A 2-Norm Condition Number for Bézier Curve Intersection

1 Introduction

The problem of intersecting two planar Bézier curves is an important one in Computer Aided Geometric Design (CAGD). Many intersection algorithms have been described in the literature, both geometric ([SP86, SN90, KLS98]) and algebraic ([MD92]). Though the general convergence properties of these algorithms have been studied (e.g. [Sch09]), no condition number for the intersection problem has been described in the CAGD literature¹¹1As far as the author has been able to tell. In many CAGD textbooks, there is a long review of methods for intersecting two planar Bézier curves (e.g. [Far01, Sed16]) but no mention of conditioning..

There are more generic condition numbers for rational polynomial systems ([HT15]) or nonlinear algebraic systems ([Hig02, Chapter 25, Section 25.4]). However, the condition numbers with an algebraic focus (rather than an analytic one) often require too much computation to be useful. The numerical analytic condition numbers are in some ways too general to be useful for planar Bézier curve intersection.

In this paper, we describe a simple relative root condition number for this intersection problem. Since tangent intersections are to transversal intersections as multiple roots are to simple roots of a function, this condition number is infinite for non-transversal intersections. We present a few examples verifying that the condition number increases as a family of intersections approach an ill-behaved intersection.

2 Preliminaries

Throughout, we’ll refer to a Bézier curve as a parametric plane curve given by

b (s) = n \sum j = 0 b_{j} B_{j, n} (s)

(2.1)

where $B_{j, n} (s) = (\frac{n}{j}) (1 - s)^{n - j} s^{j}$ is a Bernstein polynomial. When the parameter $s \in [0, 1]$ , the coefficients $B_{j, n} (s) \in [0, 1]$ as well and the evaluation is a convex combination of the set of control points $b_{j} \in R^{2}$ .

A Bézier curve intersection is a root $[\begin{matrix} α β \end{matrix}]$ of the function

F (s, t) = b_{0} (s) - b_{1} (t)

(2.2)

where $b_{0} (s)$ and $b_{1} (t)$ are Bézier curves. Note that $F : R^{2} ⟶ R^{2}$ .

Each component $x (s)$ and $y (s)$ of a Bézier curve is a polynomial in Bernstein form. For such a polynomial

p (s) = n \sum j = 0 p_{j} B_{j, n} (s)

(2.3)

the (absolute) condition number of evaluation is $˜ p (s) = \sum_{j = 0}^{n} ∣ ∣ p_{j} ∣ ∣ B_{j, n} (s)$ ([FR87]) when the parameter $s$ is in the unit interval.

3 Conditioning of Generic Root-finding

Consider a smooth function $F : R^{n} ⟶ R^{n}$ with Jacobian $F_{x} = J$ . We want to consider a special class of functions of the form $F (x) = \sum_{j} c_{j} ϕ_{j} (x)$ where the basis functions $ϕ_{j}$ are also smooth functions on $R^{n}$ and each $c_{j} \in R$ . We want to consider the effects on a root $α \in R^{n}$ of a perturbation in one of the coefficients $c_{j}$ . We examine the perturbed function

G (x, δ) = F (x) + δ ϕ_{j} (x) .

(3.1)

Since $G (α, 0) = 0$ , if $J^{- 1}$ exists at $x = α$ then the implicit function theorem tells us that we can define $x$ via

G (x (δ), δ) = 0 .

(3.2)

Taking the derivative with respect to $δ$ we find that $0 = G_{x} x_{δ} + G_{δ}$ . Plugging in $δ = 0$ we find that $0 = J (α) x_{δ} + ϕ_{j} (α)$ , hence we conclude that

x (δ) = α - J {(α)}^{- 1} ϕ_{j} (α) δ + O (δ^{2}) .

(3.3)

This gives

\frac{∥ ∥ J {(α)}^{- 1} ϕ_{j} (α) ∥ ∥}{∥ α ∥} .

(3.4)

as the relative condition number for a perturbation in $c_{j}$ . By considering perturbations in all of the coefficients: $∣ ∣ δ_{j} ∣ ∣ \leq ε ∣ ∣ c_{j} ∣ ∣$ , a similar analysis gives a root function

x (δ_{0}, \dots, δ_{n}) = α - J {(α)}^{- 1} n \sum j = 0 δ_{j} ϕ_{j} (α) + O (ε^{2}) .

(3.5)

With this, we can define a root condition number

Definition 3.1.

