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. Theorem 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 Theorem 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.