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Showing papers on "Ray tracing (graphics) published in 1998"


01 Jan 1998
TL;DR: A parallel rendering library that is being developed on the Intel iPSC/860, Paragon, and on other machines supported by the Message Passing Interface (MPI) that can be used as a rendering subprocess within a larger parallel computation.
Abstract: Many simulations that are run on parallel computers can beneet from the use of parallel graphics libraries that can do in-place rendering. This paper describes a parallel rendering library that is being developed on the Intel iPSC/860, Paragon, and on other machines supported by the Message Passing Interface (MPI). The rendering library can be used stan-dalone with a simple scene parser that is supplied , or as a rendering subprocess within a larger parallel computation. At the present time, the library supports a variety of mod-eling primitives including cylinders, patches, polygons, quadrics, spheres, and volumetric data sets. Features that set the ray tracing library apart from polygon based renderers are mirror reeection, refraction, shadows, texture mapping, and volume rendering. Current efforts are focused on improving volume rendering , overall performance, and ease-of-use.

339 citations


Proceedings ArticleDOI
18 Oct 1998
TL;DR: It is shown that it is feasible to perform interactive isosurfacing of very large rectilinear datasets with brute-force ray tracing on a conventional (distributed) shared-memory multiprocessor machine.
Abstract: We show that it is feasible to perform interactive isosurfacing of very large rectilinear datasets with brute-force ray tracing on a conventional (distributed) shared-memory multiprocessor machine. Rather than generate geometry representing the isosurface and render with a z-buffer, for each pixel we trace a ray through a volume and do an analytic isosurface intersection computation. Although this method has a high intrinsic computational cost, its simplicity and scalability make it ideal for large datasets on current high-end systems. Incorporating simple optimizations, such as volume bricking and a shallow hierarchy, enables interactive rendering (i.e. 10 frames per second) of the 1 GByte full resolution Visible Woman dataset on an SGI Reality Monster. The graphics capabilities of the Reality Monster are used only for display of the final color image.

330 citations


Journal ArticleDOI
TL;DR: In this paper, a vertical plane launch (VPL) technique was proposed to estimate a full 3D site-specific ray trace to predict propagation effects in cities for frequencies in the 300 MHz-3 GHz band.
Abstract: A vertical-plane-launch (VPL) technique for approximating a full three dimensional (3-D) site-specific ray trace to predict propagation effects in cities for frequencies in the 300 MHz-3-GHz band is described and its predictions are compared with measurements for Rosslyn, VA. The VPL technique employs the standard shoot and bounce method in the horizontal plane while using a deterministic approach to find the vertical displacement of the unfolded ray paths. This approximation is valid since building walls are almost always vertical. The VPL method shows significant improvement compared with the slant-plane/vertical-plane (SP/VP) method for rooftop antennas. For a base station located at street level, the VPL method gives better predictions than the two-dimensional (2-D) method in locations where propagation over buildings is significant.

302 citations


Journal ArticleDOI
TL;DR: This model implements state-of-the-art Monte Carlo ray-tracing techniques and is dedicated to the study of light propagation in terrestrial environments as a virtual laboratory, where scenes of arbitrary complexity can be described explicitly and the relevant radiative processes can be represented in great detail at spatial scales relevant to simulate actual measurements.
Abstract: A model of radiation transfer in three-dimensional (3D) heterogeneous media is designed and evaluated. This model implements state-of-the-art Monte Carlo ray-tracing techniques and is dedicated to the study of light propagation in terrestrial environments. It is designed as a virtual laboratory, where scenes of arbitrary complexity can be described explicitly and where the relevant radiative processes can be represented in great detail, at spatial scales relevant to simulate actual measurements. The approach capitalizes on the existing understanding of the elementary radiative processes and recognizes that the major difficulty in accurately describing the radiation field after its interaction with a typical terrestrial scene results from the complexity of the structure and the diversity of the properties of the elements of the scene. The output of the model can be customized to address various scientific investigations, including the determination of absorption profiles or of light-scattering distributions. The performance of the model is evaluated through detailed comparisons with laboratory measurements of an artificial target as well as with other established reflectance models for plant canopies.

