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Using numerical calculations of equi-energy contour plot of the Fouriertransform of the spin Hamiltonian, we study the magnetic phase diagram ($J_{2}$and $J_{3}$) of the rutile type $\beta$-MnO$_{2}$, where $J_{1}$ ($<0$) isfixed and is the antiferromagnetic interaction along the diagonal direction,$J_{2}$ is the interaction along the c axis, and $J_{3}$ is the interactionalong the a axis. The magnetic phase diagram consists of the multricriticalpoint (the intersection $J_{2}J_{3}=J_{1}^{2}$ and $J_{2}+J_{3}=2J_{1}$), thehelical order along the c axis, the $(h=1/2,k=0,l=1/2)$ phase, the helicalorder along the a axis, and the phase $(h=0,k=0,l=1)$. The shift of thelocation of the magnetic Bragg peak in the $(h,0,l)$ reciprocal lattice planeis examined with the change of $J_{2}$ and $J_{3}$ in the phase diagram. Theshift is discontinuous on the first-order phase transition, and is continuouson the second-order phase transition. The detail of our magnetic phase diagramis rather different from that reported by Yoshimori.

Ordered spin structures of $\beta$-MnO$_{2}$ (rutile-type) systems with  competing exchange interactions: numerical approach using equi-energy contour  plot