It is relatively easy to solve the composition of two perpendicular harmonic oscillations with the same frequency when the phase difference are some specific values, so are the curve shape and its variation. As for the general value of phase difference, coordinate transformation is usually required to give the curve shape and the variation on phase difference. The motion curve of the oscillating mass point is derived, then the motion at equal-amplitude and unequal-amplitude conditions is discussed. With equal amplitudes and general phase difference, the mass point moves on an ellipse whose major axis is at an angle of π/4 or 3π/4 to the x-axis. The closer the phase difference is to 0 or π, the larger is the eccentricity of ellipse. If the amplitudes are not equal, the major axis of ellipse rotates when the phase difference varies, and the eccentricity becomes larger as the phase difference is close to 0 or π. This paper can enhance the understanding on the combination of two perpendicular harmonic oscillations of the same frequency.