Are there any cubic bezier curve that cannot imitate by multiple quadratic bezier curve?

Question

I want to make a line curve system with bezier curve. And I want to use only quadratic bezier curve so it can be extend and control easily, it can add control point anywhere and more intuitive

But I'm curious that it might not be cover some kind of graph. At least I want it able to imitate cubic bezier curve

Is it any mathematically impossible? I have seen many system use cubic bezier curve over quadratic (System Font and Photoshop Path at least)

Are there any problem with quadratic bezier curve?

It can approximate cubic Bezier curve, and I don't know what you means by "imitate". — user202729, May 26 '16 at 04:20
@user202729 I mean not just approximate but really equal, by using less than four composite quadratic parts — Thaina, May 26 '16 at 04:59

score 3 · Accepted Answer · answered May 27 '16 at 01:13

You can only approximate a cubic Bezier curve by multiple quadratic Bezier curves. They will not be equal as there will always be a small difference between them.

A cubic Bezier curve has the following advantages over quadratic Bezier curve: $C^2$ continuous, can have inflection point and can have separate control over the end slopes. These are the reason they are used much more widely than quadratic Bezier curves.

score 1 · Answer 2 · answered Jul 25 '20 at 13:41

In 3D there is another fundamental difference between cubic Bezier and quadratic Bezier curves. Any quadratic Bezier segment is defined by three control points, and by the convex hull property, the whole segment is therefore flat - it belongs to the plane defined by the three control points. So if the four control points of the cubic Bezier segment are not coplanar, any part of it will not be flat, and it will never be possible to exactly represent it by a flat quadratic Bezier.

Are there any cubic bezier curve that cannot imitate by multiple quadratic bezier curve?

2 Answers2