基于斜率的圆锥曲线新定义

On a New Definition of Conic Curves by Slopes

为解决圆锥曲线对圆、椭圆、抛物线和双曲线统一的定义问题,可定义圆锥曲线是动点与二定点连线(或其中一连线为折线)斜率之积为定值的轨迹。此法不但较好地解决了圆锥曲线定义的不统一问题,而且数学推导也异常简单,有着明显的优点。此外,还论述了按此定义,用《几何画板》画各种圆锥曲线时,如何有效设置生成点的问题。

For circle,ellipse,hyperbola, and parabola, there is a problem of unified definition. Now a new definition is offered. The two moving lines with slope κ1 and κ2 are formed by one moving point connecting with two standing points. If the product κ of the two slopes is constant,the locus will be conic curve： κ〈 1 for ellipse,and κ= - 1 for circle; κ〉 0,for hyperbola. But for parabola, one of the moving lines should be substituted by broken lines, as is shown on Fig. 3. The derivation the equation, and the form of the equation are very simple. The generating point of drawing the curves by the new definition is disused in detail.