📉Graphical Representation: Curvature (Just the Quotes)

"Characterizing a two-dimensional scatterplot is relatively easy, particularly with the full range of recently developed graphical enhancements at hand. However, standard patterns to watch for in three-dimensional plots are not as well understood as they are in many two-dimensional plots. We can certainly look for very general characteristics like curvature in three-dimensional plots, but it may not be clear how or if the curvature itself should be characterized. It is also possible to obtain useful insights into higher-dimensional scatterplots, but for the most part their interpretation must rely on lower-dimensional constructions. Similar statements apply to scatterplot matrices and various linked plots." (R Dennis Cook, "Regression Graphics: Ideas for Studying Regressions through Graphics", 1998)

"The fundamental concept of Gauss’s surface theory is the curvature, a quantity that is positive (and constant) for a sphere, zero for the plane and cylinder, and negative for surfaces that are 'saddle-shaped' in the neighborhood of each point." (John Stillwell, "The Four Pillars of Geometry", 2000) 

"Even though hyperbolic trees employ the same ranking principle as radial trees, based on a series of concentric circles, they do not operate in conventional Euclidean space, but instead within a spherical negative curvature based on hyperbolic geometry. Due to their magnifying feature, hyperbolic trees are useful for displaying and manipulating large hierarchies on a limited screen size. As visualizations ideally suited for direct manipulation, hyperbolic trees are rarely depicted in print and are found almost exclusively within the confines of their natural digital domain." (Manuel Lima, "The Book of Trees: Visualizing Branches of Knowledge", 2014)

"Once a model has been fitted to the data, the deviations from the model are the residuals. If the model is appropriate, then the residuals mimic the true errors. Examination of the residuals often provides clues about departures from the modeling assumptions. Lack of fit - if there is curvature in the residuals, plotted versus the fitted values, this suggests there may be whole regions where the model overestimates the data and other whole regions where the model underestimates the data. This would suggest that the current model is too simple relative to some better model." (DeWayne R Derryberry, "Basic data analysis for time series with R", 2014)

"A visual channel is a way to control the appearance of marks, independent of the dimensionality of the geometric primitive. […]  The motion-oriented channels include the motion pattern, for in stance, oscillating circles versus straight jumps, the direction of motion, and the velocity. Angle is also a channel, sometimes called tilt. Curvature is also a visual channel. Shape is a complex phenomenon, but it is treated as a channel in this framework." (Tamara Munzner, "Visualization: Analysis & Design", 2015) 

"Tensor fields typically contain a matrix at each cell in the field, capturing more complex structure than what can be expressed in a vector field. Tensor fields can measure properties such as stress, conductivity, curvature, and diffusivity." (Tamara Munzner, "Visualization: Analysis & Design", 2015) 

"The curvature channel is not very accurate, and it can only be used with line marks. It cannot be used with point marks that have no length, or area marks because their shape is fully constrained. The number of distinguishable bins for this channel is low, probably around two or three; it is in an equivalence class with volume (3D size) at the bottom of the magnitude channel ranking." (Tamara Munzner, "Visualization: Analysis & Design", 2015)

"Important features to look for in a scatter plot are whether there is one cloud of dots or several clusters, whether there is an upward or downward slope to the cloud of dots, and whether there is any curvature to the slope." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)

"Adjusting scale is an important practice in data visualization. While the log transform is versatile, it doesn’t handle all situations where skew or curvature occurs. For example, at times the values are all roughly the same order of magnitude and the log transformation has little impact. Another transformation to consider is the square root transformation, which is often useful for count data." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)