Bezier Curves - USTCstaff.ustc.edu.cn/~renjiec/CAGD_2019S1/materials/cagd_lec03.pdf · Bezier curves • Today: Standard tool for 2D curve editing • Cubic 2D Bezier curves are everywhere:

Bezier Curves

陈仁杰

中国科学技术大学

计算机辅助几何设计2019秋学期

Bezier curves

• Bezier curves/splines developed by• Paul de Casteljau at Citroen (1959)• Pierre Bezier at Renault (1963)for free-form parts in automotive design

Bezier curves

• Today: Standard tool for 2D curve editing

• Cubic 2D Bezier curves are everywhere:• Inkscape, Corel Draw, Adobe Illustrator, Powerpoint, …• PDF, Truetype (quadratic curves), Windows GDI, …

• Widely used in 3D curve & surface modeling as well

Curve representation

• The implicit curve form 𝑓𝑓 𝑥𝑥,𝑦𝑦 = 0 suffers from several limitations:

Curve representation

• The implicit curve form 𝑓𝑓 𝑥𝑥,𝑦𝑦 = 0 suffers from several limitations:

• Multiple values for the same 𝑥𝑥-coordinates

• Undefined derivative 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

(see blue cross)

• Not invariant w.r.t axes transformations

Parametric representation

• Remedy: parametric representation 𝑐𝑐 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 ,𝑦𝑦 𝑡𝑡

• Easy evaluations

• The parameter 𝑡𝑡 can be interpreted as time

• The curve can be interpreted as the path traced by a moving particle

Modeling with the power basis, …

• Example of a parabola: 𝒇𝒇 𝑡𝑡 = 𝒂𝒂𝑡𝑡2 + 𝒃𝒃𝑡𝑡 + 𝒄𝒄

𝒇𝒇 𝑡𝑡 = 11 𝑡𝑡2 + −2

0 𝑡𝑡 + 10

Modeling with the power basis, …no thanks!• Examples of a parabola: 𝒇𝒇 𝑡𝑡 = 𝒂𝒂𝑡𝑡2 + 𝒃𝒃𝑡𝑡 + 𝒄𝒄: the coefficients of

