The standard plane conics

•             Parabola:     y = x²
   
•              Ellipse:     x² + y² = 1
   
•             Hyperbola:     xy = 4
   
   

• The projective closures have following equations:
   
       (Here, z = 0 is the equation of the line at infinity.)
   
  •             Parabola:     yz = x²      (One intersection with the line at infinity.)
     
  •             Ellipse:     x² + y² = z²      (No intersections with the line at infinity.)
     
  •             Hyperbola:     xy = 4z²      (Two intersections with the line at infinity.)
     
     

Click here to see how the projective closures intersect the line at infinity.