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QLong wrote:Do we have to insert t3 to the t knot vector of this blending function?
QLong wrote:Is it possible to have two sets of knot vectors at the same point?
Figure 9.d. shows the compatible blending function at [s3,t2]. Once t3 has been inserted into the blending function with t knot vector [t0 t1 t2 t4 t5] then we can sum the blending functions together to create a single blending function located at P2. Summing blending functions together to yield a single blending function at each point is required to have a valid mesh.
Nicholas North wrote:QLong wrote:Is it possible to have two sets of knot vectors at the same point?
As an intermediate state, yes, logically there may be multiple blending functions (with different knot vectors) at given point at once. Once all of these blending functions have matching knot vectors then we can just sum them together to get a single blending function for that point. This is implied, but not specifically stated in the paper (as far as I can tell).
Lets look at our example:
As you said, the addition of P2 requires that we insert s3 into the blending function at [s4,t2]. The portion of this blending function that is "donated" to P2 has knot vectors [s1 s2 s3 s4 s5 ], [t0 t1 t2 t4 t5].
On the other side of P2 we have the blending function centered at [s2,t2]. After inserting s3, a blending function with knot vectors [s1 s2 s3 s4 s5], [t0 t1 t2 t3 t4] is contributed to the blending function at P2.
So we have two types of blending functions that are donated to the blending function at P2, one with t knot vector [t0 t1 t2 t4 t5] and one with [t0 t1 t2 t3 t4]. To make these compatible, we must insert t3 into the blending function that lacks it. This gives rise to the required extra point at [s3,t3] (via Violation 2).
Figure 9.d. shows the compatible blending function at [s3,t2]. Once t3 has been inserted into the blending function with t knot vector [t0 t1 t2 t4 t5] then we can sum the blending functions together to create a single blending function located at P2. Summing blending functions together to yield a single blending function at each point is required to have a valid mesh.
Hopefully this helps.
 Nick
oliver wrote:Has been settled...
Nicholas North wrote:QLong wrote:Do we have to insert t3 to the t knot vector of this blending function?
The short answer is No. I'll do my best to explain a bit more.
To help, I've attached an image of Figure 9 from the paper. I'll go through the steps illustrated in Figure 9 and explain what is happening.
I hope this helps.
 Nick
simbaforrest wrote:In the 2004 paper about the local refinement, am I correct that the WHOLE purpose of ensuring no violations is just to make sure the Tspline surface's shape will NOT change after inserting the knot/control points?
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