242 citations


Journal ArticleDOI
TL;DR: The author describes how the use of volumetric textures is well suited to complex repetitive scenes containing forests, foliage, grass, hair, or fur, using a single ray per pixel.
Abstract: Complex repetitive scenes containing forests, foliage, grass, hair, or fur, are challenging for common modeling and rendering tools The amount of data, the tediousness of modeling and animation tasks, and the cost of realistic rendering have caused such kind of scene to see only limited use even in high-end productions The author describes how the use of volumetric textures is well suited to such scenes These primitives can greatly simplify modeling and animation tasks More importantly, they can be very efficiently rendered using ray tracing with few aliasing artifacts The main idea, initially introduced by Kajiya and Kay (1989), is to represent a pattern of 3D geometry in a reference volume, that is tiled over an underlying surface much like a regular 2D texture In our contribution, the mapping is independent of the mesh subdivision, the pattern can contain any kind of shape, and it is prefiltered at different scales as for MIP-mapping Although the model encoding is volumetric, the rendering method differs greatly from traditional volume rendering A volumetric texture only exists in the neighborhood of a surface, and the repeated instances (called texels) of the reference volume are spatially deformed Furthermore, each voxel of the reference volume contains a key feature which controls the reflectance function that represents aggregate intravoxel geometry This allows for ray tracing of highly complex scenes with very few aliasing artifacts, using a single ray per pixel (for the part of the scene using the volumetric texture representation) The major technical considerations of our method lie in the ray-path determination and in the specification of the reflectance function

193 citations


Book ChapterDOI
29 Jun 1998
TL;DR: A new family of image- based scene representations are introduced and corresponding image-based rendering algorithms that are capable of handling general synthetic scenes containing not only diffuse reflectors, but also specular and glossy objects are described.
Abstract: Most current image-based rendering methods operate under the assumption that all of the visible surfaces in the scene are opaque ideal diffuse (Lambertian) reflectors. This paper is concerned with image-based rendering of non-diffuse synthetic scenes. We introduce a new family of image-based scene representations and describe corresponding image-based rendering algorithms that are capable of handling general synthetic scenes containing not only diffuse reflectors, but also specular and glossy objects. Our image-based representation is based on layereddepth images. It represents simultaneously and separately both view-independent scene information and view-dependent appearance information. The view-dependent information may be either extracted directly from our data-structures, or evaluated procedurally using an image-based analogue of ray tracing. We describe image-based rendering algorithms that recombine the two components together in a manner that produces a good approximation to the correct image from any viewing position. In addition to extending image-based rendering to non-diffuse synthetic scenes, our paper has an important methodological contribution: it places image-based rendering, light field rendering, and volume graphics in a common framework of discrete raster-based scene representations.

167 citations


Proceedings ArticleDOI
Parker, Shirley, Livnat, Hansen, Sloan 
01 Jan 1998

117 citations


Journal ArticleDOI
Brian Smits1
TL;DR: Optimizations for bounding-volume hierarchies that many people use but are rarely described in the literature are presented, presenting optimizations for bounders that avoid some of the common pitfalls.
Abstract: Ray casting is the bottleneck of many rendering algorithms. Although much work has been done on making ray casting more efficient, most published work is high level. This paper discusses efficiency at a slightly lower level, presenting optimizations for bounding-volume hierarchies that many people use but are rarely described in the literature. A set of guidelines for optimization are presented that avoid some of the common pitfalls. Finally, the effects of the optimizations are shown for a set of models.

104 citations


Proceedings ArticleDOI
24 Jul 1998
TL;DR: This paper presents a novel method for interactive computation of reflections on curved objects, which can be reduced to a 2-D one that is utilized more accurately and efficiently.
Abstract: Global view-dependent illumination phenomena, in particular reflections, greatly enhance the realism of computer-generated imagery. Current interactive rendering methods do not provide satisfactory support for reflections on curved objects. In this paper we present a novel method for interactive computation of reflections on curved objects. We transform potentially reflected scene objects according to reflectors, to generate virtual objects. These are rendered by the graphics system as ordinary objects, creating a reflection image that is blended with the primary image. Virtual objects are created by tessellating scene objects and computing a virtual vertex for each resulting scene vertex. Virtual vertices are computed using a novel space subdivision, the reflection subdivision. For general polygonal mesh reflectors, we present an associated approximate acceleration scheme, the explosion map. For specific types of objects (e.g., linear extrusions of planar curves) the reflection subdivision can be reduced to a 2-D one that is utilized more accurately and efficiently. CR Categories: I.3.3 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism.

