定義
雙曲拋物面又稱馬鞍面,它在笛卡兒坐標系中的方程為:
![雙曲拋物面](/img/0/188/wZwpmLyIzN1YTM4IzM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyMzLxczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
其中x、y、z是平面直角坐標系三個坐標軸方向上的變數,a、b是常數。
幾何表示
如果把雙曲拋物面
![雙曲拋物面](/img/f/1ba/wZwpmL0EDM2IjN4YTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL2EzLzczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
順著+ z的方向鏇轉π/4的角度,則方程為:
![雙曲拋物面](/img/4/602/wZwpmL0EDMxUTO2gTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4UzLyIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![雙曲拋物面](/img/d/3c2/wZwpmL0MTNykTM3MTMzEzM1UTM1QDN5MjM5ADMwAjMwUzLzEzLwIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
如果 ,則簡化為:.
![雙曲拋物面](/img/d/cd6/wZwpmL3IDNxgTM2YzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL2czL3UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![雙曲拋物面](/img/e/193/wZwpmL3UTN0MjM1kTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL0UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
最後,設 ,我們可以看到雙曲拋物面
![雙曲拋物面](/img/b/78b/wZwpmL2EjN2gTN0kTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL5UzLxQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
與以下的曲面是全等的:
![雙曲拋物面](/img/2/f7b/wZwpmLwIDN4EDN4ADO2UzM1UTM1QDN5MjM5ADMwAjMwUzLwgzLxEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
因此它可以視為乘法表的幾何表示。
雙曲拋物面圖像
![圖1.雙曲拋物面](/img/4/006/wZwpmL0MTM4gzN1kTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL5kzLxIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![雙曲拋物面](/img/1/a31/wZwpmLwEjM0ATN3UTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzL3QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![雙曲拋物面](/img/d/74c/wZwpmL3QjNyQzM3QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![雙曲拋物面](/img/d/74c/wZwpmL3QjNyQzM3QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![雙曲拋物面](/img/0/0db/wZwpmL4MDN1MDOzMzM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzMzL3UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![雙曲拋物面](/img/7/462/wZwpmLxQTO0czMxUjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
用平面 截此曲面, 所得截痕l為平面 上的拋物線 ,此拋物線開口向下,其頂點坐標為 。當t變化時,l的形狀不變,位置只作平移,而l的頂點的軌跡L為平面y=0上的拋物線。因此,以l為母線,L為準線,母線l的頂點在準線L上滑動,且母線作平行移動,這樣得到的曲面便是雙曲拋物面。