For a smooth function $F (x) = \sum_{j} c_{j} ϕ_{j} (x)$ parameterized by the coefficients ${c_{j}}$ with root $α$ and Jacobian $J (α)$ , we define a relative root condition number

κ_{α} = limsup ε \to 0 \frac{∥ δ α ∥ / ε}{∥ α ∥} = lim ε \to 0 ⎛ ⎝ sup ∣ ∣ δ_{j} ∣ ∣ \leq ε ∣ ∣ c_{j} ∣ ∣ \frac{∥ ∥ J {(α)}^{- 1} \sum_{j} δ_{j} ϕ_{j} (α) ∥ ∥ / ε}{∥ α ∥} ⎞ ⎠ .

(3.6)

In [Hig02, Chapter 25, Section 25.4]

a similar definition is given. Instead of bounding the perturbations component-wise, it bounds the entire perturbation vector

lim ε \to 0 ⎛ ⎝ sup ∥ δ ∥ \leq ε ∥ c ∥ \frac{∥ ∥ J {(α)}^{- 1} \sum_{j} δ_{j} ϕ_{j} (α) ∥ ∥ / ε}{∥ α ∥} ⎞ ⎠ = ∥ ∥ J^{- 1} F_{c} ∥ ∥ \frac{∥ c ∥}{∥ α ∥} .

(3.7)

While this has the benefit of having a closed form that is straightforward to compute, it may be less useful than the condition number given in Definition 3.1 since the larger coefficients can dominate the rest.

3.1 One-dimensional Case

When $n = 1$ , due to the triangle inequality:

| δ α | = ∣ ∣ ∣ ∣ J^{- 1} n \sum j = 0 δ_{j} ϕ_{j} (α) ∣ ∣ ∣ ∣ \leq \frac{1}{| F^{'} (α) |} n \sum j = 0 ∣ ∣ δ_{j} ϕ_{j} (α) ∣ ∣ .

(3.8)

The sign and magnitude of each $δ_{j}$ can be chosen to make $δ_{j} ϕ_{j} (α) = ∣ ∣ c_{j} ϕ_{j} (α) ∣ ∣ ε$ , hence for these values equality holds in the triangle inequality:

∣ ∣ ∣ ∣ n \sum j = 0 δ_{j} ϕ_{j} (α) ∣ ∣ ∣ ∣ = ε n \sum j = 0 ∣ ∣ c_{j} ϕ_{j} (α) ∣ ∣ .

(3.9)

Thus we get a root condition number for a polynomial given in Bernstein form

κ_{α} = \frac{1}{| α F^{'} (α) |} n \sum j = 0 ∣ ∣ c_{j} ϕ_{j} (α) ∣ ∣ .

(3.10)

that agrees with the common definition ([FR87]) when the $ϕ_{j}$ are specialized to Bernstein basis functions.

4 Conditioning of Bézier Curve Intersection

To define a condition number for the intersection of two planar Bézier curves, we write the difference as

F (s, t) = [\begin{matrix} x_{0} (s) 0 \end{matrix}] + [\begin{matrix} 0 y_{0} (s) \end{matrix}] - [\begin{matrix} x_{1} (t) 0 \end{matrix}] - [\begin{matrix} 0 y_{1} (t) \end{matrix}] .

(4.1)

We can show that there is a closed form for the condition number given by Definition 3.1, specialized to the 2-norm.

Theorem 4.1.

Let $[\begin{matrix} α β \end{matrix}]$ be a transversal intersection of two planar Bézier curves $b_{0} (s)$ and $b_{1} (t)$ . Define the vectors

{˜ F}_{+} = [\begin{matrix} {˜ x}_{0} (α) + ˜ x_{1} (β) {˜ y}_{0} (α) + {˜ y}_{1} (β) \end{matrix}] % a n d {˜ F}_{-} = [\begin{matrix} {˜ x}_{0} (α) + {˜ x}_{1} (β) - {˜ y}_{0} (α) - {˜ y}_{1} (β) \end{matrix}] .

(4.2)

Then the root condition number of the intersection is the greater of two vector norms:

(4.3)

Proof.

Let the curve $b_{0} (s)$ be degree $m$ and $b_{1} (t)$ be degree $n$ . Then $F (s, t)$ can be written as $2 (m + 1) + 2 (n + 1)$ terms:

F (s, t) = m \sum i = 0 c_{i}^{(1)} [\begin{matrix} B_{i, m} (s) 0 \end{matrix}] + m \sum i = 0 c_{i}^{(2)} [\begin{matrix} 0 B_{i, m} (s) \end{matrix}] + n \sum j = 0 c_{j}^{(3)} [\begin{matrix} - B_{j, n} (t) 0 \end{matrix}] + n \sum j = 0 c_{j}^{(4)} [\begin{matrix} 0 - B_{j, n} (t) \end{matrix}] .