the power basis lack intuitive geometric meaning

Back to the drawing board

• A point on a parametric line

𝒃𝒃𝟏𝟏

𝒃𝒃𝟎𝟎

𝒃𝒃𝟎𝟎𝟏𝟏𝒃𝒃𝟎𝟎𝟏𝟏 = 1 − 𝑡𝑡 𝒃𝒃𝟎𝟎 + 𝑡𝑡𝒃𝒃𝟏𝟏


• Another point on a second parametric line

𝒃𝒃𝟏𝟏𝟏𝟏 = 1 − 𝑡𝑡 𝒃𝒃𝟏𝟏 + 𝑡𝑡𝒃𝒃𝟐𝟐𝒃𝒃𝟎𝟎𝟏𝟏 = 1 − 𝑡𝑡 𝒃𝒃𝟎𝟎 + 𝑡𝑡𝒃𝒃𝟏𝟏

𝒃𝒃𝟏𝟏

𝒃𝒃𝟎𝟎𝒃𝒃𝟐𝟐

𝒃𝒃𝟏𝟏𝟏𝟏

𝒃𝒃𝟎𝟎𝟏𝟏


• A third point on the line defined by the first two points

𝒃𝒃𝟏𝟏


𝒃𝒃𝟏𝟏𝟏𝟏

𝒃𝒃𝟎𝟎𝟏𝟏𝒃𝒃𝟎𝟎𝟐𝟐

𝒃𝒃𝟎𝟎𝟐𝟐 = 1 − 𝑡𝑡 𝒃𝒃𝟎𝟎𝟏𝟏 + 𝑡𝑡𝒃𝒃𝟏𝟏𝟏𝟏

𝒃𝒃𝟏𝟏𝟏𝟏 = 1 − 𝑡𝑡 𝒃𝒃𝟏𝟏 + 𝑡𝑡𝒃𝒃𝟐𝟐𝒃𝒃𝟎𝟎𝟏𝟏 = 1 − 𝑡𝑡 𝒃𝒃𝟎𝟎 + 𝑡𝑡𝒃𝒃𝟏𝟏


• And then simplify…

𝒃𝒃𝟎𝟎𝟏𝟏 = 1 − 𝑡𝑡 𝒃𝒃𝟎𝟎 + 𝑡𝑡𝒃𝒃𝟏𝟏

𝒃𝒃𝟎𝟎𝟐𝟐 = 1 − 𝑡𝑡 𝒃𝒃𝟎𝟎𝟏𝟏 + 𝑡𝑡𝒃𝒃𝟏𝟏𝟏𝟏

𝒃𝒃𝟏𝟏𝟏𝟏 = 1 − 𝑡𝑡 𝒃𝒃𝟏𝟏 + 𝑡𝑡𝒃𝒃𝟐𝟐

𝒃𝒃𝟎𝟎𝟐𝟐 = 1 − 𝑡𝑡 1 − 𝑡𝑡 𝒃𝒃𝟎𝟎 + 𝑡𝑡𝒃𝒃𝟏𝟏 + 𝑡𝑡 1 − 𝑡𝑡 𝒃𝒃𝟏𝟏 + 𝑡𝑡𝒃𝒃𝟐𝟐

𝒃𝒃𝟎𝟎𝟐𝟐 = 1 − 𝑡𝑡 2𝒃𝒃𝟎𝟎 + 2𝑡𝑡 1 − 𝑡𝑡 𝒃𝒃𝟏𝟏 + 𝑡𝑡2𝒃𝒃𝟐𝟐

𝒃𝒃𝟏𝟏


𝒃𝒃𝟏𝟏𝟏𝟏

𝒃𝒃𝟎𝟎𝟏𝟏𝒃𝒃𝟎𝟎𝟐𝟐


• We obtained another description of parabolic curves

• The coefficients 𝒃𝒃𝟎𝟎,𝒃𝒃𝟏𝟏,𝒃𝒃𝟐𝟐 have a geometric meaning

𝒃𝒃𝟎𝟎𝟐𝟐 = 1 − 𝑡𝑡 2𝒃𝒃𝟎𝟎 + 2𝑡𝑡 1 − 𝑡𝑡 𝒃𝒃𝟏𝟏 + 𝑡𝑡2𝒃𝒃𝟐𝟐

𝒃𝒃𝟏𝟏


𝒃𝒃𝟏𝟏𝟏𝟏

𝒃𝒃𝟎𝟎𝟏𝟏𝒃𝒃𝟎𝟎𝟐𝟐

Example re-visited

• Let’s rewrite our initial parabolic curve example in the new basis

𝒇𝒇 𝑡𝑡 = 11 𝑡𝑡2 + −2

0 𝑡𝑡 + 10

𝒇𝒇 𝑡𝑡 = 10 1 − 𝑡𝑡 2 + 0

0 2𝑡𝑡 1 − 𝑡𝑡 + 01 𝑡𝑡2

Example re-visited

• The coefficient have a geometric meaning

• More intuitive for curve manipulation

Another example

𝒃𝒃0 = 01 , 𝒃𝒃1 = 1

1 , 𝒃𝒃2 = 02

Going further

• Cubic approximation

• Given 4 points: 𝒑𝒑00 𝑡𝑡 = 𝒑𝒑0, 𝒑𝒑10 𝑡𝑡 = 𝒑𝒑1, 𝒑𝒑20 𝑡𝑡 = 𝒑𝒑2, 𝒑𝒑30 𝑡𝑡 = 𝒑𝒑3• First iteration

• 2nd iteration

• Curve𝒄𝒄 𝑡𝑡 = 1 − 𝑡𝑡 3𝒑𝒑0 + 3𝑡𝑡 1 − 𝑡𝑡 2𝒑𝒑1 + 3𝑡𝑡2 1 − 𝑡𝑡 𝒑𝒑2 + 𝑡𝑡3𝒑𝒑3

𝒑𝒑02 = 1 − 𝑡𝑡 2𝒑𝒑0 + 2𝑡𝑡 1 − 𝑡𝑡 𝒑𝒑1 + 𝑡𝑡2𝒑𝒑2𝒑𝒑12 = 1 − 𝑡𝑡 2𝒑𝒑1 + 2𝑡𝑡 1 − 𝑡𝑡 𝒑𝒑2 + 𝑡𝑡2𝒑𝒑3

𝒑𝒑01 = 1 − 𝑡𝑡 𝒑𝒑0 + 𝑡𝑡𝒑𝒑1𝒑𝒑11 = 1 − 𝑡𝑡 𝒑𝒑1 + 𝑡𝑡𝒑𝒑2𝒑𝒑21 = 1 − 𝑡𝑡 𝒑𝒑2 + 𝑡𝑡𝒑𝒑3

Throughout these examples, we just re-invented a primitive version of the de Casteljau algorithm

Now let’s examine it more closely …

De Casteljau algorithm

• De Casteljau Algorithm: Computes 𝑥𝑥 𝑡𝑡 for given 𝑡𝑡• Bisect control polygon in ratio 𝑡𝑡: 1 − 𝑡𝑡• Connect the new dots with lines (adjacent segments)• Interpolate again with the same ratio• Iterate, until only one points is left








• Algorithm description• Input: points 𝒃𝒃0,𝒃𝒃1, …𝒃𝒃𝑛𝑛 ∈ ℝ3

• Output: curve 𝒙𝒙 𝑡𝑡 , 𝑡𝑡 ∈ 0,1

• Geometric construction of the points 𝒙𝒙 𝑡𝑡 for given 𝑡𝑡:𝒃𝒃𝑖𝑖0 𝑡𝑡 = 𝒃𝒃𝑖𝑖 , 𝑖𝑖 = 0, … ,𝑛𝑛

