旋转变换中二次曲线系数的几何解释

Algebraiec Interpretation of Coeffients of Quadratics Undergoing Angular Transformation

二次曲线的二次项系数反映了该曲线的主要形状特征.其反映了曲线的类型,并由其决定如何旋转到标准位置.一般情况下并看不出为达到标准位置所需旋转的角度,必须先代入旋转公式将旋转角求出来.通过二次项系数构造一个圆,在该圆中旋转角度和二次项系数的大小关系一目了然.利用该圆不止可以直观地作出旋转角而且还可以看出当旋转角逐渐变化时二次项系数相应的变化情况.

The shape of a quadratic is mainly characterised by the coefficients of the second order terms. These coeffiencts tells one the type of the quadratic and also how to undergo transformation in order to be in the standard position. Under normal circumstances, one can not ascertain straight away the angle required to transform the curve into the standard position. The angle must be first calculated by substituting the transformation equations. Using the coefficients of the second order terms to form a circle. From this, the relationship between the transformation angle and the second order terms＇coeftiencts will become apparent. Furthermore, the effects of changes in the transformation angle on the second order terms＇ coeffiencts can also be seen.