Quaternion calculus as a basic tool in computer graphics

Pletinckx, Daniel

doi:10.1007/bf01901476

Cited by 95 publications

(36 citation statements)

References 10 publications

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“…A quaternion can be used to define the rotation from an arbitrary orientation of a 3D object into any other arbitrary orientation [Pletinckx, 1989]. A quaternion defining such a rotation can be decomposed into an axis of rotation and an angle of rotation about that axis.…”

Section: Results Of Experimentsmentioning

confidence: 99%

Rotating virtual objects with real handles

Ware

Rose

1999

ACM Trans. Comput.-Hum. Interact.

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Times for virtual object rotations reported in the literature are of the order of ten seconds or more and this is far longer than it takes to manually orient a "real" object, such as a cup. This is a report of a series of experiments designed to investigate the reasons for this difference and to help design interfaces for object manipulation. The results suggest that two major factors are important. Having the hand physically in the same location as the virtual object being manipulated is one. The other is based on whether the object is being rotated to a new, randomly determined orientation, or is always rotated to the same position. Making the object held in the hand have the same physical shape as the object being visually manipulated was not found to be a significant factor. The results are discussed in the context of interactive virtual environments.

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Section: Results Of Experimentsmentioning

confidence: 99%

Rotating virtual objects with real handles

Ware

Rose

1999

ACM Trans. Comput.-Hum. Interact.

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“…When represented with Euler angles, every orientation is at most 90 degrees away from a singularity. Such a singularity, known as gimbal lock from its physical manifestation in gyroscopes, occurs when two of the three rotation axes coincide and results in the loss of one degree of freedom, i.e., one rotation having no effect [44]. Since gimbal lock is a discontinuity in the Euler angle representation, it might have undesirable side-effects such as ill-conditioning or instabilities in applications involving rotation operations like iterative optimization, filtering, averaging or interpolation.…”

Section: Euler Anglesmentioning

confidence: 99%

“…The reader is referred to [16,29,44,61,64] for more detailed introductions on quaternions and their properties. A quaternion q ∈ H such that q = 1, is called a unit quaternion.…”

Section: Unit Quaternionsmentioning

confidence: 99%

“…Quaternion interpolation is ubiquitous in the fields of computer graphics, robotics and aerospace engineering [8,21,22,44,55]. Generating smooth orientation paths between key orientations is a very challenging task, primarily because we wish to attach linear interfaces onto steering mechanisms which, by definition, manipulate objects (i.e., rotations) that reside in a spherical manifold.…”

Section: Quaternion Interpolationmentioning

confidence: 99%

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Modified Rodrigues Parameters: An Efficient Representation of Orientation in 3D Vision and Graphics

2017

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Modified Rodrigues parameters (MRPs) are triplets in R 3 bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-of-freedom parameterization of orientation through novel solutions to prominent problems in the fields of 3D vision and computer graphics. In our primary contribution, we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Furthermore, we show that updates to unit quaternions from perturbations in parameter space can be computed without explicitly invoking the parameters in the computations. Based on the former, we introduce a novel approach for designing orientation splines by configuring their backprojections in 3D space. Finally, in the general topic of nonlinear optimization for geometric vision, we run performance analyses and provide comparisons on the convergence behavior of MRP parameterizations on the tasks of absolute orientation, exterior orientation and large-scale bundle adjustment of public datasets.

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“…For each light source j within the set, we compute the neighborhood integral − → Qj at each vertex during the preprocessing stage. At run time, we approximate the scattering integral − → Q using quaternion-based vector interpolation [Pletinckx 1989] of the integrals of its four closest − → Qj 's in the set. We compute the dot-product of the interpolated scattering integral − → Q with the real light source direction.…”

Section: Quantized Light Sourcesmentioning

confidence: 99%

Interactive subsurface scattering for translucent meshes

Hao¹,

Baby²,

Varshney³

2003

Proceedings of the 2003 Symposium on Interactive 3D Graphics - SI3D '03

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We propose a simple lighting model to incorporate subsurface scattering effects within the local illumination framework. Subsurface scattering is relatively local due to its exponential falloff and has little effect on the appearance of neighboring objects. These observations have motivated us to approximate the BSSRDF model and to model subsurface scattering effects by using only local illumination. Our model is able to capture the most important features of subsurface scattering: reflection and transmission due to multiple scattering.In our approach we build the neighborhood information as a preprocess and modify the traditional local illumination model into a run-time two-stage process. In the first stage we compute the reflection and transmission of light on the surface. The second stage involves bleeding the scattering effects from a vertex's neighborhood to produce the final result. We then show how to merge the run-time twostage process into a run-time single-stage process using precomputed integral. The complexity of our run-time algorithm is O(N ), where N is the number of vertices. Using this approach, we achieve interactive frame rates with about one to two orders of magnitude speedup compared with the state-of-the-art methods.

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Quaternion calculus as a basic tool in computer graphics

Cited by 95 publications

References 10 publications

Rotating virtual objects with real handles

Rotating virtual objects with real handles

Modified Rodrigues Parameters: An Efficient Representation of Orientation in 3D Vision and Graphics

Interactive subsurface scattering for translucent meshes

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