Bezier curve & B spline curve

Bezier curve & B spline curve

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  BEZIER CURVES AND B-SPLINECURVE By: Arvind Kumar Assistant Professor (Vidya College of Engineering)
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  Contents By: Arvind Kumar CG Curvesand Surface Quadric Surface Spheres Ellipsoid Blobby Objects Bezier Curves and its Numerical B-Spline Curve
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  Curves and Surface By:Arvind Kumar CG Quadric Surface: It is described with Second Degree Equation (quadratics). Quadric surfaces, particularly spheres and ellipsoids, are common elements of graphics scenes. Spheres: In Cartesian coordinates, a spherical surface with radius r centered on the coordinate origin is defined as the set of points (x, y, z) that satisfy the equation x2 + y2 + z2 = r2
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  Curves and Surface By:Arvind Kumar CG Ellipsoid: An ellipsoidal surface can be described as an extension of a spherical surface, where the radii in three mutually perpendicular directions can have different values. Blobby Objects: Irregular surfaces and change their surface characteristics in certain motion. Called blobby objects. e.g cloth, rubber melting object water droplets etc.
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  Concept of Spline By:Arvind Kumar CG Spline: Spline means a flexible strip used to produce a smooth curve through a designated set of points. A Spline curve specify by a giving set of coordinate positions called “Control Point”. Interpolation is when a curve passes through a set of “control points.” ● ● ● Approximation is when a curve approximates but doesn’t necessarily contain its control points.
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  Bezier Curve By: ArvindKumar CG Bezier Curve: Bezier curves are defined using four control points, known as knots. Two of these are the end points of the curve, while the other two points control the shape of the curve P0 P1 P2 P1 P2 P3 P0
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  Bezier Curve By: ArvindKumar CG Suppose the Curve has n + 1 control-point positions: pk = (xk, yk, zk), with k varying from 0 to n. These coordinate points can be blended to produce the following position vector P(u), which describes the path of an approximating Bezier polynomial function between p0 and pn. The position of vector can be given 𝐏 𝐮 = 𝒌=𝟎 𝒏 𝑷 𝒌 𝑩𝑬𝒁 𝒌,𝒏 𝒖 𝟎 ≤ 𝒖 ≤ 𝟏
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  Bezier Curve By: ArvindKumar CG The Bezier blending functions BEZk,n(u) are the Bemstein polynomials: 𝑩𝑬𝒁 𝒌,𝒏 (u) = C(n,k) 𝒖 𝒌(1−u) 𝒏−𝒌 where the C(n, k) are the binomial coefficients 𝐶 𝑛, 𝑘 = 𝑛! 𝑘! (𝑛 − 𝑘)! The Bezier blending functions with the recursive calculation 𝑩𝑬𝒁 𝒌,𝒏 (u) = (1−u) 𝑩𝑬𝒁 𝒌,𝒏−𝟏(u) + u 𝑩𝑬𝒁 𝒌−𝟏,𝒏−𝟏(u) ,n>k ≥ 1 Where, BEZ k,k = uk , and BEZ0,k = (1 – u)k .
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  Bezier Curve By: ArvindKumar CG A set of three parametric equations for the individual curve coordinates. 𝒙 𝒖 = 𝒌=𝟎 𝒏 𝒙 𝒌 𝑩𝑬𝒁 𝒌,𝒏(u) y 𝒖 = 𝒌=𝟎 𝒏 𝒚 𝒌 𝑩𝑬𝒁 𝒌,𝒏(u) 𝒛 𝒖 = 𝒌=𝟎 𝒏 𝒛 𝒌 𝑩𝑬𝒁 𝒌,𝒏(u)
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  Bezier Curve By: ArvindKumar CG Properties of Bezier Curve: 1. Bezier curve always passes through the first & last control points. 2. The degree of polynomial defining the curve segment is one less than the number of defining polygon points. Therefore for 4, control points the degree of polynomial is three. 3. The curve generally follows the shape of the defining polygon. 4. The curve lies entirely within the convex hull formed by control points. 5. The curve exhibits the variation diminishing property. This means that the curve does not oscillate about any straight- line move often than the defining polygon 6. The curve is invariant under an affine transformation.
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  Bezier Curve -Disadvantage By: Arvind Kumar CG It has two main disadvantages. • The number of control points is directly related to the degree. Therefore, to increase the complexity of the shape of the curve by adding control points requires increasing the degree of the curve or satisfying the continuity conditions between consecutive segments of a composite curve. • Changing any control points affects the entire curve or surface, making design of specific sections very difficult.