𝒃𝒃𝑖𝑖𝑟𝑟 𝑡𝑡 = 1 − 𝑡𝑡 𝒃𝒃𝑖𝑖𝑟𝑟−1 𝑡𝑡 + 𝑡𝑡 𝒃𝒃𝑖𝑖+1𝑟𝑟−1 𝑡𝑡

𝑟𝑟 = 1, … ,𝑛𝑛 𝑖𝑖 = 0, … ,𝑛𝑛 − 𝑟𝑟

• Then 𝒃𝒃0𝑛𝑛 𝑡𝑡 is the searched curve point 𝒙𝒙 𝑡𝑡 at the parameter value 𝑡𝑡

De Casteljau algorithm• Repeated convex combination of control points

𝒃𝒃𝑖𝑖𝑟𝑟 = 1 − 𝑡𝑡 𝒃𝒃𝑖𝑖

𝑟𝑟−1 +𝑡𝑡𝒃𝒃𝑖𝑖+1𝑟𝑟−1

𝒃𝒃00

𝒃𝒃10

𝒃𝒃20

𝒃𝒃30




𝒃𝒃00

𝒃𝒃10

𝒃𝒃20

𝒃𝒃30

𝑡𝑡 𝒃𝒃01

𝒃𝒃11

𝒃𝒃21

𝑡𝑡

𝑡𝑡




𝒃𝒃00

𝒃𝒃10

𝒃𝒃20

𝒃𝒃30

𝑡𝑡 𝒃𝒃01

𝒃𝒃11

𝒃𝒃21

𝑡𝑡

𝑡𝑡

𝑡𝑡

𝑡𝑡

𝒃𝒃02

𝒃𝒃12




𝒃𝒃00

𝒃𝒃10

𝒃𝒃20

𝒃𝒃30

𝑡𝑡 𝒃𝒃01

𝒃𝒃11

𝒃𝒃21

𝑡𝑡

𝑡𝑡

𝑡𝑡

𝑡𝑡

𝒃𝒃02

𝒃𝒃12 𝑡𝑡 𝒃𝒃0

3 = 𝑥𝑥(𝑡𝑡)De Casteljau scheme


• The intermediate coefficients 𝒃𝒃𝑖𝑖𝑟𝑟 𝑡𝑡 can be written in a triangular matrix: the de Casteljau scheme:

• 𝒃𝒃0 = 𝒃𝒃00

• 𝒃𝒃1 = 𝒃𝒃10 𝒃𝒃01

• 𝒃𝒃2 = 𝒃𝒃20 𝒃𝒃11 𝒃𝒃02

• 𝒃𝒃3 = 𝒃𝒃30 𝒃𝒃21 𝒃𝒃12 𝒃𝒃03

• ……………………

• 𝒃𝒃𝑛𝑛−1 = 𝒃𝒃𝑛𝑛−10 𝒃𝒃𝑛𝑛−21 … 𝒃𝒃0𝑛𝑛−1

• 𝒃𝒃𝑛𝑛 = 𝒃𝒃𝑛𝑛0 𝒃𝒃𝑛𝑛−11 … 𝒃𝒃1𝑛𝑛−1 𝒃𝒃0𝑛𝑛 = 𝑥𝑥 𝑡𝑡


• Algorithm:• for r=1..n• for i=0..n-r

• 𝒃𝒃𝑖𝑖𝑟𝑟 = 1 − 𝑡𝑡 𝒃𝒃𝑖𝑖

𝑟𝑟−1 + 𝑡𝑡 𝒃𝒃𝑖𝑖+1𝑟𝑟−1

• end• end

• return 𝒃𝒃0𝑛𝑛

The whole algorithm consists only of repeated linear interpolations.

De Casteljau algorithm: Properties

• The polygon consisting of the points 𝒃𝒃𝟎𝟎, … ,𝒃𝒃𝒏𝒏 is called Bezier polygon(control polygon)

• The points 𝒃𝒃𝒊𝒊 are called Bezier points (control points)• The curve defined by the Bezier points 𝒃𝒃𝟎𝟎, … ,𝒃𝒃𝒏𝒏 and the de Casteljau

algorithm is called Bezier curve• The de Casteljau algorithm is numerically stable, since only convex

combinations are applied.• Complexity of the de Casteljau algorithm

• 𝑂𝑂 𝑛𝑛2 time• 𝑂𝑂 𝑛𝑛 memory• with 𝑛𝑛 being the number of Bezier points


• Properties of Bezier curves:• Given: Bezier points 𝒃𝒃0, … ,𝒃𝒃𝑛𝑛

Bezier curve 𝒙𝒙 𝑡𝑡• Bezier curve is polynomial curve of degree 𝑛𝑛• End points interpolation: 𝒙𝒙 0 = 𝒃𝒃0, 𝒙𝒙 1 = 𝒃𝒃𝑛𝑛. The remaining Bezier

points are only approximated in general• Convex hull property:Bezier curve is completely inside the convex hull of its Bezier polygon


• Variation diminishing• No line intersects the Bezier curve more often than its Bezier polygon

• Influence of Bezier points: global but pseudo-local• Global: moving a Bezier points changes the whole curve progression

• Pseudo-local: 𝒃𝒃𝑖𝑖 has its maximal influence on 𝑥𝑥 𝑡𝑡 at 𝑡𝑡 = 𝑖𝑖𝑛𝑛

• Affine invariance:• Bezier curve and Bezier polygon are invariant under affine

transformations

• Invariance under affine parameter transformations


• Symmetry• The following two Bezier curves coincide, they are only traversed in

opposite directions:𝒙𝒙 𝑡𝑡 = 𝒃𝒃0, … ,𝒃𝒃𝑛𝑛 𝒙𝒙′ 𝑡𝑡 = 𝒃𝒃𝑛𝑛, …𝒃𝒃0

• Linear Precision:• Bezier curve is line segment, if 𝒃𝒃0, … ,𝒃𝒃𝑛𝑛 are colinear

• Invariance under barycentric combinations

Recap

de Casteljau algorithm

Bezier CurvesTowards a polynomial description

Bezier CurvesTowards a polynomial description

𝑥𝑥 𝑡𝑡 = �𝑖𝑖=0

𝑛𝑛

𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 ⋅ 𝑏𝑏𝑖𝑖

Polynomial description of Bezier curves

• The same problem as before:• Given: 𝑛𝑛 + 1 control points 𝒃𝒃0, … ,𝒃𝒃𝑛𝑛• Wanted: Bezier curve 𝒙𝒙 𝑡𝑡 with 𝑡𝑡 ∈ 0,1

