Curved Phenine Normal Vectors: Geometric Measures of Geodesic Phenine Frameworks

Tatsuru Mio1, Koki Ikemoto1,2, Hiroyuki Isobe1,2

  • 1Department of Chemistry, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan.

Related Concept Videos

Bending of Curved Members - Strain Analysis01:14

Bending of Curved Members - Strain Analysis

The mechanics of deformation in curved members, such as beams or arches, under bending moments, involve complex responses. When such a member, symmetric about the y-axis and shaped like a segment of a circle centered at point C, is subjected to equal and opposite forces, its curvature and surface lengths change significantly. This alteration results in the shift of the curvature's center from C to C', indicating a tighter curve.
The important part of bending analysis for such a member...
442
Deformations in a Symmetric Member in Bending01:18

Deformations in a Symmetric Member in Bending

When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
When the member is segmented into tiny cubic elements, it is observed that the primary stress...
424
The Distance Formula01:20

The Distance Formula

In geometry, measuring the direct distance between two points on a plane is essential in various practical and theoretical applications. Whether in navigation, engineering, or computer graphics, determining the shortest path between two locations involves using the distance formula. This formula is derived from the Pythagorean Theorem, which relates the lengths of the sides of a right triangle. On a coordinate plane, the horizontal and vertical distances between two points serve as the legs of...
504
Hyperbolas01:30

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse...
217
Geometry of Hyperbolas01:30

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
222
Reflective Property of Parabolas01:26

Reflective Property of Parabolas

A parabola is a basic type of conic section that results from the intersection of a plane with a double-napped cone in a direction parallel to one of the cone's sides. This U-shaped curve has a distinctive reflective property: all incoming rays parallel to its axis of symmetry are directed toward a single point, known as the focus. This property is widely utilized in optical and communication technologies that require precise signal concentration.In analytic geometry, a parabola is defined as...
156