Appropriate choice of colors significantly aids viewers in understanding the structures in multiclass scatterplots and becomes more important with a growing number of data points and groups. An appropriate color mapping is also an important parameter for the creation of an aesthetically pleasing scatterplot. Currently, users of visualization software routinely rely on color mappings that have been pre-defined by the software. A default color mapping, however, cannot ensure an optimal perceptual separability between groups, and sometimes may even lead to a misinterpretation of the data. In this paper, we present an effective approach for color assignment based on a set of given colors that is designed to optimize the perception of scatterplots. Our approach takes into account the spatial relationships, density, degree of overlap between point clusters, and also the background color. For this purpose, we use a genetic algorithm that is able to efficiently find good color assignments. We implemented an interactive color assignment system with three extensions of the basic method that incorporates top K suggestions, user-defined color subsets, and classes of interest for the optimization. To demonstrate the effectiveness of our assignment technique, we conducted a numerical study and a controlled user study to compare our approach with default color assignments; our findings were verified by two expert studies. The results show that our approach is able to support users in distinguishing cluster numbers faster and more precisely than default assignment methods.

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In this paper, we introduce Canis, a high‐level domain‐specific language that enables declarative specifications of data‐driven chart animations. By leveraging data‐enriched SVG charts, its grammar of animations can be applied to the charts created by existing chart construction tools. With Canis, designers can select marks from the charts, partition the selected marks into mark units based on data attributes, and apply animation effects to the mark units, with the control of when the effects start. The Canis compiler automatically synthesizes the Lottie animation JSON files [Aira], which can be rendered natively across multiple platforms. To demonstrate Canis’ expressiveness, we present a wide range of chart animations. We also evaluate its scalability by showing the effectiveness of our compiler in reducing the output specification size and comparing its performance on different platforms against D3.

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The main difference between a linear system and a nonlinear system is in the non-uniqueness of solutions manifested by the singular Jacobian matrix. It is important to be able to express the Jacobian accurately, completely, and efficiently in an algorithm to analyze a nonlinear system. For periodic response, the incremental harmonic balance (IHB) method is widely used. The existing IHB methods, however, requiring double summations to form the Jacobian matrix, are often extremely time-consuming when higher order harmonic terms are retained to fulfill the completeness requirement. A new algorithm to compute the Jacobian is to be introduced with the application of fast Fourier transforms (FFT) and Toeplitz formulation. The resulting Jacobian matrix is constructed explicitly by three vectors in terms of the current Fourier coefficients of response, depending respectively on the synchronizing mass, damping, and stiffness functions. The part of the Jacobian matrix depending on the nonlinear stiffness is actually a Toeplitz matrix. A Toeplitz matrix is a matrix whose k, r position depends only on their difference k-r. The other parts of the Jacobian matrix depending on the nonlinear mass and damping are Toeplitz matrices modified by diagonal matrices. If the synchronizing mass is normalized in the beginning, we need only two real vectors to construct the Toeplitz Jacobian matrix (TJM), which can be treated in one complex fast Fourier transforms. The present method of TJM is found to be superior in both computation time and storage than all existing IHB methods due to the simplified explicit analytical form and the use of FFT.