• Now with an algebraic approach using basis functions

Desirable Properties

• Useful requirements for a basis:• Well behaved curve

• Smooth basis functions



• Smooth basis functions

• Local control (or at least semi-local)• Basis functions with compact support



• Smooth basis functions• Local control (or at least semi-local)

• Basis functions with compact support• Affine invariance:

• Appling an affine map 𝒙𝒙 → 𝐴𝐴𝒙𝒙 + 𝑏𝑏 on• Control points• Curve

Should have the same effect• In particular: rotation, translation• Otherwise: interactive curve editing very difficult


• Useful requirements for a basis:• Convex hull property:

• The curve lays within the convex hull of its control points• Avoids at least too weird oscillations

• Advantages• Computational advantages (recursive intersection tests)• More predictable behavior

Summary

• Useful properties• Smoothness• Local control / support• Affine invariance• Convex hull property

Curve basis function control points

𝒇𝒇 𝑡𝑡 = �𝑖𝑖=1

𝑛𝑛

𝑏𝑏𝑖𝑖 𝑡𝑡 𝒑𝒑𝑖𝑖

Notations

Affine Invariance

• Affine map: 𝒙𝒙 → 𝐴𝐴𝒙𝒙 + 𝒃𝒃• Part I: Linear invariance – we get this automatically

• Linear approach: 𝒇𝒇 𝑡𝑡 = ∑𝑖𝑖=1𝑛𝑛 𝑏𝑏𝑖𝑖 𝑡𝑡 𝒑𝒑𝑖𝑖 = ∑𝑖𝑖=1𝑛𝑛 𝑏𝑏𝑖𝑖 𝑡𝑡𝑝𝑝𝑖𝑖𝑑𝑑

𝑝𝑝𝑖𝑖𝑑𝑑

𝑝𝑝𝑖𝑖𝑧𝑧

• Therefore: 𝐴𝐴 𝒇𝒇 𝑡𝑡 = 𝐴𝐴 ∑𝑖𝑖=1𝑛𝑛 𝑏𝑏𝑖𝑖 𝑡𝑡 𝒑𝒑𝑖𝑖 = ∑𝑖𝑖=1𝑛𝑛 𝑏𝑏𝑖𝑖 𝑡𝑡 𝐴𝐴𝒑𝒑𝑖𝑖

Affine Invariance

• Affine Invariance:• Affine map: 𝒙𝒙 → 𝐴𝐴𝒙𝒙 + 𝒃𝒃• Part II: Translational invariance

�𝑖𝑖=1

𝑛𝑛

𝑏𝑏𝑖𝑖 𝑡𝑡 𝒑𝒑𝑖𝑖 + 𝒃𝒃 = �𝑖𝑖=1

𝑛𝑛

𝑏𝑏𝑖𝑖 𝑡𝑡 𝒑𝒑𝑖𝑖 + �𝑖𝑖=1

𝑛𝑛

𝑏𝑏𝑖𝑖 𝑡𝑡 𝒃𝒃 = 𝒇𝒇 𝑡𝑡 + �𝑖𝑖=1

𝑛𝑛

𝑏𝑏𝑖𝑖 𝑡𝑡 𝒃𝒃

• For translational invariance, the sum of the basis functions must be one everywhere (for all parameter values 𝑡𝑡 that are used).

• This is called “partition of unity property”• The 𝑏𝑏𝑖𝑖’s form an “affine combination” of the control points 𝒑𝒑𝑖𝑖• This is very important for modeling

Convex Hull Property

• Convex combinations:• A convex combination of a set of points 𝒑𝒑1, … ,𝒑𝒑𝑛𝑛 is any point of the form:

∑𝑖𝑖=1𝑛𝑛 𝜆𝜆𝑖𝑖𝒑𝒑𝒊𝒊 with ∑𝑖𝑖=1𝑛𝑛 𝜆𝜆𝑖𝑖 = 1 and ∀𝑖𝑖 = 1 …𝑛𝑛: 0 ≤ 𝜆𝜆𝑖𝑖 ≤ 1

• (Remark: 𝜆𝜆𝑖𝑖 ≤ 1 is redundant)

• The set of all admissible convex combinations forms the convex hull of the point set

• Easy to see (exercise): The convex hull is the smallest set that contains all points 𝒑𝒑1, … ,𝒑𝒑𝑛𝑛 and every complete straight line between two elements of the set


• Accordingly:• If we have this property∀𝑡𝑡 ∈ Ω:∑𝑖𝑖=1𝑛𝑛 𝑏𝑏𝑖𝑖 𝑡𝑡 = 1 and ∀𝑡𝑡 ∈ Ω,∀𝑖𝑖: 𝑏𝑏𝑖𝑖 𝑡𝑡 ≥ 0the constructed curves / surfaces will be:

• Affine invariant (translations, linear maps)• Be restricted to the convex hull of the control points

• Corollary: Curves will have linear precision• All control points lie on a straight line⇒ Curve is a straight line segment

• Surfaces with planar control points will be flat, too


• Very useful property in practice• Avoids at least the worst oscillations

• no escape from convex hull, unlike polynomial interpolation

• Linear precision property is intuitive (people expect this)• Can be used for fast range checks

• Test for intersection with convex hull first, then the object• Recursive intersection algorithms in conjunctions with subdivision rules (more on

this later)

