Abstract: Conformal deformations are infinitesimal scale-rotations, which can be parameterized by quaternions. The condition that such a quaternion field gives rise to a conformal deformation is nonlinear and in any case only admits Mobius transformations as solutions. We propose a particular decoupling of scaling and rotation which allows us to find near to conformal deformations as minimizers of a quadratic, convex Dirichlet energy. Applied to tetrahedral meshes we find deformations with low quasiconformal distortion as the principal eigenvector of a (quaternionic) Laplace matrix. The resulting algorithms can be implemented with highly optimized standard linear algebra libraries and yield deformations comparable in quality to far more expensive approaches.