Hermit curves & beizer curves

Hermit Curves & Beizer Curves

1. HERMITE CURVES & BEIZER CURVES

2. A Hermite curve is a curve for which the user provides: The endpoints of the curve The parametric derivatives of the curve at the endpoints (tangents with length) The parametric derivatives are dx/ dt , dy / dt , dz / dt That is enough to define a cubic Hermite spline, more derivatives are required for higher order curves. Hermite curves

3. A cubic spline has degree 3, and is of the form : For some constants a, b, c and d derived from the control points Hermite curves Constraints : The curve must pass through p 1 when t=0 The derivative must be ∆p 1 when t=0 The curve must pass through p 2 when t=1 The derivative must be ∆ p 2 when t=1

4. Control point positions and first derivatives are given as constraints for each end-point. → End point constraints for each segment is given as:

5. These polynomials are called Hermite blending functions, and tells us how to blend boundary conditions to generate the position of a point P( u ) on the curve.

6. More degrees of freedom Directly transformable Dimension independent No infinite slope problems Separates dependent and independent variables Inherently bounded Easy to express in vector and matrix form Common form for many curves and surfaces Advantages of parametric forms

7. Bezier Curve

8. Bezier Curve – Parametric Equation

9. Quadratic Bezier Curve - Derivation

10. Contd..

11. De Casteljau’s Algorithm

12. Cubic Bezier Curve

13. Design of n Bezier Curve

14. Binomial Coefficient

15. Cubic Bernstein Polynomial

16. Behaviour of Bernstein Polynomial

17. Matrix Form – Bezier Curves

18. Properties of Bezier Curve

19. They generally follow the shape of the control polygon, which consists of the segments joining the control points. They always pass through the first and last control points. They are contained in the convex hull of their defining control points. The degree of the polynomial defining the curve segment is one less that the number of defining polygon point. Therefore, for 4 control points, the degree of the polynomial is 3, i.e. cubic polynomial. The direction of the tangent vector at the end points is same as that of the vector determined by first and last segments. The convex hull property for a Bezier curve ensures that the polynomial smoothly follows the control points. Bezier curves exhibit global control means moving a control point alters the shape of the whole curve. Properties of Bezier Curve

20. Examples

21. B-splines automatically take care of continuity, with exactly one control vertex per curve segment Many types of B-splines: degree may be different (linear, quadratic, cubic,…) and they may be uniform or non-uniform With uniform B-splines, continuity is always one degree lower than the degree of each curve piece Linear B-splines have C continuity, cubic have C 2 , etc B-Splines – (Basis Spline)

22. B-Spline – Analytical Definition normally called the “Knot Sequence”.

23. Contd.. The N i,k functions are described as follows

24. The sum of the B-spline basis functions for any parameter value is 1. Each basis function is positive or zero for all parameter values. Each basis function has precisely one maximum value, except for k=1. The maximum order of the curve is equal to the number of vertices of defining polygon. The degree of B-spline polynomial is independent on the number of vertices of defining polygon. B-spline allows the local control over the curve surface because each vertex affects the shape of a curve only over a range of parameter values where its associated basis function is nonzero. Properties of B-Spline Curves

25. Bezier Curve vs B-Spline - Control

26. The choice of a knot vector directly influences the resulting curve. Types of Knot vectors Uniform (periodic) Open-Uniform Non-Uniform Knot Vectors

27. Open curves expect that do not passes through the first and last control points and therefore are not tangent to the first and last segment of the control polygon. Closed curves where the first and last control points curve connected. Closed curves with the first and last control point being the same (coincident ). Open and closed B-spline curves

28. When the spacing between knot values is constant, the resulting curve is called a uniform B-spline. The spacing between knot values is not constant and hence ,any values and intervals can be specified for the knot vector the curve is called a non uniform B-spline . Different intervals which can be used to adjust spline shapes . Uniform and non uniform B-spline curves

29. Rational curves is defined as ratio of two polynomials. Non rational curves is defined by one polynomials. The most widely used rational curves are non uniform rational B-splines (NURBS) NURBS is capable of representing in a single form non –rational B-splines and Bezier curves as well as linear and quadratic analytic curves. Rational Curves

30. Add weights to each points Unweighted : Where Weighted NURBS- Non Uniform Rational B-Splines

31. NURBS

32. Rational curve can handle both analytical and synthetic curves. Rational curve represents a point with homogenous coordinate system where a 3D coordinate system is expressed as ( wi x, wi y , wi z,wi ). Using same control points with different weights ,different curves can be generated. The weight associated with each control point can affect the curve locally and the curve is pulled towards the control point with increases valued of its weight wi . Characteristics of Rational curves

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