Polynomial description of Bezier curves

• The same problem as before:• Given: 𝑛𝑛 + 1 control points 𝒃𝒃0, … ,𝒃𝒃𝑛𝑛• Wanted: Bezier curve 𝑥𝑥 𝑡𝑡 with 𝑡𝑡 ∈ 0,1

• Now with an algebraic approach using basis functions

• Need to define 𝑛𝑛 + 1 basis functions• Such that this describes a Bezier curve:

𝐵𝐵0𝑛𝑛 𝑡𝑡 , … ,𝐵𝐵𝑛𝑛𝑛𝑛 𝑡𝑡 over 0,1

𝒙𝒙 𝑡𝑡 = �𝑖𝑖=0

𝑛𝑛

𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 ⋅ 𝒃𝒃𝑖𝑖

Bernstein Basis

• Let’s examine the Bernstein basis: 𝐵𝐵 = {𝐵𝐵0𝑛𝑛 ,𝐵𝐵1

𝑛𝑛 , … ,𝐵𝐵𝑛𝑛𝑛𝑛 }

• Bernstein basis of degree 𝑛𝑛:

𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 = 𝑛𝑛

𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖 = 𝐵𝐵𝑖𝑖−th basis functiondegree

where the binomial coefficients are given by:

𝑛𝑛𝑖𝑖 = �

𝑛𝑛!𝑛𝑛 − 𝑖𝑖 ! 𝑖𝑖!

for 0 ≤ 𝑖𝑖 ≤ 𝑛𝑛

0 otherwise

Binomial Coefficients and Theorem

𝑥𝑥 + 𝑦𝑦 𝑛𝑛 = �𝑖𝑖=0

𝑛𝑛𝑛𝑛𝑖𝑖 𝑥𝑥𝑖𝑖𝑦𝑦𝑛𝑛−𝑖𝑖

1 1 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

𝑛𝑛𝑖𝑖 + 𝑛𝑛

𝑖𝑖 + 1 = 𝑛𝑛 + 1𝑖𝑖 + 1

Examples: The first few

• The first three Bernstein bases:

• 𝐵𝐵00 ≔ 1

• 𝐵𝐵01 ≔ 1 − 𝑡𝑡 𝐵𝐵1

1 ≔ 𝑡𝑡

• 𝐵𝐵01 ≔ 1 − 𝑡𝑡 2 𝐵𝐵1

2 ≔ 2𝑡𝑡 1 − 𝑡𝑡 𝐵𝐵22 ≔ 𝑡𝑡2

• 𝐵𝐵02 ≔ 1 − 𝑡𝑡 3 𝐵𝐵1

3 ≔ 3𝑡𝑡 1 − 𝑡𝑡 2 𝐵𝐵23 ≔ 3𝑡𝑡2 1 − 𝑡𝑡 𝐵𝐵3

3 ≔ 𝑡𝑡3


𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖

Examples: The first few

• 𝐵𝐵02 ≔ 1 − 𝑡𝑡 3

• 𝐵𝐵13 ≔ 3𝑡𝑡 1 − 𝑡𝑡 2

• 𝐵𝐵23 ≔ 3𝑡𝑡2 1 − 𝑡𝑡

• 𝐵𝐵33 ≔ 𝑡𝑡3


𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖

𝐵𝐵00 ≔ 1

𝐵𝐵01 ≔ 1 − 𝑡𝑡

𝐵𝐵11 ≔ 𝑡𝑡

𝐵𝐵01 ≔ 1 − 𝑡𝑡 2

𝐵𝐵12 ≔ 2𝑡𝑡 1 − 𝑡𝑡

𝐵𝐵22 ≔ 𝑡𝑡2

Bernstein Basis

• Bezier curves use the Bernstein basis: 𝐵𝐵 = 𝐵𝐵0𝑛𝑛 ,𝐵𝐵1

𝑛𝑛 , … ,𝐵𝐵𝑛𝑛𝑛𝑛

• Bernstein basis of degree 𝑛𝑛:


𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖 = 𝐵𝐵𝑖𝑖−th basis functiondegree

Bernstein Basis

• What about the desired properties?• Smoothness• Local control / support• Affine invariance• Convex hull property

Bernstein Basis: Properties

• 𝐵𝐵 = 𝐵𝐵0𝑛𝑛 ,𝐵𝐵1

𝑛𝑛 , … ,𝐵𝐵𝑛𝑛𝑛𝑛 , 𝐵𝐵𝑖𝑖

𝑛𝑛 𝑡𝑡 = 𝑛𝑛𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖

• Basis for polynomials of degree 𝑛𝑛

• Each basis function 𝐵𝐵𝑖𝑖𝑛𝑛 has its maximum at 𝑡𝑡 = 𝑖𝑖

𝑛𝑛

Smoothness

Local control (semi-local)





• Partition of unity (binomial theorem)1 = 1 − 𝑡𝑡 + 𝑡𝑡

�𝑖𝑖=0

𝑛𝑛

𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 = 𝑡𝑡 + 1 − 𝑡𝑡 𝑛𝑛 = 1

Affine invariance Convex hull property

What about the desired properties?

