Interactive 4-D Visualization of Stereographic Images From the Double Orthogonal Projection

The double orthogonal projection of the 4-space onto two mutually perpendicular 3-spaces is a method of visualization of four-dimensional objects in a three-dimensional space. We present an interactive animation of the stereographic projection of a hyperspherical hexahedron on a 3-sphere embedded in the 4-space. Described are synthetic constructions of stereographic images of a point, hyperspherical tetrahedron, and 2-sphere on a 3-sphere from their double orthogonal projections. Consequently, the double-orthogonal projection of a freehand curve on a 3-sphere is created inversely from its stereographic image. Furthermore, we show an application to a synthetic construction of a spherical inversion and visualizations of double orthogonal projections and stereographic images of Hopf tori on a 3-sphere generated from Clelia curves on a 2-sphere.Comment: ICGG 2020 submissio

Dental anxiety and behavioral problems: A study of prevalence and related factors among a group of Iranian children aged 6-12

Purpose: The aims of this study were to assess the prevalence and also some related demographic and dental factors of dental anxiety and behavioral problems in school-aged children. Subjects and Methods: A total of 150 children of 6-12 years old were selected according to the inclusion criteria. Prior to the dental visit, the mothers were asked to answer a questionnaire of dental and demographic background and a Corah dental anxiety scale (CDAS). At the same time, a faces version of the modified child dental anxiety scale (MCDAS) was completed by the child. Next, the child was guided to the operating room. According to the treatment plan, local anesthesia solution was injected and the child′s cooperative behaviors were quantified based on the Frankle index duration the injection stage. Analysis of Variance and Linear regression models were used for the statistical analysis. Results: The mean scores of the child′s dental anxiety and cooperative behavior were 20.81 (±6.97) and 3.04 (±0.86), respectively. Forty four children (29.33%) had severe dental anxiety. Child′s age and regular dental visit are predictive factors for the child′s dental anxiety (P < 0.05). Dental behavioral problems had been identified in 43 children (28.67%). Unpleasant previous dental experience is an important factor affecting the child′s cooperative behaviors (P < 0.05). Conclusion: High prevalence of severe dental anxiety may be seen in early years of school. It seems that general factors such as family factors have less impact on behavior of school aged children in a dental visit

4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: A holistic treatment of multiple disconnected anatomical structures

Image-based parcellation of the brain often leads to multiple disconnected anatomical structures, which pose significant challenges for analyses of morphological shapes. Existing shape models, such as the widely used spherical harmonic (SPHARM) representation, assume topological invariance, so are unable to simultaneously parameterize multiple disjoint structures. In such a situation, SPHARM has to be applied separately to each individual structure. We present a novel surface parameterization technique using 4D hyperspherical harmonics in representing multiple disjoint objects as a single analytic function, terming it HyperSPHARM. The underlying idea behind HyperSPHARM is to stereographically project an entire collection of disjoint 3D objects onto the 4D hypersphere and subsequently simultaneously parameterize them with the 4D hyperspherical harmonics. Hence, HyperSPHARM allows for a holistic treatment of multiple disjoint objects, unlike SPHARM. In an imaging dataset of healthy adult human brains, we apply HyperSPHARM to the hippocampi and amygdalae. The HyperSPHARM representations are employed as a data smoothing technique, while the HyperSPHARM coefficients are utilized in a support vector machine setting for object classification. HyperSPHARM yields nearly identical results as SPHARM, as will be shown in the paper. Its key advantage over SPHARM lies computationally; HyperSPHARM possess greater computational efficiency than SPHARM because it can parameterize multiple disjoint structures using much fewer basis functions and stereographic projection obviates SPHARM’s burdensome surface flattening. In addition, HyperSPHARM can handle any type of topology, unlike SPHARM, whose analysis is confined to topologically invariant structures