conics in euclidean and similarity geometries

We obtained standard forms of the equations for plane conics by choosing
particular lines as the x- and y-axes. Observe that changing the axes can
be viewed as applying a euclidean transformation t to the axes, but leaving
the conic fixed, OR as applying t-1 to the conic, but leaving the axes fixed.

Thus, our results on standard forms can be interpreted as a result about the
congruence classes of conics in euclidean geometry:

conics in euclidean geometry
Every plane conic is congruent in euclidean geometry to one of the following

  • the parabola y2 = 4ax, with a > 0,
  • the ellipse x2/a2 + y2/b2 = 1, with a > b > 0,
  • the hyperbola x2/a2 - y2/b2 = 1, with a, b > 0,
Note. The list does not include the circle.

Examination on the results on standard forms shows that the values of the
eccentricity e and the focus-directrix distance f can be recovered from those
of a and, when relevant, b. Any euclidean transformation preserves lengths,
so euclidean congruent conics have the same values of e and f. On the other
hand, since e and f determine a and b, two plane conics with the same e and f
with be congruent to the same standard conic, and hence to one another.

Two plane conics are congruent in euclidean geometry if and only if they
have the same eccentricity and focus-directrix distance.

In similarity geometry, the eccentricity is still invariant, since it is defined as
a ratio of lengths. Thus conics congruent in similarity geometry will have the
same value of e, but we may scale the figure to make f equal to any value
we wish. Thus, we have the

Two plane conics are congruent in similarity geometry if and only if they
have the same eccentricity.

Note that the classes in similarity geoemtry are fewer in number, but that
each class is larger.

main conics page