• Smoothness

• Local control / support

• Affine invariance

• Convex hull property

YesTo some extentYesYes





• Recursive computation𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 ≔ 1 − 𝑡𝑡 𝐵𝐵𝑖𝑖

𝑛𝑛−1 𝑡𝑡 + 𝑡𝑡𝐵𝐵𝑖𝑖−1𝑛𝑛−1 1 − 𝑡𝑡

with 𝐵𝐵00 𝑡𝑡 = 1,𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 = 0 for 𝑖𝑖 ∉ 0 …𝑛𝑛• Symmetry

𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 = 𝐵𝐵𝑛𝑛−𝑖𝑖𝑛𝑛 1 − 𝑡𝑡

• Non-negativity: 𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 ≥ 0 for 𝑡𝑡 ∈ [0. . 1]

𝑛𝑛 − 1𝑖𝑖 + 𝑛𝑛 − 1

𝑖𝑖 − 1 =𝑛𝑛𝑖𝑖





• Non-negativity II

𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 > 0 for 0 < 𝑡𝑡 < 1

𝐵𝐵0𝑛𝑛 0 = 1, 𝐵𝐵1𝑛𝑛 0 = ⋯ = 𝐵𝐵𝑛𝑛𝑛𝑛 0 = 0

𝐵𝐵0𝑛𝑛 1 = ⋯ = 𝐵𝐵𝑛𝑛−1𝑛𝑛 1 = 0, 𝐵𝐵𝑛𝑛𝑛𝑛 0 = 1

Derivatives𝐵𝐵𝑖𝑖


• Bernstein basis properties• Derivatives:

𝑑𝑑𝑑𝑑𝑑𝑑𝐵𝐵𝑖𝑖

𝑛𝑛 𝑡𝑡 =




𝑑𝑑𝑑𝑑𝑑𝑑𝐵𝐵𝑖𝑖

𝑛𝑛 𝑡𝑡 = 𝑛𝑛𝑖𝑖 𝑖𝑖𝑡𝑡 𝑖𝑖−1 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖 − 𝑛𝑛 − 𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖−1

=𝑛𝑛!

𝑛𝑛 − 𝑖𝑖 ! 𝑖𝑖!𝑖𝑖𝑡𝑡 𝑖𝑖−1 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖 −

𝑛𝑛!𝑛𝑛 − 𝑖𝑖 ! 𝑖𝑖!

𝑛𝑛 − 𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖−1

= 𝑛𝑛 𝑛𝑛 − 1𝑖𝑖 − 1 𝑡𝑡 𝑖𝑖−1 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖 − 𝑛𝑛 − 1

𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖−1

= 𝑛𝑛 𝐵𝐵𝑖𝑖−1𝑛𝑛−1 𝑡𝑡 − 𝐵𝐵𝑖𝑖

𝑛𝑛−1 𝑡𝑡

(Notation: {𝒌𝒌} = 𝒌𝒌 if 𝒌𝒌 > 𝟎𝟎, zero otherwise)




𝑑𝑑2

𝑑𝑑𝑑𝑑2𝐵𝐵𝑖𝑖

𝑛𝑛 𝑡𝑡 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑛𝑛 𝐵𝐵𝑖𝑖−1

𝑛𝑛−1 𝑡𝑡 − 𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡

= 𝑛𝑛 𝑛𝑛 − 1 𝐵𝐵𝑖𝑖−2𝑛𝑛−2 𝑡𝑡 − 𝐵𝐵𝑖𝑖−1

𝑛𝑛−2 𝑡𝑡 − 𝑛𝑛 − 1 𝐵𝐵𝑖𝑖−1𝑛𝑛−2 𝑡𝑡 − 𝐵𝐵𝑖𝑖

𝑛𝑛−2 𝑡𝑡

= 𝑛𝑛 𝑛𝑛 − 1 𝐵𝐵𝑖𝑖−2𝑛𝑛−2 𝑡𝑡 − 2𝐵𝐵𝑖𝑖−1

𝑛𝑛−2 𝑡𝑡 + 𝐵𝐵𝑖𝑖𝑛𝑛−2 𝑡𝑡

(Notation: {𝒌𝒌} = 𝒌𝒌 if 𝒌𝒌 > 𝟎𝟎, zero otherwise)

Bezier Curves in Bernstein form

• Bezier Curves:

𝒇𝒇 𝑡𝑡 = ∑𝑖𝑖=1𝑛𝑛 𝐵𝐵𝑖𝑖𝑛𝑛𝒑𝒑𝑖𝑖 , 𝑡𝑡 ∈ 0,1


• Bezier Curves:



• Bezier Curves:



• Bezier Curves:



• Bezier Curves, also in 3D



• Bezier curves:• Curves: 𝒇𝒇 𝑡𝑡 = ∑𝑖𝑖=1𝑛𝑛 𝐵𝐵𝑖𝑖𝑛𝑛𝒑𝒑𝑖𝑖• Considering the interval 𝑡𝑡 ∈ 0. . 1• Properties as discussed before:

• Affine invariant• Curves contained in the convex hull• Influence of control pointsMoving along the curve with index 𝑖𝑖

Largest influence at 𝑡𝑡 = 𝑖𝑖𝑛𝑛

Single curve segments: no full local control

Bezier Curve Properties: another look at derivatives

• Given: 𝒑𝒑0, … ,𝒑𝒑𝑛𝑛, 𝒇𝒇 𝑡𝑡 = ∑𝑖𝑖=0𝑛𝑛 𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 𝒑𝒑𝑖𝑖

• Then: 𝒇𝒇′ 𝑡𝑡 = 𝑛𝑛∑𝑖𝑖=0𝑛𝑛−1𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖+1 − 𝒑𝒑𝑖𝑖

