Most surfaces probably don't have C1 continuity everywhere right? If a surface were to have C1 continuity everywhere, then it'd basically be collinear everywhere, which would mean that it's just a flat plane. I was just a bit confused as first, since many functions are C1 continuous everywhere (derivative is continuous everywhere), so I would've thought that the same would apply to well-behaved surfaces.
mnicoletti15
I would like to point out to the commenter above that it is not true that if a surface is C1 then it is a plane. Either that or I am not sure what definition of a C1 surface we are using. But there are many examples, e.g. a torus is a C-infinity surface (obviously not flat).
Staffpsmanohar
Loads of surfaces are C1 continuous. Pretty much every surface that feels "smooth" is going to be C1 continuous (or even C∞ continuous). Take a sphere, for example (or a torus as mnicoletti suggests).
Keep in mind that the collinearity on this slide is for the control points, which aren't necessarily on the surface.
Staffirisli
Curve continuity is sometimes subtle but also nice to drool over and actually has aesthetic importance in industrial design. For example, Apple products have smooth curves. Even though the Apple example is 2D, you can see how it translate to higher dimensions.
Most surfaces probably don't have C1 continuity everywhere right? If a surface were to have C1 continuity everywhere, then it'd basically be collinear everywhere, which would mean that it's just a flat plane. I was just a bit confused as first, since many functions are C1 continuous everywhere (derivative is continuous everywhere), so I would've thought that the same would apply to well-behaved surfaces.
I would like to point out to the commenter above that it is not true that if a surface is C1 then it is a plane. Either that or I am not sure what definition of a C1 surface we are using. But there are many examples, e.g. a torus is a C-infinity surface (obviously not flat).
Loads of surfaces are C1 continuous. Pretty much every surface that feels "smooth" is going to be C1 continuous (or even C∞ continuous). Take a sphere, for example (or a torus as mnicoletti suggests).
Keep in mind that the collinearity on this slide is for the control points, which aren't necessarily on the surface.
Curve continuity is sometimes subtle but also nice to drool over and actually has aesthetic importance in industrial design. For example, Apple products have smooth curves. Even though the Apple example is 2D, you can see how it translate to higher dimensions.
https://medium.com/tall-west/no-cutting-corners-on-the-iphone-x-97a9413b94e