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  Bezier Curve- Numerical By:Arvind Kumar CG Q1. Construct the bezier curve of order 3 and with 4 polygon vertices A(1,1) , B(2,3) , C(4, 3),D(6,4). Solution: the equation of bezier curve is given as 𝑷 𝒖 = 𝟏 − 𝒖 𝟑P1 +𝟑𝒖(𝟏 − 𝒖) 𝟐 𝑷𝟐 + 𝟑𝒖 𝟐 𝟏 − 𝒖 𝑷𝟑 + 𝒖 𝟑P4 for 0≤u≤1 Where P(u) is the point on curve Let us take u = 0, ¼, ½, ¾, 1 𝐏 𝐮 = 𝒌=𝟎 𝟑 𝑷 𝒌 𝑩𝑬𝒁 𝒌,𝒏 𝒖
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  Bezier Curve- Numerical By:Arvind Kumar CG At u=0, P(0) = (1,1) At u =1/4 P(1/4)= 1 − 1/4 3 P1+3 ∗ 1/4(1 − 1/4)2 𝑃2 + 3 ∗ ( 1 4 )2 1 − 1/4 𝑃3 + ( 1 4 )3 = 1 − 1 4 3 (1,1)+3 ∗ 1/4(1 − 1/4)2 (2,3) + 3 ∗ 1 4 2 1 − 1 4 (4,3) + ( 1 4 )3 (6,4) = 27/64(1,1)+ 27/64 (2,3) + 9/64 (4,3) + 1/64(6,4) = [27/64 *1 +27/64 * 2 + 9/64 *4 + 1/64 *6, 27/64 *1 +27/64 * 3 + 9/64 *3 + 1/64 *4] = [123/64, 139/64] = (1.9218 , 2.1718) At u= ½ , P(1/2) = (3.125, 2.87) At u = ¾, P(3/4) = (4.5156, 3.375) At u= 1 , P(1) = (6,4) These are control points for Bezier Curve.
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  B-Spline Curve By: ArvindKumar CG The disadvantages are remedied with the B-spline (basis- spline) representation. The coordinate positions along a B-spline curve in a blending- function formulation as 𝑷 𝒖 = 𝒌=𝟎 𝒏 𝒑 𝒌 𝑩 𝒌,𝒅(𝒖) , 𝒖 𝒎𝒊𝒏 ≤ 𝐮 ≤ 𝒖 𝒎𝒂𝒙 Where pk are an input set of n + 1 control points.
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  B-Spline Curve By: ArvindKumar CG Blending functions for B-spline curves are defined by 𝑩 𝒌,𝟏(𝒖 ) = 𝟏 , 𝑖𝑓 𝑢 𝑘 ≤ 𝑢 ≤ 𝑢 𝑘+1 𝟎 , 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑩 𝒌,𝒅(𝒖 ) = 𝒖−𝒖 𝒌 𝒖 𝒌+𝒅−𝟏 𝑩 𝒌,𝒅−𝟏(𝒖 ) + 𝒖 𝒌+𝒅−𝒖 𝒖 𝒌+𝒅−𝒖 𝒌+𝒅 𝑩 𝒌+𝟏,𝒅−𝟏(𝒖 ) Where, each blending function is defined over d subintervals of the total range of u. The selected set of subinterval endpoints u, is referred to as a knot vector.
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  B-Spline Curve By: ArvindKumar CG Types of Knot vector: 1. Uniform Knot: In a Uniform Knot vector individual knot values are evenly spaced e.g. [0 1 2 3 4]. 2. Open Uniform Knot: It has multiplicity of knot values at the ends equal to the order k of B-Spline basis function. Internal knot values are evenly spaced. e.g. k=2 [0 0 1 2 3 3] k= 3[0 0 0 1 2 3 3 3 ] k= 4[0 0 0 0 1 2 2 2 2 ]
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  B-Spline Curve By: ArvindKumar CG Properties of B-spline curve: 1. The degree of B-Spline polynomial is independent of the number of vertices of the polygon. 2. The curve generally follows the shape of the defining polygon. 3. The curve lies within the convex hull of its defining polygon. 4. The curve exhibits the variation diminishing property. This means that the curve does not oscillate about any straight- line move often than the defining polygon. 5. The B-Spline allows local control over the curve surface because each vertex affects the shape of the curve only over a range of parameter values where its associated basic function is non-zero.