• Proof: 𝒇𝒇′ 𝑡𝑡 = ∑𝑖𝑖=0𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 𝒑𝒑𝑖𝑖 = 𝑛𝑛∑𝑖𝑖=0𝑛𝑛 𝐵𝐵𝑖𝑖−1𝑛𝑛−1 𝑡𝑡 − 𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖

= 𝑛𝑛�𝑖𝑖=0

𝑛𝑛

𝐵𝐵𝑖𝑖−1𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖 − 𝑛𝑛�𝑖𝑖=0

𝑛𝑛

𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖

= 𝑛𝑛 �𝑖𝑖=−1

𝑛𝑛−1

𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖+1 − 𝑛𝑛�𝑖𝑖=0

𝑛𝑛

𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖 = 𝑛𝑛�𝑖𝑖=0

𝑛𝑛−1

𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖+1 − 𝑛𝑛�𝑖𝑖=0

𝑛𝑛−1

𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖

= 𝑛𝑛�𝑖𝑖=0

𝑛𝑛−1

𝐵𝐵𝑖𝑖𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖+1 − 𝒑𝒑𝑖𝑖

Index change

Bezier Curve Properties

• Higher order derivatives:

𝑓𝑓 𝑟𝑟 𝑡𝑡 =𝑛𝑛!

𝑛𝑛 − 𝑟𝑟 !⋅ �𝑖𝑖=0

𝑛𝑛−𝑟𝑟

𝐵𝐵𝑖𝑖𝑛𝑛−𝑟𝑟 𝑡𝑡 ⋅ Δ𝑟𝑟𝒑𝒑𝑖𝑖


• Imporant for continuous concatenation:• Function value at 0,1 :


𝑛𝑛−1𝑛𝑛𝑖𝑖 𝑡𝑡𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖𝒑𝒑𝑖𝑖

⇒ 𝒇𝒇 0 = 𝒑𝒑0𝒇𝒇 1 = 𝒑𝒑1

• First derivative vector at 0,1• Second derivative vector at 0,1


First derivative vector at 0,1𝑑𝑑𝑑𝑑𝑑𝑑𝒇𝒇 𝑡𝑡 =


First derivative vector at 0,1𝑑𝑑𝑑𝑑𝑡𝑡𝒇𝒇 𝑡𝑡 = 𝑛𝑛�

𝑖𝑖=0

𝑛𝑛−1

𝐵𝐵𝑖𝑖−1𝑛𝑛−1 𝑡𝑡 − 𝐵𝐵𝑖𝑖

𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖


First derivative vector at 0,1𝑑𝑑𝑑𝑑𝑡𝑡𝒇𝒇 𝑡𝑡 = 𝑛𝑛�

𝑖𝑖=0

𝑛𝑛−1

𝐵𝐵𝑖𝑖−1𝑛𝑛−1 𝑡𝑡 − 𝐵𝐵𝑖𝑖

𝑛𝑛−1 𝑡𝑡 𝒑𝒑𝑖𝑖

= 𝑛𝑛 � −𝐵𝐵0𝑛𝑛−1 𝑡𝑡 𝒑𝒑0 + 𝐵𝐵0

𝑛𝑛−1 𝑡𝑡 − 𝐵𝐵1𝑛𝑛−1 𝑡𝑡 𝒑𝒑1 + ⋯

𝑑𝑑𝑑𝑑𝑑𝑑𝒇𝒇 0 = 𝑛𝑛 𝒑𝒑1 − 𝒑𝒑0

𝑑𝑑𝑑𝑑𝑑𝑑𝒇𝒇 1 = 𝑛𝑛 𝒑𝒑𝑛𝑛 − 𝒑𝒑𝑛𝑛−1


• Imporant for continuous concatenation:• Function value at 0,1 :

𝒇𝒇 0 = 𝒑𝒑0𝒇𝒇 1 = 𝒑𝒑1

• First derivative vector at 0,1𝒇𝒇′ 0 = 𝑛𝑛 𝒑𝒑1 − 𝒑𝒑0𝒇𝒇′ 1 = 𝑛𝑛 𝒑𝒑𝑛𝑛 − 𝒑𝒑𝑛𝑛−1

• Second derivative vector at 0,1𝒇𝒇′′ 0 = 𝑛𝑛 𝑛𝑛 − 1 𝒑𝒑2 − 𝟐𝟐𝒑𝒑1 + 𝒑𝒑0

𝒇𝒇′′ 1 = 𝑛𝑛 𝑛𝑛 − 1 𝒑𝒑𝑛𝑛 − 2𝒑𝒑𝑛𝑛−1 + 𝒑𝒑𝑛𝑛−2

Degree elevation

• Given: 𝒃𝒃0,…,𝒃𝒃𝑛𝑛 → 𝒙𝒙 𝑡𝑡

• Wanted: �𝒃𝒃0, … , �𝒃𝒃𝑛𝑛, �𝒃𝒃𝑛𝑛+1 → �𝒙𝒙 𝑡𝑡 with 𝒙𝒙 = �𝒙𝒙

• Solution:

Degree elevation

• Given: 𝒃𝒃0,…,𝒃𝒃𝑛𝑛 → 𝒙𝒙 𝑡𝑡

• Wanted: �𝒃𝒃0, … , �𝒃𝒃𝑛𝑛, �𝒃𝒃𝑛𝑛+1 → �𝒙𝒙 𝑡𝑡 with 𝒙𝒙 = �𝒙𝒙

• Solution: �𝒃𝒃0 = 𝒃𝒃𝟎𝟎�𝒃𝒃𝑛𝑛+1 = 𝒃𝒃𝑛𝑛

�𝒃𝒃𝑗𝑗 = 𝑗𝑗𝑛𝑛+1

𝒃𝒃𝑗𝑗−1 + 1 − 𝑗𝑗𝑛𝑛+1

𝒃𝒃𝑗𝑗 for 𝑗𝑗 = 1, … ,𝑛𝑛

Proof

• Let’s consider1 − 𝑡𝑡 𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 = 1 − 𝑡𝑡 𝑛𝑛

𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛−𝑖𝑖𝑡𝑡𝑖𝑖 = 𝑛𝑛𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛+1−𝑖𝑖𝑡𝑡𝑖𝑖

=𝑛𝑛 + 1 − 𝑖𝑖𝑛𝑛 + 1

𝑛𝑛 + 1𝑖𝑖 1 − 𝑡𝑡 𝑛𝑛+1−𝑖𝑖𝑡𝑡𝑖𝑖

=𝑛𝑛 + 1 − 𝑖𝑖𝑛𝑛 + 1

𝐵𝐵𝑖𝑖𝑛𝑛+1 𝑡𝑡

Similarly

𝑡𝑡𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 =𝑖𝑖 + 1𝑛𝑛 + 1

𝐵𝐵𝑖𝑖𝑛𝑛+1 𝑡𝑡

Proof

𝒇𝒇 𝑡𝑡 = 1 − 𝑡𝑡 + 𝑡𝑡 𝒇𝒇 𝑡𝑡 = 1 − 𝑡𝑡 + 𝑡𝑡 �𝑖𝑖=0

𝑛𝑛

𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 𝑷𝑷𝑖𝑖 = �𝑖𝑖=0

𝑛𝑛

1 − 𝑡𝑡 𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 + 𝑡𝑡𝐵𝐵𝑖𝑖𝑛𝑛 𝑡𝑡 𝑷𝑷𝑖𝑖

= �𝑖𝑖=0

𝑛𝑛𝑛𝑛 + 1 − 𝑖𝑖𝑛𝑛 + 1

𝐵𝐵𝑖𝑖𝑛𝑛+1 𝑡𝑡 +𝑖𝑖 + 1𝑛𝑛 + 1

𝐵𝐵𝑖𝑖+1𝑛𝑛+1 𝑡𝑡 𝑷𝑷𝑖𝑖 = �𝑖𝑖=0


𝐵𝐵𝑖𝑖𝑛𝑛+1 𝑡𝑡 𝑷𝑷𝑖𝑖 + �𝑖𝑖=0

𝑛𝑛𝑖𝑖 + 1𝑛𝑛 + 1

𝐵𝐵𝑖𝑖+1𝑛𝑛+1 𝑡𝑡 𝑷𝑷𝑖𝑖

= �𝑖𝑖=0



𝑛𝑛+1𝑖𝑖

𝑛𝑛 + 1𝐵𝐵𝑖𝑖𝑛𝑛+1 𝑡𝑡 𝑷𝑷𝑖𝑖−1

= �𝑖𝑖=0

𝑛𝑛+1𝑛𝑛 + 1 − 𝑖𝑖𝑛𝑛 + 1


𝑛𝑛+1𝑖𝑖

𝑛𝑛 + 1𝐵𝐵𝑖𝑖𝑛𝑛+1 𝑡𝑡 𝑷𝑷𝑖𝑖−1

= �𝑖𝑖=0

𝑛𝑛+1

𝐵𝐵𝑖𝑖𝑛𝑛+1 𝑡𝑡𝑛𝑛 + 1 − 𝑖𝑖𝑛𝑛 + 1

𝑷𝑷𝑖𝑖 +𝑖𝑖

𝑛𝑛 + 1𝑷𝑷𝑖𝑖−1

Using results from

last slide

Adding null terms, 𝑖𝑖 = 𝑛𝑛 + 1, 𝑖𝑖 = 0

Degree elevation: Example

• �𝒃𝒃0 = 𝒃𝒃0 �𝒃𝒃𝑗𝑗 = 𝑗𝑗𝑛𝑛+1


𝒃𝒃𝑗𝑗• �𝒃𝒃𝑛𝑛+1 = 𝒃𝒃𝑛𝑛 𝑗𝑗 = 1, … ,𝑛𝑛





1





𝟑𝟑𝟒𝟒

𝟏𝟏𝟒𝟒





𝟏𝟏𝟐𝟐

𝟏𝟏𝟐𝟐





𝟑𝟑𝟒𝟒

𝟏𝟏𝟒𝟒





𝟏𝟏





Degree elevation

For repeated degree elevation, the Bezier polygon converges to the Bezier curve. (slow convergence)

Recap

Useful properties for basis functions

• Smoothness

• Local control / support

• Affine invariance

• Convex hull property

Curve basis function control points


𝑛𝑛

𝑏𝑏𝑖𝑖 𝑡𝑡 𝒑𝒑𝑖𝑖

de Casteljau algorithm

Summary and Outlook

• Bezier curves and curve design• The rough form is specified by the position of the control points• Results: smooth curve approximating the control points• Computation / Representation:

• de Casteljau algorithm• Bernstein form

• Problems:• High polynomial degree• Moving a control point can change the whole curve• Interpolation of points• →Bezier splines

Documents

Bezier Curves - USTCstaff.ustc.edu.cn/~renjiec/CAGD_2019S1/materials/cagd_lec03.pdf · Bezier curves • Today: Standard tool for 2D curve editing • Cubic 2D Bezier curves are everywhere: