Curves and surfaces
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This document discusses techniques for modeling curves and surfaces in computer graphics. It introduces three common representations of curves and surfaces: explicit, implicit, and parametric forms. It focuses on parametric polynomial forms, specifically discussing cubic polynomial curves, Hermite curves, Bezier curves, B-splines, and NURBS. It also covers rendering curves and surfaces by evaluating polynomials, recursive subdivision of Bezier polynomials, and ray casting for implicit surfaces like quadrics. Finally, it discusses mesh subdivision techniques like Catmull-Clark and Loop subdivision for generating smooth surfaces.
Curves and surfaces
- 1. Curves and Surface Alzaiem Alazhari University College of computer Science and Information Technology Chapter 10 – Advanced Computer Graphics 1
- 2. Curves and Surface The world around us is full of objects of remarkable shapes. Nevertheless, in computer graphics, we continue to populate our virtual worlds with flat objects. We have a good reason for such persistence. Graphics systems can render flat three-dimensional polygons at high rates, including doing hidden-surface removal, shading, and texture mapping.. 2
- 3. Curves and Surface We introduce three ways to model curves and surfaces, paying most attention to the parametric polynomial forms. We also discuss how curves and surfaces can be rendered on current graphics systems, a process that usually involves subdividing the curved objects into collections of flat primitives. 3
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- 5. REPRESENTATION OF CURVES AND SURFACES Explicit Representation The explicit form of a curve in two dimensions gives the value of one variable, the dependent variable, in terms of the other, the independent variable. In x, y space, we might write y = f (x). a surface represented by an equation of the form z = f (x, y) 5
- 6. REPRESENTATION OF CURVES AND SURFACES Implicit Representations In two dimensions, an implicit curve can be represented by the equation f (x, y) = 0 The implicit form is less coordinate-system dependent than is the explicit form. In three dimensions, the implicit form f (x, y, z) = 0 Curves in three dimensions are not as easily represented in implicit form. We can represent a curve as the intersection, if it exists, of the two surfaces: f (x, y, z) = 0, g(x, y, z) = 0. 6
- 7. REPRESENTATION OF CURVES AND SURFACES Parametric Form The parametric form of a curve expresses the value of each spatial variable for points on the curve in terms of an independent variable, u, the parameter. In three dimensions, we have three explicit functions: x = x(u) , y = y(u) , z = z(u). One of the advantages of the parametric form is that it is the same in two and three dimensions. In the former case, we simply drop the equation for z. Parametric surfaces require two parameters. We can describe a surface by three equations of the form : x = x(u, v) , y = y(u, v) , z = z(u, v), 7
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- 9. DESIGN CRITERIA There are many considerations that determine why we prefer to use parametric polynomials of low degree, including: Local control of shape Smoothness and continuity Ability to evaluate derivatives Stability Ease of rendering 9
- 10. CROSSE SECTION Approximation of cross-section curve Derivative discontinuity at join point 10
- 11. PARAMETRIC CUBIC POLYNOMIAL CURVES Once we have decided to use parametric polynomial curves, we must choose the degree of the curve. if we choose a high degree, we will have many parameters that we can set to form the desired shape, but evaluation of points on the curve will be costly. In addition, as the degree of a polynomial curve becomes higher, there is more danger that the curve will become rougher. On the other hand, if we pick too low a degree, we may not have enough parameters with which to work. 11
- 12. PARAMETRIC CUBIC POLYNOMIAL CURVES However, if we design each curve segment over a short interval, we can achieve many of our purposes with low- degree curves. Although there may be only a few degrees of freedom these few may be sufficient to allow us to produce the desired shape in a small region. For this reason, most designers, at least initially, work with cubic polynomial curves 12
- 13. Cubic interpolating polynomial • First example of a cubic parametric polynomial. • Although we rarely used • Illustrates the steps we must follow for our other types . 13
