Computer graphics(parametric cubic curves)

Cubic curves are commonly used in graphics because curves of lower order commonly have too little flexibility, while curves of higher order are usually considered unnecessarily complex and make it easy to introduce undesired wiggles.

1. PARAMETRIC CUBIC CURVES Presented by SINTHIA SARKER Dept. of CSE GREEN UNIVERSITY OF BANGLADESH 7/21/2020 1

2. CONTENTS : Curve and Surface Parametric cubic curves Types Mathematical complexity 7/21/2020 2

3. CURVE AND SURFACE Curve: In computer graphics, we often need to draw different types of objects onto the screen. Objects are not flat all the time and we need to draw curves many times to draw an object. Types: Implicit curves Explicit curves Parametric curves 7/21/2020 3

4. CNTD….. Surface : Objects are represented as a collection of surfaces. Most common representation for surfaces: Polygon mesh Parametric surfaces Quadric surfaces 7/21/2020 4

5. REPRESENTATION There are many ways to represent curves and surfaces . We want a representation that is - Stable - Smooth - Easy to evaluate 7/21/2020 5

6. APPLICATIONS We need smooth curves and surfaces in many applications: - Model real world objects - Computer-aided design (CAD) - High quality fonts - Data plots - Artists sketches 7/21/2020 6

7. PARAMETRIC CUBIC CURVE P olylines and polygons: Large amounts of data to achieve good accuracy. Interactive manipulation of the data is tedious. Higher-order curves: More compact (use less storage). Easier to manipulate interactively. Possible representations of curves: explicit, implicit, and parametric 7/21/2020 7

8. TYPES There are Three Types of Parametric Cubic Curves. Hermite Curves: Defined by two endpoints and two endpoint tangent vectors (used 1 st order) Bézier Curves: Defined by two endpoints and two control points which control the endpoint’ tangent vectors Splines : Defined by four control points 7/21/2020 8

9. PARAMETRIC CUBIC CURVE General form: c= coefficient matrix T= parameter matrix 7/21/2020 9

10. Tangent Vector If cubic the derivation…. 7/21/2020 10

11. CONTD….. Rewrite the coefficient matrix as where M is a 4  4 basis matrix , G is called the geometry matrix . So, where is called the blending function 7/21/2020 11

12. MATHEMATICAL COMPLEXITY Why cubic? lower-degree polynomials give too little flexibility in controlling the shape of the curve higher-degree polynomials can introduce unwanted wiggles and require more computation lowest degree that allows specification of endpoints and their derivatives lowest degree that is not planar in 3D 7/21/2020 12

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