(4.4)

Since $F (s, t) = b_{0} (s) - b_{1} (t)$ we have Jacobian . Since a transversal intersection, we have $det J (α, β) \neq 0$ . In a perturbation, we replace each $c_{j}^{(k)}$ with a $δ_{j}^{(k)}$ bounded by $∣ ∣ δ_{j}^{(k)} ∣ ∣ \leq ε ∣ ∣ c_{j}^{(k)} ∣ ∣$ . Writing $J^{- 1} = [\begin{matrix} v & w \end{matrix}]$ , we have

J {(α)}^{- 1} \sum j δ_{j} ϕ_{j} (α) = [m \sum i = 0 δ_{i}^{(1)} B_{i, m} (α) + n \sum j = 0 δ_{j}^{(3)} B_{j, n} (β)] v + [m \sum i = 0 δ_{i}^{(2)} B_{i, m} (α) + n \sum j = 0 δ_{j}^{(4)} B_{j, n} (β)] w = ν_{1} v + ν_{2} w

(4.5)

where

	$\| ν_{1} \| / ε$	$\leq m \sum i = 0 ∣ ∣ c_{i}^{(1)} ∣ ∣ B_{i, m} (α) + n \sum j = 0 ∣ ∣ c_{j}^{(3)} ∣ ∣ B_{j, n} (β) = {˜ x}_{0} (α) + {˜ x}_{1} (β) = μ_{1}$		(4.6)
	$\| ν_{2} \| / ε$	$\leq m \sum i = 0 ∣ ∣ c_{i}^{(2)} ∣ ∣ B_{i, m} (α) + n \sum j = 0 ∣ ∣ c_{j}^{(4)} ∣ ∣ B_{j, n} (β) = {˜ y}_{0} (α) + {˜ y}_{1} (β) = μ_{2} .$		(4.7)

As in (3.8), the bound can be attained by choosing the sign and magnitude of each perturbation so that $δ_{j}^{(k)} B_{j, d} = ε ∣ ∣ c_{j}^{(k)} ∣ ∣ B_{j, d}$ . The factor of $ε$ can be cancelled to give condition number

	$κ_{α, β}$	$= \frac{1}{\sqrt{α^{2} + β^{2}}} sup \| ν_{k} \| \leq μ_{k} {∥ ν_{1} v + ν_{2} w ∥}_{2}$		(4.8)
		$= \sqrt{\frac{{sup}_{\| ν_{k} \| \leq μ_{k}} ν_{1}^{2} (v \cdot v) + 2 ν_{1} ν_{2} (v \cdot w) + ν_{2}^{2} (w \cdot w)}{α^{2} + β^{2}}} .$		(4.9)

Now we seek to maximize the objective function $θ (ν_{1}, ν_{2}) = ν_{1}^{2} (v \cdot v) + 2 ν_{1} ν_{2} (v \cdot w) + ν_{2}^{2} (w \cdot w)$ in the rectangle $[- μ_{1}, μ_{1}] \times [- μ_{2}, μ_{2}]$ .

To find interior critical points, we solve the system $θ_{ν_{1}} = θ_{ν_{2}} = 0$ :

[\begin{matrix} 2 v \cdot v & 2 v \cdot w 2 v \cdot w & 2 w \cdot w \end{matrix}] [\begin{matrix} ν_{1} ν_{2} \end{matrix}] = [\begin{matrix} 00 \end{matrix}] .

(4.10)

This system has the unique solution $ν_{1} = ν_{2} = 0$ unless $∥ v ∥_{2} ∥ w ∥_{2} = | v \cdot w |$ . By the Cauchy-Schwarz inequality, this can only occur if $v$ and $w$ are parallel; since $J^{- 1}$ is invertible, we know they are not. Hence $θ (0, 0) = 0$ is the only interior critical point and it is the global minimum.

Along the boundary of the rectangle, we fix one of $ν_{1}$ or $ν_{2}$ and the resulting univariate function is an up-opening parabola. For example, fixing $ν_{2} = c$ gives $θ (ν_{1}, c) = ν_{1}^{2} (v \cdot v) + ν_{1} [2 c (v \cdot w)] + c^{2} (w \cdot w)$ which has positive lead coefficient $∥ v ∥_{2}^{2}$ . The lead coefficient cannot be $0$ since if $v$ were the zero vector we’d have $det J = 0$ . Since $θ$ is an up-opening parabola along the boundary, any critical point must be a local minimum.