98 citations


Journal ArticleDOI
TL;DR: In this article, a simplified model for medium and high frequencies in structures or acoustics is presented, which does not take into account interferences between propagative waves, is asymptotic in the sense that it is more accurate as frequency increases.

82 citations


Journal ArticleDOI
TL;DR: A modified Monte Carlo method to calculate multipath dispersion due to wall reflections on indoor wireless diffuse optical channels is presented in this article, which allows evaluation of not only the Lambertian but also specular reflections and can be used to validate previous simulation schemes.
Abstract: A modified Monte Carlo method to calculate multipath dispersion due to wall reflections on indoor wireless diffuse optical channels is presented. As with other Monte Carlo methods, it allows evaluation of not only the Lambertian but also specular reflections and can be used to validate previous simulation schemes. The main difference is that for each ray traced and for each rebound, a contribution to the receiver is calculated. This leads to a faster and more accurate simulation.

Proceedings Article
01 Jan 1998
TL;DR: This paper describes a novel way of ray-tracing procedural displacement shaders directly, that is, without introducing intermediate geometry, and Affine arithmetic is used to compute bounding boxes for the shader over any range in the parameter domain.
Abstract: Displacement maps and procedural displacement shaders are a widely used approach of specifying geometric detail and increasing the visual complexity of a scene. While it is relatively straightforward to handle displacement shaders in pipeline based rendering systems such as the Reyes-architecture, it is much harder to efficiently integrate displacement-mapped surfaces in ray-tracers. Many commercial ray-tracers tessellate the surface into a multitude of small triangles. This introduces a series of problems such as excessive memory consumption and possibly undetected surface detail. In this paper we describe a novel way of ray-tracing procedural displacement shaders directly, that is, without introducing intermediate geometry. Affine arithmetic is used to compute bounding boxes for the shader over any range in the parameter domain. The method is comparable to the direct ray-tracing of Bézier surfaces and implicit surfaces using Bézier clipping and interval methods, respectively.

Journal ArticleDOI
TL;DR: This paper examines the effectiveness of load balancing strategies for ray tracing on large parallel computer systems and cluster computers and finds a hybrid strategy that combines static and dynamic approaches produces nearly optimal performance on a variety of images and computer systems.
Abstract: This paper examines the effectiveness of load balancing strategies for ray tracing on large parallel computer systems and cluster computers. Popular static load balancing strategies are shown to be inadequate for rendering complex images with contemporary ray tracing algorithms, and for rendering NTSC resolution images on 128 or more computers. Strategies based on image tiling are shown to be ineffective except on very small numbers of computers. A dynamic load balancing strategy, based on a diffusion model, is applied to a parallel Monte Carlo rendering system. The diffusive strategy is shown to remedy the defects of the static strategies. A hybrid strategy that combines static and dynamic approaches produces nearly optimal performance on a variety of images and computer systems. The theoretical results should be relevant to other rendering and image processing applications.

Proceedings ArticleDOI
01 Jan 1998

Book ChapterDOI
29 Jun 1998
TL;DR: An efficient and simple algorithm is presented to trace a ray recursively through the forest of triangles emerging from adaptive refinement of an envelope mesh.
Abstract: We present the necessary theory for the integration of subdivision surfaces into general purpose rendering systems. The most important functionality that has to be provided via an abstract geometry interface are the computation of surface points and normals as well as the ray intersection test. We demonstrate how to derive the corresponding formulas and how to construct tight bounding volumes for subdivision surfaces. We introduce envelope meshes which have the same topology as the control meshes but tightly circumscribe the limit surface. An efficient and simple algorithm is presented to trace a ray recursively through the forest of triangles emerging from adaptive refinement of an envelope mesh.

Proceedings ArticleDOI
01 Aug 1998
TL;DR: The nature of data access patterns and often the sheer size of the scene to be rendered, means that a straightforward parallelisation, if one exists, may not always lead to good performance.
Abstract: Global illumination is an area of research which tries to develop algorithms and methods to render images of artificial models or worlds as realistically as possible. Such algorithms are known for their unpredictable data accesses and their high computational complexity. Rendering a single high quality image may take several hours, or even days. For this reason parallel processing must be considered as a viable option to compute images in a reasonable time. The nature of data access patterns and often the sheer size of the scene to be rendered, means that a straightforward parallelisation, if one exists, may not always lead to good performance. This holds for all three rendering techniques considered in this report: ray tracing, radiosity and particle tracing.

