# Web based app for differential geometry visualization

Is there by any chance a web based app for viewing parametric curves and surfaces that arise in differential geometry to help visualization? Thanks.

• You can use Wolframalpha – HallaSurvivor Jan 3 at 18:48
• Are these curves and surfaces already immersed in $\mathbb R^3$? – Mark S. Jan 3 at 23:26

I assume you are talking about curves/surfaces that are immersed in $$\mathbb R^2$$ or $$\mathbb R^3$$. More abstract or higher dimensional curves/surfaces would be a bit harder to work with.

## Static Plots

As HallaSurvivor mentioned in a comment, Wolfram|Alpha's plotting and graphics capabilities include static plots of parametric surfaces and curves in 3d, or curves in 2d.

## 2D Animations/Interactives

If you want something more interactive, Desmos's graphing calculator works well for 2D. The Desmos learn page on "parametric equations" links to an example page including a family of parametric curves with an extra parameter $$a$$ that can be modified with a slider to display different curves. And the Desmos learn page on "points" links to an example page including a parametric curve shown one point at a time where the parameter can be controlled with a slider. Pressing the play button ▶ will have the parameter increase at a fixed speed so the nature of the parametrization can be understood in motion.

## 3D Animations/Interactives

Finally, the Math3D app has many features similar to Desmos, plus some additional features built in (e.g. vectors), and can handle interactive/animated parametrized curves and surfaces in 3D, as shown in the Math3D example for a ruled hyperboloid (of one sheet).

Edit: Thanks to Semiclassical's comment for reminding me about the GeoGebra 3d calculator which has similar functionality (and can do a lot more with some complicated inputs). Geogebra has curve and surface commands for parametric constructions. You can see both with some sliders in a demonstration by matheagle. You can even code a 2d slider to control a pair of parameters for a surface, as exemplified in Lenore Horner's demonstration.

## Programming

I didn't mention this originally because there is a much steeper barrier to entry, but CoCalc has a free plan and could be used for graphing on the web if you're already comfortable with a relevant programming language. See one collection of examples (including a graph of a parametrized surface) using SageMath and one collection (including a parametrized triangulated Möbius strip) using matplotlib in Python.

• Geogebra also has a 3D mode in addition to the more familiar 2D options. – Semiclassical Jan 4 at 0:24
• Wow... thank you very much! I will explore these options. – user9106985 Jan 4 at 7:32