Lecture 20: Geodesics (Discrete Differential Geometry)

[AerysBat]

This lecture series is monumental. It makes differential geometry feel easy and natural instead of getting bogged down in technical language. The slides and lecture notes are gorgeous. And if I need to learn how to do this stuff in code, I know exactly the exercises to follow. Thank you so much for making this material public.

[maximanurov]

The best explanation I've ever heard, thanks for your work!

[columbusmyhw]

Wow - just discovered this channel. This is such a useful and well taught lecture! Thank you!

[EmanuelSygal]

Amazing Lecture! I learned so much, thank you!
In 1:43:12 the first g() term has a missing second input g(*, W)

[diakorudd]

[00:06] INTRODUCTION TO GEODESICS

Basic concept: Generalization of straight lines to curved spaces
Two key properties:

Straightest possible curves with no acceleration
Locally minimize length between points



[00:47] HISTORICAL PERSPECTIVE: EUCLID'S POSTULATES

Five basic postulates:

Two points can be connected by straight line
Straight lines can be extended indefinitely
Circles can be drawn with any center and radius
All right angles are equal
Parallel postulate (unique parallel through point)


Historical controversy around parallel postulate

[06:26] HYPERBOLIC GEOMETRY

Alternative to Euclidean geometry
Violates parallel postulate
Multiple parallels through a point
Visualized using models (circular arcs)

[08:57] EXAMPLES OF GEODESICS

Great arcs on sphere

Shortest paths between points
Example: Flight paths across Earth


Shortest paths in graphs

Navigation through street networks
Maze solving


Applications in general relativity

Light bending around massive objects
Gravitational lensing



[16:26] ISOMETRY INVARIANCE

Key property: Geodesics preserved under isometries
Isometries: Deformations preserving intrinsic geometry
Example: Folding a map doesn't change distances
Preserves Riemannian metric

[18:14] DIFFERENT PERSPECTIVES ON GEODESICS

Straightest curves (zero acceleration)
Locally shortest curves
Zero geodesic curvature
Harmonic maps
Gradient of distance function

[20:05] LOCALLY SHORTEST PROPERTY

Distinction between locally and globally shortest
Example: Multiple geodesics on sphere between points
Definition of minimal geodesic
Connection to Dirichlet energy

[24:26] RELATIONSHIP TO DIRICHLET ENERGY

Connection between curve length and Dirichlet energy
Reparameterization and unit speed curves
Minimization of energy yields geodesics
Harmonic function interpretation

[30:04] GEODESICS ON CURVED SURFACES

Extension of Dirichlet energy to curved surfaces
Role of Riemannian metric
Multiple critical points vs unique minimum
Example: Multiple geodesics on sphere

[34:35] DISCRETE SHORTEST PATHS

Limitations of Dijkstra's algorithm
Need for paths through triangles
Iterative straightening approaches
Global vs local shortest paths

[37:30] LOCAL BEHAVIOR NEAR VERTICES

Dependence on vertex curvature
Three cases:

Flat vertices (angle sum = 2π)
Cone vertices (angle sum < 2π)
Saddle vertices (angle sum > 2π)



[41:29] DYNAMIC PERSPECTIVE ON GEODESICS

Zero tangential acceleration
Introduction to covariant derivative
Parallel transport along curves
Challenge of comparing vectors in different tangent spaces

[45:38] COVARIANT DERIVATIVE

Formal definition and properties
Relationship to Christoffel symbols
Metric compatibility
Torsion-free property

[50:54] EXPONENTIAL AND LOGARITHMIC MAPS

Definition of exponential map
Relationship to geodesics
Injectivity radius
Cut locus concept

[1:00:06] DISCRETE EXPONENTIAL MAP

Implementation on triangle meshes
Unfolding triangles into plane
Handling vertex crossings
Examples on various surfaces

[1:10:34] AVERAGING ON SURFACES

Karcher mean definition
Iterative algorithm using exp/log maps
Applications to rotation averaging
Examples on different surfaces

[1:24:26] NUMERICAL COMPUTATION

Two approaches:

Discrete surface approximation
Smooth ODE integration


Trade-offs between methods
Pros and cons discussion

[1:52:51] SUMMARY AND CONCLUSIONS

Multiple characterizations of geodesics
Disconnect between smooth and discrete cases
Importance of choosing right definition for task
Limitations of different approaches
Value of multiple perspectives

[apurvamodak]

Professor, thanks for the lectures!! Any plans to upload previous ones? It would be very helpful.

[shadowkryans]

Ooooh Boiii. That was a video marathon, just went through all 20 of these lectures in 1 day. I guess I enjoyed learning about stuff in this topic a lil too much. XD 🥴👌

[leoyoung]

Sorry, I wonder why the lecture 19 is lost.

[emmpati]

1:21:00 there is no '1/n' term in Karcher mean

[emmpati]

1:41:38 I think 'i' at last partial derv. term of Lie bracket formulation should be subscript instead of superscript

[farissaadat]

Does anyone have a good explanation for why the number of points of contact of the medial ball with the boundary is the same as the number of branches at that point of the medial axis?

[wipperod]

The illustrations shown on the slides are beautiful! Can you share what kind of software are you using to plot them?

[Null_Simplex]

It seems like between any two points on a discrete surface, there is always a curve which has both the property of locally minimizing distance and being a discrete straightest geodesic. Around vertices whose angles sum to less than 360 degrees, the shortest path between any two points is already a straightest path. Around vertices whose angles sum to 360 degrees, geometry is identical to Euclidean geometry so shortest paths between points is already straightest. Lastly, around vertices whose angles sum to more than 360 degrees, while there are infinitely many paths that locally minimize distance, there is always exactly one of those shortest paths that is also a discrete straightest geodesic. If my assumption about these "True" discrete geodesics always existing is true, I wonder if they would in some sense be the "ideal" approximations for geodesics on a smooth surface which is being approximated by discrete surface.

[Astro-X]

Hey Keenan! Thank you for this wonderful series. I would like to leave you a tip because this playlist has helped me immensely with my Differential Geometry course at the University of Auckland, NZ. I would not have passed if it wasn't for you. Please let me know if there is any way I could donate to you!

[andresross]

I would like to know where to go from here to learn more about this topic?

[nikkyu]

is there code that we can use to put these into concrete terms?

[朱镝中]

Tremendous! Anybody knows where to get these slides?