In this research the direct singular positions of the parallel manipulator Tricept are determined. An alternative 3×3 Jacobian matrix, simpler than the existing one, is obtained in this study. For a given moving platform’s orientation, the determinant of the Jacobian matrix may be expressed as a cubic polynomial in moving platform’s extension. Direct singular positions may thus be obtained by solving cubic polynomial equations. For an arbitrarily chosen moving platform’s orientation, there exists at least one moving platform’s extension that causes direct kinematic singularity. It is found that in two regions within the moving platform’s workspace direct kinematic singularities can only occur at positions impossible to reach.