arXiv subjects
Publications and source records attributed to Chamberlain Fong.
The squircle is an intermediate shape between the square and the circle. In this paper, we examine and discuss equations for different types of squircles. We then build upon these 2D shapes to come-up with various 3D surfaces based on squircles.
We present and discuss different algorithms for converting rectangular imagery into elliptical regions. We mainly focus on methods that use mathematical mappings with explicit and invertible equations. The key idea is to start with invertible mappings between the square and the circular disc then extend it to handle rectangles and ellipses. This extension can be done by simply removing the eccentricity and reintroducing it back after using a chosen square-to-disc mapping.
We present an algorithm for converting an indoor spherical panorama into a photograph with a simulated overhead view. The resulting image will have an extremely wide field of view covering up to 4π steradians of the spherical panorama. We argue that our method complements the stereographic projection commonly used in the "little planet" effect. The stereographic projection works well in creating little planets of outdoor scenes; whereas our method is a well-suited counterpart for indoor scenes. The main innovation of our method is the introduction of a novel azimuthal map projection that can smoothly blend between the stereographic projection and the Lambert azimuthal equal-area projection. Our projection has an adjustable parameter that allows one to control and compromise between distortions in shape and distortions in size within the projected panorama. This extra control parameter gives our projection the ability to produce superior results over the stereographic projection.
The Fernandez-Guasti squircle is a plane algebraic curve that is an intermediate shape between the circle and the square. It has qualitative features that are similar to the more famous Lam\'e curve. However, unlike the Lam\'e curve which has unbounded exponents, the Fernandez-Guasti squircle is a low degree quartic curve. This makes it more amenable to algebraic manipulation and simplification. In this paper, we will analyze this squircle and derive formulas for its area, arc length, and polar form. We will also provide several parametric equations of this squircle. Finally, we extend the Fernandez-Guasti squircle to three dimensions by coming up with an analogous surface that is an intermediate shape between the sphere and the cube.
The Peirce quincuncial projection is a mapping of the surface of a sphere to the interior of a square. It is a conformal map except for four points on the equator. These points of non-conformality cause significant artifacts in photographic applications. In this paper, we propose an algorithm and user-interface to mitigate these artifacts. Moreover, in order to facilitate an interactive user-interface, we present a fast algorithm for calculating the Peirce quincuncial projection of spherical imagery. We then promote the Peirce quincuncial projection as a viable alternative to the more popular stereographic projection in some scenarios.
We present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphasis on mappings with desirable properties. These include conformal, equiareal, and radially-constrained mappings. Finally, we present applications to logo design, panoramic photography, and hyperbolic art.
We propose a modification of a key component in the Doomsday Algorithm for calculating the day of the week of any calendar date. In particular, we propose to replace the calculation of the required term: \lfloor \frac{x}{12} \rfloor + x \bmod 12 + \lfloor \frac{x \bmod 12}{4} \rfloor with -[ \frac{x+11(x \bmod 2)}{2} + 11 (\frac{x+11(x \bmod 2)}{2}\bmod 2)] \bmod 7 for a 2-digit input year x; Although our expression looks daunting and complicated, we will explain why it is actually easy to calculate mentally.
We propose a modification of a key component in the Doomsday Algorithm for calculating the day of the week of any calendar date. In particular, we propose to replace the calculation of the required term: \lfloor \frac{x}{12} \rfloor + x \bmod 12 + \lfloor \frac{x \bmod 12}{4} \rfloor with the term 2y + 10 \, (y \bmod 2) + z + \lfloor \frac{2 \, (y \bmod 2) + z}{4} \rfloor where x is an input 2-digit year; y is the tens digit of x; z is the ones digit of x; We argue the fact that our modification operates on individual base-10 digits makes the algorithm easier to calculate mentally.