Splines III – Bézier Curves

Splines III – Bézier Curvesbased on:

Michael Gleicher: Curves, chapter 15 inFundamentals of Computer Graphics, 3rd ed.

(Shirley & Marschner)Slides by Marc van Kreveld

Interpolation vs. approximation

• Interpolation means passing through given points, approximation means getting “close” to given points

• Bézier curves and B-spline curves

p1 and p2 are interpolated

p1 and p2 are approximated

Bezier curves

• Polynomial of any degree• A degree-d Bézier curve has d+1 control points• It passes through the first and last control point, and

approximates the d – 1 other control points• Cubic (degree-3) Bézier curves are most common;

several of these are connected into one curve

Bézier curves

• Cubic Bézier curves are used for font definitions• They are also used in Adobe Illustrator and many

other illustration/drawing programs

Bézier curves

• Parameter u, first control point p0 at u=0 and last control point pd at u=1

• Derivative at p0 is the vector p0p1 , scaled by d

• Derivative at pd is the vector pd-1pd , scaled by d

• Second, third, …, derivatives at p0 depend on the first three, four, …, control points

Cubic Bézier curve example

Quintic Bézier curve example

p1p0 p1

p4 p55

p0 p15

Cubic Bézier curves

• p0 = f(0) = a0 + 0 a1 + 02 a2 + 03 a3 p3 = f(1) = a0 + 1 a1 + 12 a2 + 13 a3

3(p1 – p0) = f’(0) = a1 + 20 a2 + 302 a3 3(p3 – p2) = f’(1) = a1 + 21 a2 + 312 a3

1CBbasis matrix

Cubic Bezier curves

• f(u) = (1 – 3u + 3u2 – u3) p0

+ ( 3u – 6u2 + 3u3) p1

+ ( 3u2 – 3u3) p2

+ ( u3) p3

• Bézier blending functions b0,3 = (1 – u)3

b1,3 = 3 u (1 – u)2

b2,3 = 3 u2 (1 – u)

b3,3 = u3

ibu pf

Bézier curves

• In general (degree d): bk,d(u) = C(d,k) uk (1 – u)d-k

where , for 0 k d

(binomial coefficients)

• The bk,d(u) are called Bernstein basis polynomials

Bézier curves

degrees 2 (left) up to 6 (right)

Properties of Bézier curves

• The Bézier curve is bounded by the convex hull of the control points intersection tests with a Bézier curve can be avoided if there is no intersection with the convex hull of the control points

• Any line intersects the Bézier curve at most as often as that line intersects the polygonal lie through the control points (variation diminishing property)

• A Bézier curve is symmetric: reversing the control points yields the same curve, parameterized in reverse

• A Bézier curve is affine invariant: the Bézier curve of the control points after an affine transformation is the same as the affine transformation applied to the Bézier curve itself (affine transformations: translation, rotation, scaling, skewing/shearing)

• There are simple algorithms for Bézier curves– evaluating– subdividing a Bézier curve into two Bézier curves allows

computing (approximating) intersections of Bézier curves

kdk uukdCu pf

)1(),()(

the point at parameter value u on the Bézier curve

Bézier curves in PowerPoint

• The curve you draw in PowerPoint is a Bézier curve; however you don’t specify the intermediate two control points explicitly– Select draw curve– Draw a line segment (p0 and p3)

– Right-click; edit points– Click on first endpoint and move the appearing marker (p1)

– Click on last endpoint and move the appearing marker (p2)

Splines from Bézier curves

• To ensure continuity– C0 : last control point of first piece must be same as first

control point of second piece

– G1 : last two control points of first piece must align with the first two control points of the second piece

– C1 : distances must be the same as well

Intuition for Bézier curves

• Keep on cutting corners to make a “smoother” curve• In the limit, the curve becomes smooth

Intuition for Bézier curves

• Suppose we have three control points p0 , p1 , p2;a linear connection gives two edges

