Lecture 7: Intro to Geometry, Splines, and Bezier Curves (44)

prannaypradeep999

Would the approach for higher order hermite basis just include a larger matrix but the same general process? We probably would have to specific another constraint equation, or else we would have too many unknowns.

Staffjamesfobrien

Add constraints on the second derivatives at each end and use a degree 5 polynomial. The derivation is basically the same as for cubic. Try it!

lwg0320

Is my understanding correct? For higher order hermite basis, we will have a more "accurate" curve due to being able to use second, third... derivatives. The cubic hermite basis is the baseline for forming a curve approximation so you cannot have a hermite basis that include a smaller matrix. Is there an exception? For example when the derivative of the start and end is equal to 0?

longh2000

@lwg0320 I think your understanding is right. We cannot have a smaller matrix. remember that we need to inverse matrix during derivation of the hermite basis so the matrix must be a square. If it is 3 by 3 then it can only be expressing 3 constraints on our desired curve. However, even when derivative at both points are the same, we still have four equations to constrain the polynomial, making the matrix at least 4 by 4.

Would the approach for higher order hermite basis just include a larger matrix but the same general process? We probably would have to specific another constraint equation, or else we would have too many unknowns.

Add constraints on the second derivatives at each end and use a degree 5 polynomial. The derivation is basically the same as for cubic. Try it!

Is my understanding correct? For higher order hermite basis, we will have a more "accurate" curve due to being able to use second, third... derivatives. The cubic hermite basis is the baseline for forming a curve approximation so you cannot have a hermite basis that include a smaller matrix. Is there an exception? For example when the derivative of the start and end is equal to 0?

@lwg0320 I think your understanding is right. We cannot have a smaller matrix. remember that we need to inverse matrix during derivation of the hermite basis so the matrix must be a square. If it is 3 by 3 then it can only be expressing 3 constraints on our desired curve. However, even when derivative at both points are the same, we still have four equations to constrain the polynomial, making the matrix at least 4 by 4.