In these slides, C(n) continuity means that the nth derivative is continuous. For example, C1 continuity here means that the 1st derivative of the parametric equation for this curve is continuous. Likewise, C2 continuity means that the 2nd derivative is continuous, and C0 continuity means that the equation itself is continuous.
jchen12197
Thank you for the explanation! I wasn't sure what the "C(n) continuity" was specifically referring to. I wasn't really understanding the differences between C0 and C1 continuity, but it makes sense that it's referring to the continuity of the nth derivative! I'm still not sure what the a and b in the equation represent though?
GitMerlin
Why does a_n has to be in the middle of a_{n-1} and b_1 in order to achieve C1 continuity? I thought Bezier curve guarantees the derivative at endpoint to be towards the second point. In that way, don't we just need a_{n-1}, a_n, b_1 to be collinear to achieve C1 continuity?
I found a note on the continuity of Bezier curve, https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/Bezier/bezier-der.html. In order to achieve C1 continuity, two tangent vectors of two adjacent cubic Bezier curves need to be identical.
In these slides, C(n) continuity means that the nth derivative is continuous. For example, C1 continuity here means that the 1st derivative of the parametric equation for this curve is continuous. Likewise, C2 continuity means that the 2nd derivative is continuous, and C0 continuity means that the equation itself is continuous.
Thank you for the explanation! I wasn't sure what the "C(n) continuity" was specifically referring to. I wasn't really understanding the differences between C0 and C1 continuity, but it makes sense that it's referring to the continuity of the nth derivative! I'm still not sure what the a and b in the equation represent though?
Why does a_n has to be in the middle of a_{n-1} and b_1 in order to achieve C1 continuity? I thought Bezier curve guarantees the derivative at endpoint to be towards the second point. In that way, don't we just need a_{n-1}, a_n, b_1 to be collinear to achieve C1 continuity?
I found a note on the continuity of Bezier curve, https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/Bezier/bezier-der.html. In order to achieve C1 continuity, two tangent vectors of two adjacent cubic Bezier curves need to be identical.