• Take the middle p3 of p0p1, and the middle p4 of p1p2 and place p’1 in the middle of p3 and p4

• Recurse on p0, p3, p’1 and also on p’1 , p4, p2

gives a quadratic Bézier curve 20

De Casteljau algorithm

• Generalization of the subdivision scheme just presented; it works for any degree– Given d+1 points p0, p1, …, pd

– Choose the value of u where you want to evaluate

– Determine the u-interpolation for p0 p1 , for p1 p2 , … ,

and for pd-1 pd , giving d points

– If one point remains, we found f(u), otherwise repeat the previous step with these d points (one less than before)

p2u = 1/3

one point remains, the point on the curve at u = 1/3

Splitting a Bézier curve

• The De Casteljau algorithm can be used to split a Bezier curve into two Bézier curves that together are the original Bézier curve

Question: Recalling that Bézier splines are C1 only if (in this case) the vector q2q3 is the same as r0r1 , does this mean that the spline is no longer C1 after splitting?!?

Answer: q0q1q2q3 parameterizes the part u [0, 1/3] and r0r1r2r3 parameterizes the part u [1/3, 1] The condition for C1 continuity, q2q3 = r0r1 , applies only for equal parameter-length parameterizations

• Splitting a Bézier curve is useful to find line-Bézier or Bézier-Bézier intersections

u = ½

Intersecting a Bézier curve

• To test if some line L intersects a Bézier curve with control points p0, p1, …, pd , test whether L intersects the poly-line p0, p1, …, pd – If not, L does not intersect the Bézier curve either– Otherwise, split

the Bézier curve (with u = ½ ) andrepeat on the two pieces

• If the line L separates the two endpoints (p0 and p3 for cubic) of a Bézier curve, then they intersect

• Repeating the split happens often only if the line L is nearly tangent to the Bézier curve

• If the line L separates the two endpoints (p0 and p3 for cubic) of a Bézier curve, then they intersect

• Repeating the split happens often only if the line L is nearly tangent to the Bézier curve

• When determining intersection of a line segment and a Bézier curve we must make some small changes

Splitting a Bézier curve for rendering

• Splitting a Bézier curve several times makes the new Bézier curve pieces be closer and closer to their control polygons

• At some moment we can draw the sequence of control polygons of the pieces and these will approximate the Bézier curve well (technically this approximation is only C0)

u = ½

p3p2p0 p1

u = 1/4

u = 1/2u = 3/4

p3p2p0 p1

u = 1/4

u = 1/2u = 3/4

p3u = 1/4

u = 1/2u = 3/4

De Casteljau on quadratic, cubic and quartic Bézier curves

3D Bézier surfaces

The 16 blending functions for cubic Bezier surfaces

Summary

• Bézier curves are elegant curves that pass through the start and end points and approximate the points in between

• They exist of any order (degree) but cubic is most common and useful

• Continuity between consecutive curves can be ensured

• The De Casteljau algorithm is a simple way to evaluate or split a Bézier curve

Questions

1. Consider figure 15.11, bottom left. It looks like a circular arc, but it is not. Determine whether the quadratic Bézier curve shown here goes around the midpoint of a circular arc or not

2. Can we ensure higher degrees of continuity than C1 with cubic Bézier splines? Discuss your answer

3. Suppose we want to define a closed Bezier curve of degree d. What properties must the control points have to make a C1 continuous curve? What is the minimum degree of the Bézier curve that is needed for this? What if we want a closed Bézier curve with an inflection point (boundary of a non-convex region)?

Questions

4. Apply the De Casteljau algorithm on the points (0,0), (4,0), (6,2), and (4,6) with u = ½ by drawing the construction (note that this is a cubic Bézier curve)

5. Apply the De Casteljau algorithm on the points (0,0), (4,0), (6,2), (6,8), and (10,4) with u = ½ by drawing the construction (note that this is a quartic Bézier curve)

Splines III – Bézier Curves

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