Proceedings ArticleDOI
18 May 1998
TL;DR: This work quantifies the prediction accuracy and computation time for a 3-D ray-tracing tool Wireless Systems Engineering Tool (WiSE) by comparing predictions to a large database of measurements collected in three different office buildings located in New Jersey, USA.
Abstract: Ray-tracing based prediction of indoor microwave propagation has great potential as a practical and accurate technique to engineer wireless systems. The prediction accuracy of such methods increases with the number of reflections included in a single propagation path. At the same time, the computation time grows exponentially with the number of reflections. Thus, it is critically important to understand the minimum number of reflections needed to achieve the desired level of accuracy. We quantify the prediction accuracy and computation time for a 3-D ray-tracing tool Wireless Systems Engineering Tool (WiSE) by comparing predictions to a large database of measurements collected in three different office buildings located in New Jersey, USA.

Journal ArticleDOI
TL;DR: It is shown that a three-dimensional ASAR image of the platform can be formed by inverse Fourier transforming the multifrequency, multiaspect far-field radiation data from an antenna mounted on the platform.
Abstract: The antenna synthetic aperture radar (ASAR) imaging concept is introduced. We present the ASAR imaging algorithm to pinpoint the locations of secondary scattering off a platform from antenna radiation data. It is shown that a three-dimensional (3-D) ASAR image of the platform can be formed by inverse Fourier transforming the multifrequency, multiaspect far-field radiation data from an antenna mounted on the platform. This concept is demonstrated using the computed radiation data from the code Apatch, which employs the shooting and bouncing ray (SBR) technique. Furthermore, we develop a fast ASAR algorithm specially tailored for the SBR approach. By taking advantage of the ray tracing information within the SBR engine, we demonstrate that the fast approach can result in the same quality of image as the frequency-aspect algorithm at only a fraction of the computation time.

Journal ArticleDOI
TL;DR: In this article, a multiparameter optimization method is developed for use with terrestrial and space reflector antenna electromechanical systems and other metallic and composite engineering structures, which incorporates the objectives from various structural and electromagnetic (EM) performances of the system at many working/loading cases simultaneously.
Abstract: A novel multiparameter optimization method is developed for use with terrestrial and space reflector antenna electromechanical systems and other metallic and composite engineering structures. To satisfy extremely high design requirements, the proposed approach incorporates the objectives from various structural and electromagnetic (EM) performances of the system at many working/loading cases simultaneously. A finite element method is used for structural analysis. Optical ray tracing, spline function aperture field interpolation, geometric optics aperture integration, and FFT techniques are employed to analyse the EM performances of distorted reflector antennas. A systematic method is used for parameter profile analysis of the system. The optimization involves member size, structural geometric and material design variables. Various terrestrial and orbital working environments and loading cases which affect antenna performances can be included in the optimization model. The optimization of an 8 m antenna system, as an example, is discussed and the results are given.

Journal ArticleDOI
01 Aug 1998
TL;DR: In this article, a deterministic approach to model radio propagation channels in complex environments and use the model to study diversity techniques is described, which is based on geometric optics and uses ray-tracing techniques.
Abstract: A deterministic approach to model radio propagation channels in complex environments and use the model to study diversity techniques is described. The simulation is based on geometric optics and uses ray-tracing techniques. The effects of all rectangular obstacles are considered by the model. Both reflection and refraction effects are taken into account. Simulation results are presented and the reliability of these results is examined by comparing them with those obtained from measurements at 60 GHz. Also it is shown that it is possible to minimise the simulation time of the model by an appropriate choice of cell size. The model is seen to be able to investigate delay spread in an environment in considerable detail. The performance of space, frequency and polarisation diversity is investigated by use of the model. The results show that space diversity is better than polarisation or frequency diversity in the environments examined.

