Bezier Curve and Spline Curve
- 1. Bezier Curve & Spline Curve Presented By: Md. Asif Shahariar Shovan Banik Iftekher Aziz 162-15-7772 162-15-7790 162-15-8182
- 2. • Parametric curve • Known form 1912 • Pierre Bezier • Renault Cars • Study developed 1959 by Paul de Castelju Bezier Curve
- 3. A Bézier curve is defined by its order (quadratic, cubic, etc.) and a set of control points PO through P1 , the number n of which depends on the order (n = 2 for linear, 3 for quadratic, etc.). The first and last control points are always the end points of the curve; however, the intermediate control points (if any) generally do not lie on the curve. What is Bezier Curve?
- 4. Spline Curve • Computer aided design & Computer Graphics • Frequently refers to a piecewise polynomial • Data interpolation and/or data smoothing
- 5. A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces. What is Spline Curve?
- 6. Bezier Curve
- 7. Properties of Bezier Curve • The first and last control points are interpolated. • The tangent to the curve at the first control point is along the line joining the first and second control points. • The tangent at the last control point is along the line joining the second last and last control points • The curve lies entirely within the convex hull of its control points. • The Bernstein polynomials (the basis functions) sum to 1 and are everywhere positive. • They can be rendered in many ways.
- 8. Properties of Spline Curve • It provide local control of the curve shape. • It also provide the ability to add control points without increasing the degree of the curve • It have the ability to interpolate or approximate a set of given data points. • The form of a spline curve is very similar to that of a Bezier curve.
- 9. Bezier Curve Construction Linear curves: The t in the function for a linear Bézier curve can be thought of as describing how far B(t) is from P to P . For example when t=0.25, B(t) is one quarter of the way from point P0 to P1 . As t varies from 0 to 1, B(t) describes a straight line from P to P .
- 10. Bezier Curve Construction Con. Quadratic curves: For quadratic Bézier curves one can construct intermediate points Q0 and Q1 such that as t varies from 0 to 1. Point Q0 varies from P0 to P1 and describes a linear Bézier curve. Point Q1 varies from P1 to P2 and describes a linear Bézier curve. oint B(t) varies from Q0 to Q1 and describes a quadratic Bézier curve.
- 11. Bezier Curve Construction Con. Higher-Order curves Fourth order curves
- 12. Bezier Curve Construction Con. Fifth order curves
- 13. Bezier Curve Application • Bezier curves are used in the time domain, particularly in animation and interface design, e.g., a Bezier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step.
- 14. Bezier Curve Application Con. • Bezier curves are widely used in computer graphics to model smooth curves. • Quadratic and cubic Bézier curves are most common; higher degree curves are more expensive to evaluate. When more complex shapes are needed, low order Bézier curves are patched together.
- 15. Design Techniques Bezier Curve (Weights) • Multiple control points at a single coordinate position gives more weight to that position.
- 16. Bezier Curve Design Techniques Con. • When Complicated curves are to be generated, they can be formed by piecing several Bezier sections of lower degree together. • Piecing together smaller sections gives us better control over the shape of the shape of the curve in small region. • It is easy to match curve section
- 17. Bezier Curve Design Techniques Con. • Since the tangent to the curve at an endpoint is along the line joining that endpoint to the adjacent control point.
- 18. Thank You