CurvedLand: An Applet for Illustrating Curved Geometry without Embedding

TL;DR

CurvedLand is a Java applet that visually demonstrates curved geometry through interactive mapping, showing how objects and paths appear distorted in positively curved spaces from different viewpoints.

Contribution

It introduces an interactive tool that visualizes curved geometry without embedding in higher-dimensional space, aiding intuitive understanding of curvature effects.

Findings

Objects appear distorted in the applet as expected in curved spaces.

The applet allows navigation and placement of objects, illustrating curvature effects dynamically.

It provides an accessible educational resource for visualizing curved geometry.

Abstract

We have written a Java applet to illustrate the meaning of curved geometry. The applet provides a mapping interface similar to MapQuest or Google Maps; features include the ability to navigate through a space and place permanent point objects and/or shapes at arbitrary positions. The underlying two-dimensional space has a constant, positive curvature, which causes the apparent paths and shapes of the objects in the map to appear distorted in ways that change as you view them from different relative angles and distances.