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从二次曲线的一般方程 ax~2+2hxy+by~2+2gx+2fy+c=0 (1)直接确定曲线相对于坐标轴的位置,即不经过坐标变换,直接得到标准坐标系下的标准方程,并直接确定标准坐标系在原坐标系中的位置,当(1)表示中心型曲线时,这个问题已经解决了(例如见[2]第五章§4)。本文讨论(1)为抛物线时位置的直接确定问题。按一般教科书(例如[1])中的记号,基本不变量记为当I_2=0,I_2(?)0时,(1)表示抛物线,我们已经知道可以利用不变量直接写出化简后的方程([1]中称之并且还可以求出对称轴(x~*轴)的方向,但[2]中说
From the general equation of the quadratic curve, ax~2+2hxy+by~2+2gx+2fy+c=0 (1), the position of the curve relative to the coordinate axis is directly determined, that is, the coordinate system is directly obtained without coordinate transformation. The standard equation and the position of the standard coordinate system in the original coordinate system are directly determined. This problem has been solved when (1) represents a center curve (see, for example, [5], Chapter 5, § 4). This article discusses (1) the direct determination of the position of a parabola. According to the symbols in general textbooks (eg [1]), the basic invariants are recorded as I_2=0, I_2(?)0, and (1) represents a parabola. We already know that the invariants can be written directly after the reduction. Equation ([1] calls it and can also find the direction of the axis of symmetry (x~* axis), but [2] says