Domain

The points

Types of points

Figures

$D_1:f_1<0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Four-hyperbolic points

Figure 2

$D_2:f_1>0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Three-hyperbolic points and one elliptic point

Figure 3

$D_3:f_1>0,f_2<0,f_3<0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)$

Two-hyperbolic points

Figure 4

$D_4:f_1>0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Three-hyperbolic points and one elliptic point

Figure 5

$D_5:f_1<0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Two-hyperbolic points and Two-elliptic points

Figure 6

$D_6:f_1>0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Three-hyperbolic points and one elliptic point

Figure 7

$D_7:f_1>0,f_2>0,f_3<0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Two-hyperbolic points and two-elliptic points

Figure 8

$D_8:f_1>0,f_2>0,f_3<0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Two-hyperbolic points and two-elliptic points

Figure 9, Figure 10

$D_9:f_1<0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Two-hyperbolic points and two-elliptic points

Figure 11

$D_{10}:f_1<0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)$

One-hyperbolic point and one-elliptic point

Figure 12

$D_{11}:f_1>0,f_2>0,f_3<0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Two-hyperbolic points and two-elliptic points

Figure 13

$D_{12}:f_1>0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)\mathrm{,}\left(-k\mathrm{,}\pm \sqrt{\frac{2k^2-2hk+1}{2k}}\right)$

Two-hyperbolic points and two-elliptic points

Figure 14, Figure 15

$D_{13}:f_1<0,f_2<0,f_3>0$

$\left(k\mathrm{,}\pm \sqrt{\frac{-2k^2-2hk-1}{2k}}\right)$

Two-hyperbolic points

Figure 16

$D_{14}:f_1>0,f_2<0,f_3>0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)$

One-hyperbolic point and one elliptic point

Figure 17

$D_{15}:f_1>0,f_2<0,f_3<0$

$\left(\frac{-h\pm \sqrt{h^2+3\left(k^2-1\right)}}{3}\mathrm{,0}\right)$

Two-hyperbolic points

Figure 18

$D_{16}:f_1>0,f_2<0,f_3>0$

Two-hyperbolic points

Figure 19, Figure 20