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Is it bad that I don't understand this after reading it? I'm not even sure why "lie" he's referring to about bezier curves.

by rosstex 1mo ago

yorwba 1mo ago
The "lie" is that some Bézier-spline drawing programs let you restrict on-curve points to a "symmetric" configuration where the control points are at an equal distance on opposite sides of the on-curve point, but this doesn't necessarily make the curve itself more symmetric or more smooth. The proposed alternative is to control the bend radius instead, and ensure that the curve bends the same on both sides.
- adrian17 1mo ago
  It’s trading one property for another, isn’t it? Standard beziers with symmetric handles are C1 continuous but not necessarily G2 continuous; author’s scheme is the other way around, which makes them better for some use cases and worse for others.
  - unconed 1mo ago
    The C1 continuity you refer to is relative to the parametrization t=0..1 per segment. This is useless in practice. For e.g. animation purposes, you need to reparametrize to arc length which destroys the apparent C1 continuity.
dahart 1mo ago
The author was having a little fun calling it a “lie”, it’s more like a popular simplification. The question is how to join Bézier curves smoothly, and it’s fairly common for people to join them by matching the velocity, which means if you join two cubic segments (for example) with control points [a,b,c,d] and [d,e,f,g] where you’re joining at point d, then setting d-c == e-d, which puts c,d,e in a line with d as the midpoint, will connect the two curves smoothly. Lots of Bézier curve editing software will have a mode that enforces symmetric control points around the join point. The problem, or “lie”, is that this symmetric strategy doesn’t match the 2nd derivative, so there can be an abrupt change in curvature at point d. You can have a smoother join by adding constraints like curvature or 2nd or 3rd derivative matching, and that’s what the article goes into - one way of achieving smoother matches. This WP entry mentions several other alternatives to smooth joins besides matching curvature: https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#Smoothly_joining_cubic_B%C3%A9ziers https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#Sm...

Is it bad that I don't understand this after reading it? I'm not even sure why "lie" he's referring to about bezier curves.

by rosstex 1mo ago

Is it bad that I don't understand this after reading it? I'm not even sure why "lie" he's referring to about bezier curves.

yorwba 1mo ago

The "lie" is that some Bézier-spline drawing programs let you restrict on-curve points to a "symmetric" configuration where the control points are at an equal distance on opposite sides of the on-curve point, but this doesn't necessarily make the curve itself more symmetric or more smooth. The proposed alternative is to control the bend radius instead, and ensure that the curve bends the same on both sides.

adrian17 1mo ago
It’s trading one property for another, isn’t it? Standard beziers with symmetric handles are C1 continuous but not necessarily G2 continuous; author’s scheme is the other way around, which makes them better for some use cases and worse for others.
- unconed 1mo ago
  The C1 continuity you refer to is relative to the parametrization t=0..1 per segment. This is useless in practice. For e.g. animation purposes, you need to reparametrize to arc length which destroys the apparent C1 continuity.

dahart 1mo ago

The author was having a little fun calling it a “lie”, it’s more like a popular simplification. The question is how to join Bézier curves smoothly, and it’s fairly common for people to join them by matching the velocity, which means if you join two cubic segments (for example) with control points [a,b,c,d] and [d,e,f,g] where you’re joining at point d, then setting d-c == e-d, which puts c,d,e in a line with d as the midpoint, will connect the two curves smoothly. Lots of Bézier curve editing software will have a mode that enforces symmetric control points around the join point. The problem, or “lie”, is that this symmetric strategy doesn’t match the 2nd derivative, so there can be an abrupt change in curvature at point d. You can have a smoother join by adding constraints like curvature or 2nd or 3rd derivative matching, and that’s what the article goes into - one way of achieving smoother matches. This WP entry mentions several other alternatives to smooth joins besides matching curvature: https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#Smoothly_joining_cubic_B%C3%A9ziers https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#Sm...