The infinity as a tool (El infinito como herramienta)

One of my university professors said that projective geometry is the most democratic, because all the points, the infinite ones and the ordinary ones are the same. We take advantage of this to consider some line in a problem as line at infinity and, in this way the problem becomes very easy to solve.

Uno de mis profesores en la universidad decía que la geometría proyectiva es la más democrática, ya que todos los puntos, los infinitos y ordinarios eran los mismos. Hacemos uso de esto para considerar una determinada línea en un problema como recta del infinito, y entonces el problema resulta muy fácil de resolver.

A problem by Van Khea

Area of a conic section

This problem was proposed by Shafiqur Rahman on Facebook: Find the area and length of the semi axes of the section of the paraboloid $2x^2+y^2=z$ by the plane $x + 2 y + z = 4$.

We find a rigid motion that maps the given plane to the plane $z = 0$. We apply the same transformation to the given paraboloid and make $z=0$ then we get a equation of the conic on the $z=0$ plane. Next we get the reduced equation of the conic and its semiaxes.

Read the details here: Area of a conic section