Thus we know the maximum occurs at two of the four corners of the rectangle. Due to sign cancellation, this leads to one of two values $θ = μ_{1}^{2} (v \cdot v) \pm 2 μ_{1} μ_{2} (v \cdot w) + μ_{2}^{2} (w \cdot w)$ , the largest of which is $μ_{1}^{2} (v \cdot v) + 2 μ_{1} μ_{2} | v \cdot w | + μ_{2}^{2} (w \cdot w)$ . Thus

κ_{α, β} = \sqrt{\frac{μ_{1}^{2} (v \cdot v) + 2 μ_{1} μ_{2} | v \cdot w | + μ_{2}^{2} (w \cdot w)}{α^{2} + β^{2}}} .

(4.11)

Note that ${˜ F}_{\pm} = [\begin{matrix} μ_{1} \pm μ_{2} \end{matrix}]$ , so

{∥ ∥ J {(α, β)}^{- 1} {˜ F}_{\pm} ∥ ∥}_{2}^{- 1} = {∥ ∥ [\begin{matrix} v & w \end{matrix}] {˜ F}_{\pm} ∥ ∥}_{2} = \sqrt{μ_{1}^{2} (v \cdot v) \pm 2 μ_{1} μ_{2} (v \cdot w) + μ_{2}^{2} (w \cdot w)}

(4.12)

gives each of the two values that produce the maximum value. $■$

4.1 Transversal Intersection

Consider the line $b_{0} (s) = [\begin{matrix} 2 s 2 s \end{matrix}]$ and improperly parameterized line $b_{1} (t) = [\begin{matrix} 4 t^{2} 2 - 4 t^{2} \end{matrix}]$ which intersect at $α = β = 1 / 2$ . At the intersection we have $J^{- 1} = \frac{1}{8} [\begin{matrix} 2 & 2 - 1 & 1 \end{matrix}]$ , so that $v \cdot v = w \cdot w = 5 / 64$ and $v \cdot w = 3 / 64$ . Since the $x$ -component of $F (s, t)$ can be written as $2 s - 4 t^{2} = 2 B_{1, 1} (s) - 4 B_{2, 2} (t)$ and the $y$ -component as $2 s + 4 t^{2} - 2 = 2 B_{1, 1} (s) - 2 B_{0, 2} (t) - 2 B_{1, 2} (t) + 2 B_{2, 2} (t)$ we have

	$μ_{1}$	$= 2 B_{1, 1} (α)$	$+ 4 B_{2, 2} (β) = 2$		(4.13)
	$μ_{2}$	$= 2 B_{1, 1} (α) + 2 B_{0, 2} (β) + 2 B_{1, 2} (β)$	$+ 2 B_{2, 2} (β) = 3.$		(4.14)

Following (4.11), this gives $κ_{α, β} = \sqrt{202} / 8 \approx 1.78$ . This low condition number is somewhat unexpected since when using the resultant to eliminate each parameter, one of the two roots is a double root:

	${Res}_{t} (x_{0} (s) - x_{1} (t), y_{0} (s) - y_{1} (t))$	$= 64 (2 s - 1)^{2}$		(4.15)
	${Res}_{s} (x_{0} (s) - x_{1} (t), y_{0} (s) - y_{1} (t))$	$= 4 (2 t - 1) (2 t + 1) .$		(4.16)

4.2 Line-line Intersection with Poorly Behaved Coefficients

Consider the intersection of the lines $y = x$ and $y = 1 - x$ when $x \in [0, 1]$ . These correspond to the Bézier curves

b_{0} (s) = [\begin{matrix} s s \end{matrix}], b_{1} (t) = [\begin{matrix} t 1 - t \end{matrix}] .

(4.17)

By adding a scalar $D > 0$ to each component, we leave $F (s, t)$ and hence the solution unchanged. However, the coefficients of the curves change:

b_{0} (s) = [\begin{matrix} D (1 - s) + (1 + D) s D (1 - s) + (1 + D) s \end{matrix}], b_{1} (t) = [\begin{matrix} D (1 - t) + (1 + D) t (1 + D) (1 - t) + D t \end{matrix}] .