- 14. Interpolating Curve • Given 4 control points P0, P1, P2, P3 • Space 0 <= u <= 1 evenly • P0 = P(0), P1 = P(1/3), P2 = P(2/3), P3 = P(1) 14
- 15. Interpolation Equations Apply the interpolating conditions at u=0, 1/3, 2/3, 1 15
- 16. Interpolation Equations We can write these equations in matrix form as 16
- 17. Interpolation Matrix Solving for c we find the interpolation matrix 17
- 18. Blending Functions Rewriting the equation for p(u) . 18
- 19. The Cubic Interpolating Patch Shows that we can build and analyze surfaces from our knowledge of curves 19
- 20. HERMITE CURVES AND SURFACES Another cubic polynomial curve Specify two endpoints and their tangents 20
- 21. The Hermite Form As Before Calculate derivative Yields 21
- 22. Bezier Curves Widely used in computer graphics Approximate tangents by using control points 22
- 23. Analysis Bezier form Is much better than the interpolating form But the derivatives are not continuous at join points What shall we do to solve this ? 23
- 24. B-Splines Basis Splines Allows us to apply more continuity the curve must lie in the convex hull of the control points 24
- 25. Spline Surfaces B-spline surfaces can be defined in a similar way 25
- 26. GENERAL B-SPLINES We can extend to splines of any degree Data and conditions to not have to given at equally spaced values (the knots) Nonuniform and uniform splines Can have repeated knots Cox-deBoor recursion gives method of evaluation 26
- 27. NURBS Nonuniform Rational B-Spline curves and surfaces add a fourth variable w to x,y,z Can interpret as weight to give more importance to some control data Can also interpret as moving to homogeneous coordinate Requires a perspective division NURBS act correctly for perspective viewing Quadrics are a special case of NURBS 27
- 28. Rendering Curves and Surfaces Introduce methods to draw curves For explicit and parametric: we can evaluate the curve or surface at a sufficient number of points that we can approximate it with our standard flat objects For implicit surfaces: we can compute points on the object that are the intersection of rays from the center of projection through pixels with the object 28
- 29. Evaluating Polynomials Simplest method to render a polynomial curve is to evaluate the polynomial at many points and form an approximating polyline For surfaces we can form an approximating mesh of triangles or quadrilaterals Use Horner’s method to evaluate polynomials p(u)=c0+u(c1+u(c2+uc3)) 29
- 30. Recursive Subdivision of Be´zier Polynomials The most elegant rendering method performs based on the use of the convex hull الهياكل المحدبة never requires explicit evaluation of the polynomial ال يتطلب عرض واضح لكثيرة الحدود 30
- 31. THE UTAH TEAPOT Most famous data set in computer graphics Widely available as a list of 306 3D vertices and the indices that define 32 Bezier patches 31
- 32. THE UTAH TEAPOT - con We can shows the teapot as a wireframe and with constant shading 32
- 33. ALGEBRAIC SURFACES - Quadrics Although quadrics can be generated as special case of NURBS curves Quadrics are described by implicit algebraic equations Quadric can be written in the form : 33
- 34. Quadrics This class of surfaces includes ellipsoids, parabaloids, and hyperboloids We can write the general equation 34
- 35. Rendering of Surfaces by Ray Casting Quadrics are easy to render we can find the intersection of a quadric with a ray by solving a scalar quadratic equation We represent the ray from p0 in the direction d parametrically as scalar equation for α: 35
- 36. SUBDIVISION CURVES AND SURFACES 36
- 37. Mesh Subdivision A theory of subdivision surfaces has emerged that deals with both the theoretical and practical aspects of these ideas. We have two type of meshes: triangles meshes. quadrilaterals meshes. 37
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- 39. Meshes methods Catmull Clark method: use to form a quadrilateral mesh. produces a smoother surface This method tends to move edge vertices at corners more than other outer vertices. 39
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- 41. 2- Loop subdivision method:- 41
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- 44. Seminar Team: Theoretical : Mawada Sayed Mohammed Mohammed Mohammed Mahmoud Ibrahim Musa Hams Ibrahim Mohammed Idris Abdallah Ahmed Modawi Mohammed Ethar Abasher Musa Hamad Practical : Mujahid Ahmed Mohammed Babeker Eltayb Babeker Mohammed Ahmed Salah Eldeen Mohammed Ismail Ibrahim 44