We introduce two probabilistic algorithms to determine the motion parameters of a planar shape without knowing a priori the point-to-point correspondences. If the target is limited to rigid objects, an Euclidean transformation can be expressed as a linear equation with six parameters, i.e. two translational parameters and four rotational parameters (the axis of rotation and the rotational speed about the axis). These parameters can be determined by applying the randomized Hough transform. One remarkable feature of our algorithms is that the calculations of the translation and rotation parameters are performed by using points randomly selected from two image frames that are acquired at different times. The estimation of rotation parameters is done using one of two approaches, which we call the triangle search and the polygon search algorithms respectively. Both methods focus on the intersection points of a boundary of the 2D shape and the circles whose centers are located at the shape's centroid and whose radii are generated randomly. The triangle search algorithm randomly selects three different intersection points in each image, such that they form congruent triangles, and then estimates the rotation parameter using these two triangles. However, the polygon search algorithm employs all the intersection points in each image, i.e. all the intersection points in the two image frames form two polygons, and then estimates the rotation parameter with aid of the vertices of these two polygons.