student at the Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 1914, 3 Jean Gallier is Faculty at the Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 1914, The problem of motion interpolation has been studied extensively both in the robotics and computer graphics communities. student at the Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 1914, 2 Christine Allen Blanchette is a Ph.D. As an application, we consider the problem of interpolation between multiple similarity transformations in R 3. To the best of our knowledge, we are the first to prove the surjectivity of the exponential map for the group of similarity transformations. every similarity transformation is the exponential of some matrix in the corresponding Lie algebra. We prove the surjectivity of the exponential map, i.e. In this paper, we investigate the exponential map and its (mutivalued inverse, the logarithm map, for the group of similarity transformations. For a more rigorous treatment of the above notions we refer the reader to the standard references of robotics and computer vision. Similarly, for any rigid body (Euclidean transformation there is a matrix consisting of a skew-symmetric part and a real vector part, the so called twist, whose exponential is the Euclidean transformation. The vector space of skewsymmetric matrices provide a linearization of the group of rotations around the identity and since the group of rotations is a Lie group, the set of skew-symmetric matrices is the Lie algebra of this Lie group. It is a well established result, that any rotation matrix is the matrix exponential of a skew-symmetric matrix, called the logarithm of the rotation. The exponential map is of particular importance in the fields of robotics and vision in analyzing rotational and rigid body motions. INTRODUCTION The exponential map is ubiquitous in engineering applications ranging from dynamical systems and robotics to computer vision and computer graphics. Given a sequence of similarity transformations, we compute a sequence of logarithms, then fit a cubic spline that interpolates the logarithms and finally, we compute the interpolating curve in SIM(3. As an application, we use these algorithms to perform motion interpolation. #WHICH COMPOSITION OF SIMILARITY TRANSFORMATIONS MAPS HOW TO#We give an explicit formula for the case of n 3 and show how to efficiently compute the logarithmic map. We give a formula for the exponential map and we prove that it is surjective. The steps above give a similarity transformation that maps ABCD to PQRS, so these two quadrilaterals are similar.1 The exponential map for the group of similarity transformations and applications to motion interpolation Spyridon Leonardos 1, Christine Allen Blanchette 2 and Jean Gallier 3 Abstract In this paper, we explore the exponential map and its inverse, the logarithm map, for the group SIM(n of similarity transformations in R n which are the composition of a rotation, a translation and a uniform scaling. īoth R and C ′ ′ ′ is the intersection of Q R and S R. Since rigid motions and dilations preserve angles, this means that ∠ D ′ ′ ′ ≅ ∠ S. Since rigid motions and dilations preserve angles, this means that ∠ B ′ ′ ′ ≅ ∠ Q. It is assumed that P Q / A B = P S / A D, so the dilation that moves B ′ ′ to Q, also moves D ′ ′ to S. Since translations and rotations are rigid motions, A B = A ′ ′ B ′ ′ and A D = A ′ ′ D ′ ′. This is how the scale factor of the dilation was chosen. The translation moves A to P and since this is the center of rotation and also the dilation, it stays there. The following table contains some observations about the position of points A ′ ′ ′, B ′ ′ ′, C ′ ′ ′, and D ′ ′ ′ relative to the image quadrilateral.
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