In MRI, $s(t)=\iint m(x,y)e^{-j2\pi[K_x(t)x+K_y(t)y]}dx\,dy$, where Kₓ and Kᵧ are proportional to time integrals of gradients Gₓ and Gᵧ. The gradients are in phase and have identical pulse periods, while their amplitudes vary between acquisitions. What k-space trajectories result?
For each acquisition the two in-phase gradients have proportional waveforms, so their integrals obey Kᵧ=aKₓ for a constant amplitude ratio a. Thus the trajectory is a straight line through the origin. Changing the amplitude ratio changes its angle, giving radial lines, A, rather than a parabola or only one coordinate axis.
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