Newton's method for the matrix square root
Mathematics of Computation
Smooth interpolation of orientations with angular velocity constraints using quaternions
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
Matrix animation and polar decomposition
Proceedings of the conference on Graphics interface '92
A general construction scheme for unit quaternion curves with simple high order derivatives
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Matrix computations (3rd ed.)
Interactive arrangement of botanical L-system models
I3D '99 Proceedings of the 1999 symposium on Interactive 3D graphics
Animating rotation with quaternion curves
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
As-rigid-as-possible shape interpolation
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Spherical averages and applications to spherical splines and interpolation
ACM Transactions on Graphics (TOG)
Linear combination of transformations
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Turning to the masters: motion capturing cartoons
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Approximating the Logarithm of a Matrix to Specified Accuracy
SIAM Journal on Matrix Analysis and Applications
General Construction of Time-Domain Filters for Orientation Data
IEEE Transactions on Visualization and Computer Graphics
Minimizing the distortion of affine spline motions
Graphical Models - Pacific graphics 2001
Practical parameterization of rotations using the exponential map
Journal of Graphics Tools
Interpolating and Approximating Moving Frames Using B-splines
PG '00 Proceedings of the 8th Pacific Conference on Computer Graphics and Applications
Twister: a space-warp operator for the two-handed editing of 3D shapes
ACM SIGGRAPH 2003 Papers
ACM SIGGRAPH 2003 Papers
A Schur-Parlett Algorithm for Computing Matrix Functions
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Computer Graphics and Geometric Modelling: Mathematics
Computer Graphics and Geometric Modelling: Mathematics
Spherical blend skinning: a real-time deformation of articulated models
Proceedings of the 2005 symposium on Interactive 3D graphics and games
Matlab Guide
Wall grammar for building generation
Proceedings of the 4th international conference on Computer graphics and interactive techniques in Australasia and Southeast Asia
Boundary of the volume swept by a free-form solid in screw motion
Computer-Aided Design
ScrewBender: Smoothing Piecewise Helical Motions
IEEE Computer Graphics and Applications
Discovering structural regularity in 3D geometry
ACM SIGGRAPH 2008 papers
Geometric skinning with approximate dual quaternion blending
ACM Transactions on Graphics (TOG)
Computer-Aided Design
OCTOR: Subset selection in recursive pattern hierarchies
Graphical Models
Geometric Transformations fo 3D Modeling
Geometric Transformations fo 3D Modeling
Temporal noise control for sketchy animation
Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Non-Photorealistic Animation and Rendering
SAMBA: steadied choreographies
CAe '12 Proceedings of the Eighth Annual Symposium on Computational Aesthetics in Graphics, Visualization, and Imaging
Technical Section: Planar shape interpolation using relative velocity fields
Computers and Graphics
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We propose to measure the quality of an affine motion by its steadiness, which we formulate as the inverse of its Average Relative Acceleration (ARA). Steady affine motions, for which ARA&equal;0, include translations, rotations, screws, and the golden spiral. To facilitate the design of pleasing in-betweening motions that interpolate between an initial and a final pose (affine transformation), B and C, we propose the Steady Affine Morph (SAM), defined as At∘ B with A &equal; C ∘ B−1. A SAM is affine-invariant and reversible. It preserves isometries (i.e., rigidity), similarities, and volume. Its velocity field is stationary both in the global and the local (moving) frames. Given a copy count, n, the series of uniformly sampled poses, A&fracin;∘ B, of a SAM form a regular pattern which may be easily controlled by changing B, C, or n, and where consecutive poses are related by the same affinity A&frac1n. Although a real matrix At does not always exist, we show that it does for a convex and large subset of orientation-preserving affinities A. Our fast and accurate Extraction of Affinity Roots (EAR) algorithm computes At, when it exists, using closed-form expressions in two or in three dimensions. We discuss SAM applications to pattern design and animation and to key-frame interpolation.