Journal ArticleDOI
TL;DR: This work introduces a new approach to three important problems in ray tracing: antialiasing, distributed light sources, and fuzzy reflections of lights and other surfaces by considering pyramidal rays (pyrays) formed by the viewpoint and the pixel.
Abstract: We introduce a new approach to three important problems in ray tracing: antialiasing, distributed light sources, and fuzzy reflections of lights and other surfaces. For antialiasing, our approach combines the quality of supersampling with the advantages of adaptive supersampling. In adaptive supersampling, the decision to partition a ray is taken in image-space, which means that small or thin objects may be missed entirely. This is particularly problematic in animation, where the intensity of such objects may appear to vary. Our approach is based on considering pyramidal rays (pyrays) formed by the viewpoint and the pixel. We test the proximity of a pyray to the boundary of an object, and if it is close (or marginal), the pyray splits into 4 sub-pyrays; this continues recursively with each marginal sub-pyray until the estimated change in pixel intensity is suciently small. The same idea also solves the problem of soft shadows from distributed light sources, which can be calculated to any required precision. Our approach also enables a method of defocusing reflected pyrays, thereby producing realistic fuzzy reflections of light sources and other objects. An interesting byproduct of our method is a substantial speedup over regular supersampling even when all pixels are supersampled. Our algorithm was implemented on polygonal and circular objects, and produced images comparable in quality to stochastic sampling, but with greatly reduced run times.

Proceedings ArticleDOI
Steven Fortune1
18 May 1998
TL;DR: A propagation model based on geometric optics has become popular for simulating indoor radio propagation and two algorithms that implement the model are presented that utilize a spatial data structure that allows very efficient path tracing.
Abstract: A propagation model based on geometric optics has become popular for simulating indoor radio propagation. We present and compare two algorithms that implement the model; both algorithms utilize a spatial data structure that allows very efficient path tracing.

Journal ArticleDOI
TL;DR: A genetic algorithm (GA) optimization method is developed and used in the design of a refractive-beam profile-shaping system and the calculation of the intensity profile is based on geometrical optics and is accomplished exclusively through ray tracing, giving this method broad applicability.
Abstract: We develop a genetic algorithm (GA) optimization method and use it in the design of a refractive-beam profile-shaping system. In this application, we employ the GA to determine the shape of one surface of the primary beam profile-shaping element in our system. The GA is instructed to vary the shape of this surface such that the output intensity profile is flat on a spherical surface some distance away. The GA does this while insuring that only a specified area of the output surface is illuminated. The calculation of the intensity profile is based on geometrical optics and is accomplished exclusively through ray tracing, giving this method broad applicability.

01 Dec 1998
TL;DR: A new, flexible computer code in the FORTRAN computer language to make ray calculations for ocean acoustic tomography is described, and the classical 4th order Runge-Kutta method appears to be the most efficient and best method for this integration problem.
Abstract: : This report describes a new, flexible computer code in the FORTRAN computer language to make ray calculations for ocean acoustic tomography. The Numerical Recipes software package provided the basis for much of this computer code. The ray equations are reviewed, and ray equations that include the effects of ocean current are derived. Methods are derived for rapidly integrating those equations to obtain time front and eigenray information for long-range, deep-water acoustic transmissions. These methods include a look-up table for sound speed, sound speed gradient, second derivative of sound speed, and range-dependent information. Cubic spline methods are used to interpolate sound speed with depth and to obtain the derivatives of sound speed. The choice of the step size increments used to integrate the equations is a critical aspect of the integration, affecting both the accuracy of the prediction and the speed of computation. A predetermined, user-specified step size appears to allow more efficient calculations than"adaptive step" methods. "Adaptive step" methods adjust the step size automatically to maintain a given accuracy in the integration of the ray equations, while user-specified step sizes allow one to use prior knowledge of the integration problem to achieve the desired accuracy with much less computational overhead. Several integration methods were explored, but the classical 4th order Runge-Kutta method appears to be the most efficient and best method for this integration problem. Appendices describe detailed aspects of the computer code, as well as the methods used for deriving eigenray information and for parallelizing the ray calculations. The computer code is designed to be unstable so that the user can easily modify it to his or her own purposes.

Proceedings ArticleDOI
01 Oct 1998
TL;DR: This work presents a method to accurately and efficiently perform perspective volumetric ray casting of uniform regular datasets, called Exponential-Region (ER) Perspective, and qualifies the algorithm by its filtering characteristics and demonstrates its effectiveness by contrasting its antialiasing quality and timing with other perspective ray casting methods.
Abstract: We present a method to accurately and efficiently perform perspective volumetric ray casting of uniform regular datasets, called Exponential-Region (ER) Perspective. Unlike previous methods which undersample, oversample, or approximate the data, our method near uniformly samples the data throughout the viewing volume. In addition, it gains algorithmic advantages from a regular sampling pattern and cache-coherent read access, making it an algorithm well suited for implementation on hardware architectures for volume rendering. We qualify the algorithm by its filtering characteristics and demonstrate its effectiveness by contrasting its antialiasing quality and timing with other perspective ray casting methods.

