This intermediate-level tutorial provides a comprehensive approach to the visualization of quaternions and their relationships to computer graphics and scientific visualization. The introduction focuses on a selection of everyday phenomena involving rotating objects whose explanation for an audience that is technically trained but not pure mathematicians is essentially impossible without a quaternion visualization. The course will then pursue selected examples of quaternion-based visualization methods to help explain the behavior of quaternion manifolds: quaternion representations of orientation frames attached to curves, surfaces, and volumes.

Visualizing Quaternions

Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method, uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error—the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF.

/pdf/computation-of-rotation-minimizing-frames-2ig7mkgj0j.pdf

Computation of rotation minimizing frames

The paper is part of a research effort that focuses on the provision of more efficient and effective design methods for broadcast modelling systems. It presents an interactive deformation technique called AxDf (Axial Deformations). Based on the paradigm of the modelling tool, the axial-deformations technique allows deformations, such as bending, scaling, twisting and stretching, that can be controlled with a 3D axis to be easily specified. Moreover, AxDf can easily be combined with other existing deformation techniques.

Axial deformations: an intuitive deformation technique

The spatial deformation methods are a family of modeling and animation techniques for indirectly reshaping an object by warping the surrounding space, with results that are similar to molding a highly malleable substance. They have the virtue of being computationally efficient (and hence interactive) and applicable to a variety of object representations.In this article we survey the state of the art in spatial deformation. Since manipulating ambient space directly is infeasible, deformations are controlled by tools of varying dimension—points, curves, surfaces and volumes—and it is on this basis that we classify them. Unlike previous surveys that concentrate on providing a single underlying mathematical formalism, we use the user-centered criteria of versatility, ease of use, efficiency and correctness to compare techniques.

A survey of spatial deformation from a user-centered perspective

A broadly applicable formulation for representing the boundary of swept geometric entities is presented. Geometric entities of multiple parameters are considered. A constraint function is defined as one entity is swept along another. Boundaries in terms of inequality constraints imposed on each entity are considered which gives rise to an ability of modeling complex solids. A rank-deficiency condition is imposed on the constraint Jacobian of the sweep to determine singular sets. Because of the generality of the rank-deficiency condition, the formulation is applicable to entities of any dimension. The boundary to the resulting swept volume, in terms of enveloping surfaces, is generated by substituting the resulting singularities into the constraint equation. Singular entities (hyperentities) are then intersected to determine sub-entities that may exist on the boundary of the generated swept volume. Physical behavior of singular entities is discussed. A perturbation method is used to identify the boundary envelope. Numerical examples illustrating this formulation are presented. Applications to NC part geometry verification, robotic manipulators, and computer modeling are discussed.

Geometric representation of the swept volume using Jacobian rank-deficiency conditions

Two moving coordinate frames for sweeping along a 3D trajectory

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Two moving coordinate frames for sweeping along a 3D trajectory