Gonna take a stab at this question, since 3D surfaces can not be continuous, for example a jagged point, is it fair to say we will not always get tangent continuity?
keenan
Try thinking about it in terms of this slide: how many conditions need to hold in order for two Bézier patches to line up at the boundary, with both positions and tangents matching?
For curves we said:
positions at the two endpoints have to agree (two constraints)
tangents at the two endpoints have to agree (two more constraints)
the shape of the curve is determined by the four control points p0, p1, p2, p3 (four degrees of freedom)
So in total we have four degrees of freedom, and four (linear) constraints. Hence, we can always ensure that two cubic Bézier curves meet with tangent continuity.
What about Bézier patches? You should try playing the same game here:
How many constraints are needed for control points to match at patch boundaries?
How many constraints are needed to ensure tangents agree at control points along the boundary?
How many degrees of freedom total do you have for a patch?
This slide should help. You may also want to think about why ensuring that positions and tangents agree at control points is enough to ensure that they agree all along the boundary.
genericname
At each point, the boundary is specified by 4 control points multiplied by 4 sides for 16 constraints. From the second slide, we see there are 16 degrees of freedom, 4 for each patch. Is that the logic?
keenan
Careful: (i) you're double-counting the boundary control points at the four corners and (ii) you're not considering the additional question of whether the tangents agree along the boundary (rather than just the positions).
Gonna take a stab at this question, since 3D surfaces can not be continuous, for example a jagged point, is it fair to say we will not always get tangent continuity?
Try thinking about it in terms of this slide: how many conditions need to hold in order for two Bézier patches to line up at the boundary, with both positions and tangents matching?
For curves we said:
So in total we have four degrees of freedom, and four (linear) constraints. Hence, we can always ensure that two cubic Bézier curves meet with tangent continuity.
What about Bézier patches? You should try playing the same game here:
This slide should help. You may also want to think about why ensuring that positions and tangents agree at control points is enough to ensure that they agree all along the boundary.
At each point, the boundary is specified by 4 control points multiplied by 4 sides for 16 constraints. From the second slide, we see there are 16 degrees of freedom, 4 for each patch. Is that the logic?
Careful: (i) you're double-counting the boundary control points at the four corners and (ii) you're not considering the additional question of whether the tangents agree along the boundary (rather than just the positions).