Differential Geometry: From Mapmaking to Conformal Geometry (Ch. 4 part 2 of2)

### Why Did Ship Captains Care About Conformal Maps?

One of the original motivations for differential geometry was surprisingly practical:
How do you draw a flat map of the Earth that preserves the angles sailors need for navigation?

As we finish Chapter 4 of Visual Differential Geometry and Forms by Tristan Needham, we’ll explore conformal maps, curvature from the metric, and the geometric ideas behind cartography itself.

What makes Needham especially interesting is not just the visual approach, but the historical context he brings to the mathematics — showing how real problems in navigation, mapping, and geometry led to the development of the subject.

Last meeting: metrics, curvature, and how geometry is encoded in (ds^2).

This meeting: conformal maps, complex viewpoints, curvature from scaling, and selected Chapter 7 problems that deepen the ideas from Chapter 4.

### What We’ll Cover

• Conformal maps and angle preservation
• Cartography and the geometry of navigation
• Curvature from the conformal scale factor
• Why every surface locally admits conformal coordinates
• Selected Chapter 7 problems related to Chapter 4 (Problems 1–13)

### What to Expect

• Calculus required (partial derivatives, multivariable thinking)
• Reading Chapter 4 encouraged (especially 4.5–4.6)
• Discussion-based (not a lecture)
• Problems worked selectively for conceptual insight

### Why It Matters

These ideas connect directly to:
• General relativity
• Complex analysis
• Differential forms and modern geometry
• The deeper idea of “structure-preserving” maps in mathematics and physics

### Important

Discussion will stay focused on the agenda.
For casual physics conversation, see other group meetings.

### Optional References

Weeks • Baez & Muniain • Greenberg • Penrose

### More in Physics With Friends

This event is one of many collaborative study tracks in our Physics With Friends community.
Explore other topics and join additional study groups here:
https://www.meetup.com/physicswithfriends/events/

Join anytime — come prepared to think.

Note: This meeting was previously listed under the placeholder title “Course on General Relativity and Differential Geometry,” carried over from last year’s MIT relativity course review series. The focus has now shifted fully to Tristan Needham’s differential geometry text.