Editing Quadric (section)

===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.