Computer Graphics

1. Bezier Curve
Amiya Kumar Dash

2. Splines – Old Days
Duck
Spline

3. Spline – Old Days
 Draftsman use ‘ducks’ and strips
of wood (splines) to draw curves
 Wood splines have second-order
continuity
A Duck (weight)
Ducks trace out curve

4. Spline – Some definition
 Control Points
 A set of points that influence the
curve’s shape
 Knots
 Control points that lie on the
curve
 Interpolating Splines
 Curves that pass through the
control points (knots)
 Approximating Splines
 Control points only influence
shape

5. Spline Specification
 Specified by a sum of smaller curve segments represented
by the function ‘Φ’ known as basis or blending function.
 For curve modeling, polynomials are often the blending
function of choice.
 Mathematically,
 Where, i= 0, 1, 2, 3, … , n
 and, p0, p1, … , pn are weights.
p(u) = piFi (u)
i=0
n
å

6. Bézier Curves
 Similar to Hermite, but more intuitive definition of
endpoint derivatives.
 Instead of using control points and slopes (as in Hermite),
Bézier curve can be generated only when control points
are given.
 Given (n+1)-control points, the basis function of Bézier
curve is a n-degree polynomial.

7. Bézier Curves
 Given (n+1)-control points, the basis function of Bézier
curve is a n-degree polynomial.
 The parametric equation of the Bézier curve is:
 Expanding it:
 P(u) = p0BEZ0,n(u) + p1BEZ1,n(u) + … + pnBEZn,n(u)
p(u) = pk BEZk,n (u)
k=0
n
å , where,0£u£1
Point on the curve
Control point
Basis or Coefficient

8. Bézier Curves
 The Bezier blending functions BEZk,n(u) are the Bernstein
polynomials.
BEZk,n (u) = n
Ckuk
(1-u)n-k
Where, n
Ck =
n!
k!(n - k)!
Binomial Coefficient

9. Linear Bézier Curve
 Two control points: p0, p1
 Represents a straight line between these points
 P(u) = p0BEZ0,n(u) + p1BEZ1,n(u) + … + pnBEZn,n(u)
 Here n=1, k=0,1
p(u) = p0BEZ0,1(u) + p1BEZ1,1(u)
= p0
1C0 u0 (1-u)1-0 + p1
1C1 u1 (1-u)1-1
= p0 . 1 . 1. (1-u) + p1 . 1 . u . 1
p(u)= p0(1-u) + p1u
Parametric equation of line

10. Quadratic Bézier Curve
 Three control points: p0, p1, p2
 Represents a parabolic segment
 P(u) = p0BEZ0,n(u) + p1BEZ1,n(u) + … + pnBEZn,n(u)
 Here n=2, k=0, 1, 2
p(u) = p0BEZ0,2(u) + p1BEZ1,2(u) + p2BEZ2,2(u)
= p0
2C0 u0 (1-u)2-0 + p1
2C1 u1 (1-u)2-1 + p2
2C2 u2 (1-u)2-2
= p0 . 1 . 1. (1-u)2 + p1 . 2 . u . (1-u) + p2 . 1 . u2 . 1
p(u)= p0(1-u)2 + 2 p1u(1-u) + p2 u2
Where, 0 ≤ u ≤ 1

11. Cubic Bézier Curve
 Most popular Bézier curve is the cubic Bézier curve.
 Basis function is a cubic polynomial.
 Four control points: p0, p1, p2, p3
 Here, n = 3, k= 0, 1, 2, 3.
p(u) = pk BEZk,3(u)
k=0
3
å
P(u) = p0BEZ0,3(u) + p1BEZ1,3(u) + p2BEZ2,3(u) + p3BEZ3,3(u)
= p0
3C0 u0 (1-u)3-0 + p1
3C1 u1 (1-u)3-1 + p2
3C2 u2 (1-u)3-2 +p3
3C3 u3 (1-u)3-3
= p0 . (1-u)3 + 3 . P1 . u . (1-u)2 + 3 . P2 . u2 . (1-u) + p3 . u3
= p0(1 – u3 – 3u + 3u2) + 3p1u(1 + u2 - 2u) + 3p2u2(1 - u) + p3 u3

12. Cubic Bézier Curve
P(u) = p0BEZ0,3(u) + p1BEZ1,3(u) + p2BEZ2,3(u) + p3BEZ3,3(u)
= p0
3C0 u0 (1-u)3-0 + p1
3C1 u1 (1-u)3-1 + p2
3C2 u2 (1-u)3-2 +p3
3C3 u3 (1-u)3-3
= p0 . (1-u)3 + 3 . P1 . u . (1-u)2 + 3 . P2 . u2 . (1-u) + p3 . u3
= p0(1 – u3 – 3u + 3u2) + 3p1u(1 + u2 - 2u) + 3p2u2(1 - u) + p3 u3
= p0 – p0u3 – 3p0u + 3p0u2 + 3p1u + 3p1u3 – 6p1u2 + 3p2u2 – 3p2u3 + p3u3
p(u) = u3 (- p0 + 3p1 - 3p2 + p3)
+ u2 (3p0 – 6p1 + 3p2)
+ u (-3p0 + 3p1)
+ p0

13. Cubic Bézier Curve
p(u) = u3 (- p0 + 3p1 - 3p2 + p3)
+ u2 (3p0 – 6p1 + 3p2)
+ u (-3p0 + 3p1)
+ p0
p(u) = u3
u2
u 1
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p0
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p2
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MBEZ

14. Blending functions with 4 control points (n=3)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BEZ0,3(u)

15. Blending functions with 4 control points (n=3)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BEZ0,3(u)
BEZ1,3(u)

16. Blending functions with 4 control points (n=3)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BEZ1,3(u)
BEZ0,3(u)
BEZ2,3(u)

17. Blending functions with 4 control points (n=3)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BEZ1,3(u)
BEZ0,3(u)
BEZ2,3(u)
BEZ3,3(u)

18. Properties of Bézier Curve
1) It always passes through the two end points i.e. first and last
control point.
2) The Bézier curve is a straight line iff the control points are
collinear.
3) The slope (tangent) at the beginning of the curve is along
the line joining the first two control points and the slope at
the end of the curve is along the line joining the last two end
points.

19. Properties of Bézier Curve
4) The degree of the polynomial defining the curve segment is
one less than the number of control points.
5) All basis functions are non-negative and their sum is always
1 i.e.
6) The curve generally follows the shape of the defining
polygon.
7) The Bézier curves always lie within the convex hull.
BEZk,n (u)
k=0
n
å =1