle and the projective line.

For <math>n>2,</math> there are many points with <math>T_n=0,</math> and thus many parameter values for the point <math>A.</math> On the other hand, the other points of the quadric for which <math>X_n=-X_0</math> (and thus <math>x_n=-1</math>) cannot be obtained for any value of the parameters. These points are the points of the intersection of the quadric and its tangent plane at <math>A.</math> In this specific case, these points have nonreal complex coordinates, but it suffices to change one sign in the equation of the quadric for producing real points that are not obtained with the resulting parametrization.