"香港大学 OpenGL课件：第12B章 曲线和曲面详解"

Curves and surfaces play a crucial role in computer graphics, allowing for the creation of complex and realistic shapes. In the field of computer graphics, curves are often approximated with straight line segments, which can be defined using various mathematical functions such as ellipses, parabolas, trigonometric functions, exponential functions, and rational functions. One common method for defining curves is through control points, which are used to determine the shape and trajectory of the curve. This can involve using polynomials that pass through the points, piecewise polynomials such as cubic splines, Bézier curves, B-splines, and Non-Uniform Rational B-Splines (NURBS). B-Splines, in particular, are a popular method for defining curves in computer graphics. The kth degree B-spline curve over a knot vector T = {t0, t1, …, tn+k-1} is defined using a set of control points and basis functions. The basis functions, denoted as Nk,i(t), determine the influence of each control point on the curve at a given parameter value t. By manipulating the control points and basis functions, designers and artists can create smooth and visually appealing curves and surfaces that can be used in a variety of applications such as 3D modeling, animation, and visualization. B-Splines offer flexibility and versatility in creating complex shapes while ensuring mathematical precision and smoothness. In conclusion, curves and surfaces are essential tools in computer graphics for creating visually stunning and realistic imagery. By understanding and utilizing mathematical functions, control points, and basis functions, designers and artists can bring their creative visions to life in the digital world. B-Splines, along with other curve and surface modeling techniques, provide a powerful framework for generating intricate and aesthetically pleasing shapes that enhance the visual quality of digital content.