Chapter 5

1. Chapter 5 Curves and Fractals

2. Curve Generation • Incomputergraphics,weoftenneedtodrawdifferenttypesofobjectsontothescreen. • Objects are not flat all the time and we need to draw curves many times to draw an object.

3. Curve • A curve is an infinitely large set of points. Each point has two neighbors except endpoints in specific direction.

4. Bezier curve • Bezier curve is discovered by the French engineer Pierre Bézier. These curves can be generated under the control of other points.

5. Bezier curve • The simplest Bézier curve is the straight line from the point P0 to P1. • A quadratic Bezier curve is determined by three control points. • AcubicBeziercurveisdeterminedbyfour controlpoints.

6. Properties of Bezier Curves • They always pass through the first and last control points. • The degree of the polynomial 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.

7. Properties of Bezier Curves • A Bezier curve generally follows the shape of the defining polygon. • The direction angle of the end points is same as that of the first and last segments • A given Bezier curve can be subdivided into two Bezier segments which join together to form a new shapes.

8. Fractals

9. Fractals • Fractals are very complex pictures generated by a computer from a single formula. • They are created using iterations. • This means one formula is repeated with slightly different values over and over again, taking into account the results from theprevious iteration.

10. Fractals are used in many areas • Astronomy: For analyzing galaxies , rings of Saturn • Biology/Chemistry: For depicting bacteria cultures, Chemical reactions, Human anatomy, molecules, Plants etc

11. Koch Fractals (Snowflakes) 1/3 1/3 1 1/3 1/3 Generator Iteration 0 Iteration 1 Iteration 2 Iteration 3

12. Fractal Tree Generator Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5

13. Fractal Fern Generator Iteration 0 Iteration 1 Iteration 2 Iteration 3

14. Fractals

15. GenerationofFractals Fractals can be generated by repeating the same shape over and over again as shown in the following figure. In figure (a) shows an equilateral triangle. Fig: (a)

16. Fractals Infigure(b),wecan see that the triangle is repeated to create a star-like shape. Fig:(b)

17. Fractals In figure (c), we can see that the star shape in figure (b) is repeated again and again to create a newshape. Fig: (c)

18. 3D Transformation In the 2D system, we use only two coordinates X and Y but in 3D, an extracoordinate Z is added. 3D graphics techniques and their application are fundamental to the entertainment, games, and computer-aided design industries.

19. Translation In 3D translation, we transfer the Z coordinate along with the X and Y coordinates. The process for translation in 3D is similar to 2D translation. A translation moves an object into a different position on the screen. Figure: 3DTranslation

20. Rotation 3D rotation is not same as 2D rotation. In 3D rotation, we have to specify the angleof rotation along with the axis of rotation. We can perform 3D rotation about X, Y, and Z axes. Figure: 3DRotation

21. Rotation The following figure explains the rotation about variousaxis:

22. Scaling You can change the size of an object using scaling transformation. In the scaling process, you either expand or compress the dimensions of the object. Scaling can be achieved by multiplying the original coordinates of the object with the scaling factorto get the desired result. Figure: 3DScaling

23. Shear A transformation that slants the shape of an object is called the shear transformation. Like in 2D shear, we can shear an object along the X-axis, Y-axis, or Z-axis in 3D. Figure: 3DShearing

24. Thank u..