Outline

• Blossoms
• Derivatives of B-spline Curves
• Knot Insertion

Introduction

• Last Tuesday, we proved basic facts about the blending functions of B-spline curves such as that they are a partition of unity.
• We then discussed the de Boor Algorithm as an analog of de Casteljau's method for B-splines.
• Today, we begin by looking at how to get control points of a B-spline from the polynomial that defines it between two knots.
• We will then look at how to determine the derivative of a B-spline and knot insertion.

Blossoms

• Suppose `\vec q(u)` is a degree `k` B-spline curve and `u_s < u_(s+1)` are two knots. Between these knots the curve `\vec q(u)` is defined by a single polynomial which we will call `\vec f(u)`.
• We are going to define a new function `\vec b(x_1, x_2, \ldots, x_k)` that takes `k` real numbers as arguments but has the diagonal property that
  `\vec b(u,u, \ldots, u) = \vec f(u)`.
• This function `\vec b(x_1, x_2, \ldots, x_k)` will be called the blossom of `\vec f`.
• Blossoms will be useful for determining control points based on the defining polynomials of B-splines between knots. They will also be useful for calculating derivatives of B-splines, doing degree elevation, etc.
• In fact, it is possible to develop the theory of B-splines using Blossoms as the starting point.

Two Properties Blossoms will Satisfy

• Symmetry Property: Changing the order of the inputs to `\vec b` will not change `\vec b`'s value. i.e., if `\pi` is a permutation (a 1-1, onto function from `{1,\ldots, k}` to `{1,\ldots, k}`) on the indices, then
  `\vec b(x_\pi(1), x_\pi(2),\ldots, x_\pi(k)) = \vec b(x_1, x_2, \ldots, x_k)`.
  A function with this property is called a symmetric function.
• Multiaffine Property: For any scalars `\alpha, \beta` with `\alpha+\beta = 1`, the blossom will satisfy:
  `\vec b(\alpha x_1 + \beta x'_1, x_2, \ldots, x_k) = \alpha\vec b(x_1, x_2, \ldots, x_k)+ \beta \vec b(x'_1, x_2, \ldots, x_k)`.
  By the symmetry property, the same property holds for the other inputs `x_2, \ldots, x_k`.

Defining Blossoms

• Before defining what exactly a blossom is, we need some notation.
• For `k > 0`, we write `[k]` for the set `{1,2, \ldots, k}`. For `J` a subset of `[k]`, we define the term `x_J` to be the product
  `x_J = \prod_(j\in J)x_j`.
  So if `J = {1, 3, 6}` then `x_J = x_1x_3x_6`. For the empty set, `x_{\emptyset} = 1`.

Definition. Let `\vec q` have degree `\leq k` so that
`\vec q(u) = \vec r_k u^k + \vec r_(k-1) u^(k-1) + \cdots + \vec r_2 u^2 + \vec r_1 u + \vec r_0`
We define the degree k blossom of `\vec q(u)` to be the `k` variable polynomial
`\vec b(x_1, \ldots, x_k) = \sum_(i=0)^k \sum_((J\subset [k]),(|J|=i))((k),(i))^(-1)\vec r_ix_J`,
where `|J|` denotes the cardinality of J.

Remarks

• It follows from the definition of blossom that it will be symmetric: A permutation will just map one subset of `i` elements of [k] to another. But since we are summing over all subsets, we will end up with the same sum.
• Also, each term `x_J` has at most one factor of any `x_i`, so `\vec b` is degree one in each variable separately and thus is affine in that variable. So the multi-affine property will hold.
• Since there are `((k),(i))` many subsets `J` of `[k]` of size `i`, the degree `i` term in `\vec b(u, u, \ldots, u)` will be
  `\sum_((J\subset [k]),(|J|=i))((k),(i))^(-1) \vec r_i u^i=((k),(i))\cdot ((k),(i))^(-1) \vec r_i u^i = \vec r_i u^i`.
  Hence, `\vec b(u, u, \ldots, u) = \vec q(u)`.
• As an example of blossoms, let `\vec q(u)` be the quadratic curve
  `\vec q(u) = \vec au^2 +\vec bu +\vec c.`
  Then, the degree two blossom of `\vec q(u)` is the polynomial `b(x_1, x_2) = \vec ax_1x_2 + 1/2\vec b(x_1 + x_2) + \vec c`.
• We can also define if we like the degree 3 blossom of `\vec q(u)`, we just view `\vec q(u)` as being a degree three polynomial with leading coefficient zero. The blossom is then
  `b(x_1, x_2, x_3) = 1/3\vec a(x_1x_2 + x_1x_3 + x_2x3) + 1/3vec b(x_1 + x_2 +x_3) + \vec c`.

