Interactive Visualizations of Projective Varieties in Spherical Geometry

Quartic Surfaces Containing a Line

Sun, 01 Mar 2026 00:00:00 +0000

Here we are visualizing a quartic surface containing a line in the spherical geometry of $\mathbb{RP}^3$ .

In the space of all quartic surfaces it is exceedingly rare that one contains a line, specifically it occurs with measure zero. For an explantion see here

We are further refining our interest in quartic surfaces defined by two cubic curves.

Symmetric Cubic Surfaces

Sun, 01 Mar 2026 00:00:00 +0000

Here we are plotting a symmetric cubic surface in the spherical geometry of $\mathbb{RP}^3$ .

To start, we are illustrating the classic result of the existence of 27 straight lines on a cubic surface. Finding these lines is a more involved task for a general cubic but luckily we are using results from (Brazelton and Raman 2025) that allow us to easily compute and visualize the lines.

Overview

Fri, 27 Feb 2026 00:00:00 +0000

Introduction

The notion of creating a deductive formalization of mathematical properties through visualizations and intuitions of geometry dates back to ancient Greece. Notably, in his 299 B.C. book, The Elements, Euclid posited as an axiom that parallel lines do not meet. In a painitng, lines that might be parallel in the real world appear to meet at a vanishing point.

Projective geometry provides a natural and conceptually complete setting for many geometric theorems. Statements about parallel lines, conic sections, and algebraic varieties often become simpler and more unified when expressed projectively. However, standard visualizations often treat points at infinity as artifacts rather than intrinsic components of the geometry.

Presentation

Tue, 24 Feb 2026 00:00:00 +0000

On February 24th we gave a presentation on our work. You can peruse the slides below

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