[atoav]

A very similar thing is done in the creation of reflective surfaces on car bodies (typically in CAD software).

They call these constraints by G and a number.

  G1 would be a positional constraint: the two surfaces meet each other at the same point
  
  G2 tangential: same as G1, but the surfaces are tangential
  
  G3: same as G2, but the curvature (radius^-1) of the surfaces is the same at the point where the two meet. This essentially means the curvature combs of the surfaces shall meet at the same position (G1)
  
  G4: same as G3, only now the meeting curvature combs have to be tangential as well
  
  G5: same as G4, only now the curvature combs of the curvature combs have to meet at the same position

And so on. The goal is to create smooth transitions between two separate mathematical surfaces that cannot be seen in the reflections in the sheet metal. E.g. if you think about the connection of straight sheet of metal (curvature: 0) and a cylindrical surface (curvature: 1/radius) the curvature will go from zero to some different value immidiately on the transation you will definitly see this as a hard corner on the reflection or when light falls onto the surface.

[johnwalkr]

A simple example of this is the squircle. This page [1] has a couple of nice images that are easy to understand.

https://99percentinvisible.org/article/circling-square-desig...

[danachow]

Minor point (well in this case at least) but you have an off by one error. Your G1 is G0.

Here is how it is defined in terms of basis vectors. https://people.eecs.berkeley.edu/~jfc/cs184f98/lec19/lec19.h...

[atoav]

Ah it was a while ago thanks for the correction.

[thaumasiotes]

> G2 tangential: same as G1, but the surfaces are tangential

This makes me think "tangential to what?".

Do you mean that, along the seam between G1 and G2, the tangent plane to G1 at a given point is equal to the tangent plane to G2 at the same point?

[swimfar]

Here's a good visual explanation: https://blogs.lt.vt.edu/2ndyearidlab/author/hohsuan/page/2/

[avianlyric]

Yep, exactly that. Removes the appearance of a “fold” or “crease” at the surface transition, and makes the surface smooth and continuous.