Central Property of Blossoms

Theorem. Let `\vec q(u)` be a degree `k`, order `m=k+1` B-spline curve with knot vector `[u_0, \ldots, u_(n+m)]` and control points `\vec p_0, \ldots, p_n`. Suppose `u_s < u_(s+1)`, where `k \leq s \leq n`. Let `\vec q(u)` be equal to the polynomial `\vec q_s(u)` for `u\in[u_s, u_(s+1))`. Let `\vec b(x_1, \ldots, x_k)` be the blossom of `\vec q_s(u)`. Then the control points `\vec p_(s-k), \ldots, \vec p_s` are equal to
`\vec p_i = \vec b(u_(i+1),u_(i+2), \ldots, u_(i+k))`,
for `i=s-k, \ldots, s`.

The proof of this result makes use of the de Boor algorithm of last day. Its proof makes use of the next lemma, which is proven by induction. Let `u^(\langle i \rangle)` be used to denote `i` occurrences of the parameter `u`. Let `u_([i,j])` denote the sequence of values `u_i, u_(i+1), \ldots, u_j`.

Lemma. Suppose `\vec p_(s-k+l) = \vec b(u_([s-k+l+1, s+l]))` (this is just the result of the theorem re-indexed) holds for all `l= 0, \ldots, k`. Then for `j= 0, \ldots, k` and `l = 0, \ldots, k-j`,
`\vec p_(s-k+j+l)^((j)) = \vec b(u_([s-k+l+1, s+l]), u^(\langle j \rangle))`.

Derivative and Smoothness of B-spline curves

We next give the formula for the derivative of a degree `k` B-spline curve as a B-spline curve of degree `k-1`.

Theorem. Let `\vec q(u)` be a degree `k= m - 1` B-spline curve with control points `\vec p_0, \ldots, \vec p_n`. Then its first derivative is
`\vec q'(u) = \sum_(i=1)^(n) k N_(i,k)\frac{\vec p_i - \vec p_(i-1)}{u_(i+k) - u_i}.`
In particular, `\vec q'(u)` is the degree `k-1` B-spline curve with control points equal to
`p_i^* = \frac{k}{u_(i+k) - u_i}(\vec p_i - \vec p_(i-1))`.

The proof of this is in stages and I give just the barest outline: We first get a formula for the derivative of a polynomial curve `\vec f(u)` in terms of a degree `k-1` blossom `\vec b^*`. Then one uses the Theorem of the previous slide to use this blossom to define control points for this curve. This result is then applied to each piece of `\vec q(u)` to get the results for points which are not knots. To handle the knot case one uses the limit results we discussed last day.

Knot Insertion

• It is often useful to be able to add knots to a B-spline curve without initially changing its shape.
• For instance, in a CAD program one, might add knots to make additional adjustments on a portion of the curve.
• We will consider the Bohm method of knot insertion...
• Suppose `\vec q(u)` is an order `m`, degree `k = m-1`, B-spline curve defined with knot vector `[u_0, \ldots, u_(n+m)]` and control points `\vec p_0, \ldots, \vec p_n`.
• Suppose we want to insert a knot `\hat u` where `u_s \leq \hat u \leq u_(s+1)`.
• We will write `[\hat u_0, \ldots, \hat u_(n+m+1)]` for the new knot vector. So `\hat u_i` will equal `u_i` if `i\leq s`, will equal `\hat u` when `i=s+1` and will equal `u_i-1` for `i > s+1`.
• Then the Bohm algorithm says the following control points will give the same curve:
  `\vec\hat p_i = {(\vec p_i, \text{ if }i\leq s-k), (\frac{u_(i+k) - \hat u}{u_(i+k) - u_i}\vec p_(i-1) + \frac{\hat u - u_i}{u_(i+k) - u_i}\vec p_i, \text{ if } s-k < i \leq s), (\vec p_(i-1), \text{otherwise}) :}`

This is proven by showing the blossom of `\vec q(u)` on `[u_s, u_(s+1))` is equal to the blossom of `\vec\hat q(u)` on the intervals `[u_s, \hat u)` and `[\hat u, u_(s+1))`.