00:00 - Introduction 00:35 - What is a Rotation? Invariant properties (length, distance, orientation) Fixed origin requirement [Relevant to gaussian splatting: Understanding how 3D rotations work is crucial for properly orienting gaussian splats in 3D space] 02:27 - Degrees of Freedom in 3D Rotations Pittsburgh to São Paulo example Need for three degrees of freedom [Gaussian splats need these 3 degrees of freedom to properly orient in space] 06:13 - Commutativity of Rotations 2D rotations commute 3D rotations don't commute Water bottle demonstration [Critical for understanding how to sequence rotations of gaussians in 3D] 10:35 - Representing Rotations in 2D Function S(θ) introduction Matrix construction from first principles Basis vector rotations 15:42 - Euler Angles Introduction to 3D rotation representation Problems with Euler angles Gimbal lock explanation [Gaussian splatting avoids Euler angles due to these limitations] 24:01 - Complex Numbers Introduction Geometric interpretation Avoiding "square root of -1" confusion Quarter-turn interpretation 32:15 - Complex Multiplication Geometric interpretation Polar form Applications to 2D rotations [Complex numbers provide elegant ways to handle 2D projections of gaussians] 40:23 - Quaternions Hamilton's discovery Four-dimensional representation Why 3D rotations need four components [Modern gaussian splatting implementations often use quaternions for rotation] 48:56 - Quaternion Operations Product rules Relationship to cross/dot products Application to 3D rotations 54:32 - Practical Applications SLERP interpolation Texture mapping Conformal maps [These concepts are crucial for smooth interpolation between gaussian splats] 59:01 - Summary and Conclusion Different rotation representations Importance of choosing right representation Preview of next lecture
9:34 In Chinese language, the yellow sign for the slippery floor is signed with"小心地滑". This sentence can be explained in 2 different ways. 小心 means watch out, then 地 means floor, 滑 means slippery. In this way, it means watch out the slippery floor. But it can be explained in the second way. 小心地means carefully. 滑 is now after an adv, and becomes a verb, which means slide. In this way, it means slide carefully.
What's the point to explain this? Here is not the Chinese Fun Fact channel... In addition, I've never seen any sign in yellow telling people to slide carefully...
I have a confusion at 18:45. I understand that we lost one degree of freedom but why is this matrix a rotation around only one axis? The axis of rotation is changing as the parameters are changed. On the Wikipedia page on gimbal lock, I also found the same statement about the same matrix in this video. It's unlikely that both the Wikipedia and you are wrong, that's why I'm confused. Thanks!
Great lesson as usual! How about geometric algebra, which unifies complex numbers, quarternions, vector spaces. We should learn more about Clifford/geometric algebra in computer graphics courses.
The rotor in geometric algebra serves the same function as quaternion in rotating 3D objects. Rotors can be composed just like quaternions to combine a series of rotations together. It can interpolate between two rotations as well.
Everything except Lectures 8, 9, 10, 11 are already up here: th-cam.com/play/PL9_jI1bdZmz2emSh0UQ5iOdT2xRHFHL7E.html The last few will be uploaded according to this schedule: 15462.courses.cs.cmu.edu/fall2020/
It's a common abbreviation for "with" in North American English: english.stackexchange.com/questions/115367/are-w-o-w-b-c-common-abbreviations-in-the-us
Professor, could you plz make a video to show how the last 2 of 5 ways of rotation representation work. I even tried the group theory tutorial, but it's literally too far beyond me.
00:00 - Introduction
00:35 - What is a Rotation?
Invariant properties (length, distance, orientation)
Fixed origin requirement
[Relevant to gaussian splatting: Understanding how 3D rotations work is crucial for properly orienting gaussian splats in 3D space]
02:27 - Degrees of Freedom in 3D Rotations
Pittsburgh to São Paulo example
Need for three degrees of freedom
[Gaussian splats need these 3 degrees of freedom to properly orient in space]
06:13 - Commutativity of Rotations
2D rotations commute
3D rotations don't commute
Water bottle demonstration
[Critical for understanding how to sequence rotations of gaussians in 3D]
10:35 - Representing Rotations in 2D
Function S(θ) introduction
Matrix construction from first principles
Basis vector rotations
15:42 - Euler Angles
Introduction to 3D rotation representation
Problems with Euler angles
Gimbal lock explanation
[Gaussian splatting avoids Euler angles due to these limitations]
24:01 - Complex Numbers Introduction
Geometric interpretation
Avoiding "square root of -1" confusion
Quarter-turn interpretation
32:15 - Complex Multiplication
Geometric interpretation
Polar form
Applications to 2D rotations
[Complex numbers provide elegant ways to handle 2D projections of gaussians]
40:23 - Quaternions
Hamilton's discovery
Four-dimensional representation
Why 3D rotations need four components
[Modern gaussian splatting implementations often use quaternions for rotation]
48:56 - Quaternion Operations
Product rules
Relationship to cross/dot products
Application to 3D rotations
54:32 - Practical Applications
SLERP interpolation
Texture mapping
Conformal maps
[These concepts are crucial for smooth interpolation between gaussian splats]
59:01 - Summary and Conclusion
Different rotation representations
Importance of choosing right representation
Preview of next lecture
9:34 In Chinese language, the yellow sign for the slippery floor is signed with"小心地滑". This sentence can be explained in 2 different ways. 小心 means watch out, then 地 means floor, 滑 means slippery. In this way, it means watch out the slippery floor. But it can be explained in the second way. 小心地means carefully. 滑 is now after an adv, and becomes a verb, which means slide. In this way, it means slide carefully.
What's the point to explain this? Here is not the Chinese Fun Fact channel... In addition, I've never seen any sign in yellow telling people to slide carefully...
I have a confusion at 18:45. I understand that we lost one degree of freedom but why is this matrix a rotation around only one axis? The axis of rotation is changing as the parameters are changed. On the Wikipedia page on gimbal lock, I also found the same statement about the same matrix in this video. It's unlikely that both the Wikipedia and you are wrong, that's why I'm confused. Thanks!
Great lesson as usual!
How about geometric algebra, which unifies complex numbers, quarternions, vector spaces. We should learn more about Clifford/geometric algebra in computer graphics courses.
It's cool and simple
The rotor in geometric algebra serves the same function as quaternion in rotating 3D objects. Rotors can be composed just like quaternions to combine a series of rotations together. It can interpolate between two rotations as well.
this is beyond fantastic!
Where are the dots for i and j?
These lessons are so good! Thank you so much. Will there be uploaded videos for lesson 7 and beyond?
Everything except Lectures 8, 9, 10, 11 are already up here: th-cam.com/play/PL9_jI1bdZmz2emSh0UQ5iOdT2xRHFHL7E.html
The last few will be uploaded according to this schedule: 15462.courses.cs.cmu.edu/fall2020/
Keenan Crane Lesson 7 was uploaded shortly after my request. Thank you so much. You are really good at explaining things.
Prof crane trolling us with the water bottle exercise :P
Thank you so much for these lectures!
Video on Lie group and geometric algebra, plz.
can anyone tell me what does “w/" mean ?
It's a common abbreviation for "with" in North American English: english.stackexchange.com/questions/115367/are-w-o-w-b-c-common-abbreviations-in-the-us
@@keenancrane thank you very much
Professor, could you plz make a video to show how the last 2 of 5 ways of rotation representation work. I even tried the group theory tutorial, but it's literally too far beyond me.
nice water bottle example🤣