基本介紹
- 中文名:共焦二次曲面族
- 外文名:familyofconfocalquadricsurface
- 屬性:有重要意義的二次曲面族
- 分類:橢球面、單葉雙曲面等
- 別名:共焦族
基礎知識,橢球,單葉雙曲面族,雙葉雙曲面族,
基礎知識
設
是空間
的點的直角坐標,由方程
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![](/img/6/798/wZ2NnL3IDMlRTZ1Y2YzMmZlNGMkJmY5MDM2QGO4ADO0QGMllzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/2/702/wZ2NnLjVWO1YGNlNzYkVjMkBTYxQTY5MDNzIWZxAjZhVmZlBzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/f/b43/wZ2NnLjVWYiljYlZjM1MGN2Y2N1YjMxczYjJ2NiFjYyQWNjV2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
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4.當
介於
與
之間時,上列方程表示虛曲面。
![](/img/f/b43/wZ2NnLjVWYiljYlZjM1MGN2Y2N1YjMxczYjJ2NiFjYyQWNjV2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/f/541/wZ2NnL4EGOwIWYyYDO3kzM4UWNkZjYwM2YwETMkdzYhJzM5czLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/d/a9d/wZ2NnLjRWZmNmYhZTM1cDN5QzYjJ2NxMzM5UjNkVTN5EWY4gzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
橢球
現在我們將闡明共焦曲面族的一個明晰的表示,而且首先從橢球著手,它的三個半軸之長顯然是
![](/img/1/9e3/wZ2NnLmVWZhJmYjZTOxQGMxgTZ1IDOwIjZzUWY0YDNzEGOmJ2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/3/689/wZ2NnLmBDN2QmY1E2NhBzYiNGOzMmZ3IWMxUjM0gTOjBTY4MzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/b/4dc/wZ2NnLyMWN3ETO1kDN2kDZ0ITN1IzN2QWY1UTZ5EjZ0ETMhVzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/4/969/wZ2NnL0MmZhZGOmZmM3E2M0AjM4cjYhZjZhRGM4YDOlhDMkF2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/2/90f/wZ2NnLiRmYxYWMwQ2YiZGZ4gjZ0gzM1EmY1EjMhFWO3UDZhRzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/e/dc4/wZ2NnLwQWY1IGZkNTYzgjYmZjZkJDOjlzNyQWNyIjMjZ2YmJzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/3/9c0/wZ2NnLidDN1EGMyYzN2cDM2ATZ0QTNwkzYkV2M5IGN1UWYxE2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![共焦二次曲面族 共焦二次曲面族](/img/3/061/AOxUTYhJDNkRmM1ITO1ImNhhDMyIzYykjZwETNjRTYxgjZiVGN0YDOvMWaw9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
圖1
單葉雙曲面族
![共焦二次曲面族 共焦二次曲面族](/img/a/5f5/wNzUWO5Q2Y3M2YzEGZkVjMmBDN1QjNhNTOwYzNmVmYjRWYxMDNmRGMvMWaw9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
圖2
這個雙曲面族一方是由一個作為
時的極限的雙曲面塊所圍成,後者並充滿著焦橢圓的外部;另一方是由一條作為
時的極限的焦雙曲線外部所構成,就是圖2中附上陰影線部分,這條雙曲線落在
平面上而且具有下列方程:
![](/img/b/4dc/wZ2NnLyMWN3ETO1kDN2kDZ0ITN1IzN2QWY1UTZ5EjZ0ETMhVzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/e/298/wZ2NnL0cDN0Y2N4I2MlZGNjZ2YiFWZkZWZyQjYmZjZmBDN3kzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/f/a6e/wZ2NnL2YDOiVWY3ATMyQDOiV2YwgjYxMTM1cTNkRTZlFzMhZ2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/c/067/wZ2NnLjBzNxcTZ5ITOxkjYmJGO0UGZmJDOyUTY0EjZxcDMwY2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
在上列兩極限位置之間充滿著其餘的單葉雙曲面,使其全體圍繞著焦雙曲線而被焦橢圓所局限著。
雙葉雙曲面族
最後,我們還要觀察雙葉雙曲面族,各曲面的實軸重合於
軸,這族圍成
平面中,焦雙曲線內部的陰影部分,作為在
時的極限情況,但是當
時的第二極限情況是作為雙重計算的
平面,在其上已不存在焦曲線,因為方程
![](/img/5/602/wZ2NnLmFmZ5QDN5UmY3cTMmBzYyMWYwgDZ0EjYhRDOhZTO0Q2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/e/dc4/wZ2NnLwQWY1IGZkNTYzgjYmZjZkJDOjlzNyQWNyIjMjZ2YmJzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/d/07c/wZ2NnLmJWN3UGNyQzMkVmYiJGO1ImYxQDOlJGNyYWNiZWMzQzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/9/47d/wZ2NnL0AzYxgjZ4cTZkJjMwkTM1cjNmVDZkFGZ1YTNmNDN0Q2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/d/282/wZ2NnLmVWM0cjN4IGO5YzMiJWYyQTMhZTO3kTYmJjZzIzMiJzLhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
![](/img/0/f6a/wZ2NnL2YWZ2UGOmFTO5UmN5QWMiRTY3UTZxMmZzU2NkdzY3Y2LhxWdtJ3bm9SbvNmLz9mYlNmYu4GZj5yZtl2ai9yL6MHc0RHa.jpg)
此外,雙葉雙曲面族也是恰恰一度地充滿著整個空間,而實際上是這樣:一個外殼總是圍繞著焦雙曲線的右支,而且另一個外殼則圍繞著左支。