Can we use rigid motions to show that the corresponding angles are congruent? Then once we’ve allowed those, it’s not too bad to prove that alternate interior angles are congruent when parallel lines are cut by a transversal.īut we wonder whether we have to let corresponding angles in as a postulate. It makes sense to students that the corresponding angles are congruent. We use dynamic geometry software to explore Parallel Lines and Transversals:Īnd then traditionally, we have allowed corresponding angles congruent when parallel lines are cut by a transversal as the postulate in our deductive system. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles. We make sense of Euclid’s 5 th Postulate (wording below from Cut the Knot): My students come to high school geometry having experience with angle measure relationships when parallel lines are cut by a transversal. Theorems include: vertical angles are congruent when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.Īfter proving that vertical angles are congruent, we turned our attention towards angles formed by parallel lines cut by a transversal. Preliminary results on prospectively undersampled data acquired with a 2D golden angle acquisition during free-breathing demonstrate the practical feasibility of the .9. Results show that MR-MOTUS reconstructs in vivo 3D rigid head motion from 474-fold retrospectively downsampled k-space data, and in vivo non-rigid 3D respiratory motion from 63-fold retrospectively undersampled k-space data. A comparison is made with state-of-the-art image registration performed on images reconstructed from the same undersampled data. ![]() The signal model is validated through numerical experiments with a digital 3D phantom and motion-fields are reconstructed from retrospectively undersampled in vivo head and abdomen data using various undersampling strategies. Using an a priori available reference image and the fact that internal body motion exhibits a high level of spatial correlation, we represent the motion-fields in a low-dimensional space and reconstruct them from minimal k-space data that can be acquired very rapidly. MR-MOTUS consists of two main components: (1) a signal model that explicitly relates the k-space signal of a deforming object to non-rigid motion-fields and a reference image, and (2) model-based reconstructions of the non-rigid motion-fields directly from k-space data. In light of this application, we propose MR-MOTUS, a framework to estimate non-rigid 3D motion from minimal k-space data. ![]() ![]() ![]() However, time-resolved estimation of this motion from MRI data still remains a challenge. The combination of an MRI scanner and a linear accelerator enables radiation plan adaptation based on internal organ motion estimated from MRI data. Time-resolved motion estimation from MRI data has received an increasing amount of interest due to the advent of the MR-Linac.
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