(4.18)

At the solution $α = β = 1 / 2$ , we have

{˜ F}_{\pm} = [\begin{matrix} 2 D + 1 \pm (2 D + 1) \end{matrix}]

(4.19)

and in either case ${∥ ∥ J^{- 1} {˜ F}_{\pm} ∥ ∥}_{2}^{- 1} = 2 D + 1$ .

So, we see the condition number $κ_{α, β} = \sqrt{2} (2 D + 1)$ increases towards infinity as $D$ does. This is what we expect as the coefficients grow so large that their ratio approaches $1$ .

4.3 Family of Lines Approaching Coincident

Consider a family of intersections in which one of the lines approaches the other:

b_{0} (s) = [\begin{matrix} s 1 \end{matrix}], b_{1} (t) = [\begin{matrix} t (1 + r) (1 - t) + t \end{matrix}] .

(4.20)

These lines $y = 1$ and $r x + y = 1 + r$ intersect when $α = β = 1$ . However as $r ⟶ 0^{+}$ , the lines become coincident: if $r = 0$ the single intersection becomes infinitely many.

At the solution, we have ${˜ F}_{\pm} = {[\begin{matrix} 2 & \pm 2 \end{matrix}]}^{T}$ and again have a condition number

κ_{α, β} = \sqrt{\frac{4}{r^{2}} + \frac{4}{r} + 2} = \frac{2}{r} + 1 + \frac{}{r} 4 + O (r^{2}) .

(4.21)

that increases towards infinity as the parameter $r ⟶ 0^{+}$ .

References

[Far01] Gerald Farin. Curves and Surfaces for CAGD, Fifth Edition: A Practical Guide (The Morgan Kaufmann Series in Computer Graphics). Morgan Kaufmann, 2001.
[FR87] R.T. Farouki and V.T. Rajan. On the numerical condition of polynomials in Bernstein form. Computer Aided Geometric Design, 4(3):191–216, Nov 1987.
[Hig02] Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Jan 2002.
[HT15] Aaron Herman and Elias Tsigaridas. Bounds for the condition number of polynomials systems with integer coefficients. In Computer Algebra in Scientific Computing, pages 210–219. Springer International Publishing, 2015.
[KLS98] Deok-Soo Kim, Soon-Woong Lee, and Hayong Shin. A cocktail algorithm for planar Bézier curve intersections. Computer-Aided Design, 30(13):1047–1051, Nov 1998.
[MD92] Dinesh Manocha and James W. Demmel. Algorithms for Intersecting Parametric and Algebraic Curves. Technical Report UCB/CSD-92-698, EECS Department, University of California, Berkeley, Aug 1992.
[Sch09] Christian Schulz. Bézier clipping is quadratically convergent. Computer Aided Geometric Design, 26(1):61–74, Jan 2009.
[Sed16] Thomas W Sederberg. Lecture notes: Computer aided geometric design, Sep 2016.
[SN90] T.W. Sederberg and T. Nishita. Curve intersection using Bézier clipping. Computer-Aided Design, 22(9):538–549, Nov 1990.
[SP86] Thomas W Sederberg and Scott R Parry. Comparison of three curve intersection algorithms. Computer-Aided Design, 18(1):58–63, Jan 1986.

A 2-Norm Condition Number for Bézier Curve Intersection

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condition-number-bezier-curve-intersection

1 Introduction

2 Preliminaries

3 Conditioning of Generic Root-finding

Definition 3.1.

3.1 One-dimensional Case

4 Conditioning of Bézier Curve Intersection

Theorem 4.1.

Proof.

4.1 Transversal Intersection

4.2 Line-line Intersection with Poorly Behaved Coefficients

4.3 Family of Lines Approaching Coincident

References

A 2-Norm Condition Number for Bézier Curve Intersection

Authors

Related Research

Intersections between the norm-trace curve and some low degree curves

Bezier Curves Intersection Using Relief Perspective

Embeddings of k-complexes into 2k-manifolds

Computing the intersection of two quadrics through projection and lifting

On Intersecting IFS Fractals with Lines

Discriminants of complete intersection space curves

The Saturated Subpaths Decomposition in Z 2 : a short note on generalized Tangential Cover

Code Repositories

condition-number-bezier-curve-intersection

This week in AI

1 Introduction

2 Preliminaries

3 Conditioning of Generic Root-finding

Definition 3.1.

3.1 One-dimensional Case

4 Conditioning of Bézier Curve Intersection

Theorem 4.1.

Proof.

4.1 Transversal Intersection

4.2 Line-line Intersection with Poorly Behaved Coefficients

4.3 Family of Lines Approaching Coincident

References