Journal ArticleDOI
TL;DR: In this paper, a projection space image reconstruction algorithm is used to correct the spatial varying geometric response of the collimator-detector system in single photon emission computed tomography (SPECT) which causes loss in resolution, shape distortions, reconstructed density nonuniformity, and quantitative inaccuracies.
Abstract: The spatially varying geometric response of the collimator-detector system in single photon emission computed tomography (SPECT) causes loss in resolution, shape distortions, reconstructed density nonuniformity, and quantitative inaccuracies. A projection space image reconstruction algorithm is used to correct these reconstruction artifacts. The projectors F use strip functions to calculate pixels more "natural" for modeling the two-dimensional (2-D) geometric response of the SPECT collimator transaxially to the axis of rotation. These projectors are defined by summing the intersection of an array of multiple strips rotated at equal angles to approximate the ideal system geometric response of the collimator. Two projection models were evaluated for modeling the system geometric response function. For one projector each strip is of equal weight, for the other projector a Gaussian weighting is used. Parallel beam and fan beam projections of a physical three-dimensional (3-D) Hoffman brain phantom and a Jaszczak cold rod phantom were used to evaluate the geometric response correction. Reconstructions were obtained by using the singular value decomposition (SVD) method and the iterative conjugate gradient algorithm to solve for q in the imaging equation FGq=p, where p is the projection measurement. The projector F included the new models for the geometric response, whereas, the backprojector G did not always model the geometric response in order to increase the computational speed. The final reconstruction was obtained by sampling the backprojection Gq at a discrete array of points. Reconstructions produced by the two proposed projectors showed improved resolution when compared against a unit-strip "natural" pixel model, the conventional image pixelized model with ray tracing to calculate the geometric response, and the filtered backprojection algorithm. When the reconstruction is displayed on fine grid points, the continuity and resolution of the image is preserved without the ring artifacts seen in the unit-strip "natural" pixel model. With present computing power, the geometric response correction using the proposed projection space reconstruction approach is not yet feasible for routine clinical use.

Journal ArticleDOI
TL;DR: This paper presents a method taking global illumination into account in a ray tracing environment, which allows to deal with all the types of light paths and the directional properties of materials and also allows to take uniform area light sources into account.
Abstract: This paper presents a method taking global illumination into account in a ray tracing environment. A vector approach is introduced, which allows to deal with all the types of light paths and the directional properties of materials. Three types of vectors are defined: Direct Light Vectors associated to light sources, Indirect Light Vectors which correspond to light having been diffusely reflected at least once and Caustic Light Vectors which are associated to light rays emitted by sources and reflected and/or transmitted by specular surfaces. These vectors are estimated at a small number of points in the scene. A weighted interpolation between known values allows to reconstruct these vectors for the other points, with the help of a gradient computation for the indirect component. This approach also allows to take uniform area light sources (spherical, rectangular and circular) into account for all the types of vectors. Computed images are thus more accurate and no discretizing of the geometry of the scene is needed.

Journal ArticleDOI
TL;DR: In this paper, two-dimensional analyses of the distribution of optically generated charge carriers in textured crystalline silicon solar cells of arbitrary geometry have been performed based on geometrical optics and ray tracing.

Journal ArticleDOI
TL;DR: A high quality beam tracer that provides a robust and general antialiasing for polyhedral scenes and is entirely compatible with the usual improvements of classical ray tracing (spatial subdivisions or hierarchical bounding volumes).

Journal ArticleDOI
TL;DR: Formulas for the calculation of the direction cosines of refracted and internally reflected rays in anisotropic uniaxial crystals are presented and a numerical example for the case of refraction and internal reflection in calcite is given.
Abstract: Formulas for the calculation of the direction cosines of refracted and internally reflected rays in anisotropic uniaxial crystals are presented. The method is based on a transformation to a nonorthonormal coordinate system in which the normal surface associated with the extraordinary ray is of spherical shape. A numerical example for the case of refraction and internal reflection in calcite is given.