Suppose I splitted a cubic bezier curve at the point corresponding to the parameter t=0.5. Call the splitted parts C and D. Then I connect ends of C by a straight line segment. Can we say that this straight line segment never touches C apart from the endpoints?
Splitted Bezier curve
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No. This is because the split portion C and D are also a cubic Bezier curve, which can be of any shape (that a cubic polynomial curve can be). A straight line connecting two ends of a cubic Bezier curve could have intersected the curve at the interior if the cubic Bezier curve is not convex.
If you want to produce an example, you will have to make the control points going up and down drastically as shown in the following example (where the green dots are control points and the yellow dot is where t=0.5).
A close-up on where t=0.5 shows
No, because as I understand it, a straight line is a degenerate case of a Bezier curve.
So you would need to ensure, for example, that the control points are not on the line.
If the also happens to be that both points are on the same side of the line (e.g. either both above, both below, both to the left or both to the right), then that would do it.
If you don't have the control points, but have two points near the endpoints of the line, you can do the same test.
Or in other words the case where you might need to worry about is when either the control points or points near the endpoints are on in different side of the line between the endpoints.