Computer Graphics and Geometric Modeling

Illumination and Shading (Basic Computer Graphics) Part 4

The Rendering Pipeline In this and previous topics we have covered many different topics that deal with getting a real picture onto a computer’s monitor screen. We started be describing the basic graphics (projection) pipeline and then moved on to visible surface algorithms and illumination and shading models. Everything might have made sense in isolation, […]

Rendering Techniques (Basic Computer Graphics) Part 1

Introduction As computers get faster and faster the goal of producing very realistic images in real time is becoming attainable even on a PC. This topic describes ray tracing and the radiosity method, methods that generate the most impressive images. We also briefly discuss volume rendering. In this topic, we describe ray tracing in more […]

Rendering Techniques (Basic Computer Graphics) Part 2

Ray Tracing CSG Objects Ray tracing CSG objects is much simpler than constructing them. The basic idea, described in [Roth82], only involves computing ray intersections with the primitives used in the CSG tree that defines the object and then combining the resulting onedimensional segments with the set operations from that CSG tree. Let T be […]

Rendering Techniques (Basic Computer Graphics) Part 3

Volume Rendering Up to now, the type of rendering of three-dimensional objects we have been discussing is sometimes called surface rendering, because it assumed that objects were represented by their boundary and so that was what we had to display. The “interior” of an object was never considered. This approach works fine for objects that […]

Rendering Techniques (Basic Computer Graphics) Part 4

The Marching Cubes Algorithm The marching cubes algorithm was developed independently by [WyMW86] and [LorC87]. We follow the outline of the algorithm presented in [LorC87]. For an implementation of the algorithm we refer the interested reader to [WatW92]. Algorithm 10.4.1.2. The 6-connected tripod line drawing. The setting for the algorithm is the following: It is […]

Curves in Computer Graphics (Geometric Modeling) Part 1

Terminology. The terms “curve" and "surface" get used in different ways. Sometimes people use them to refer to sets, other times, to functions. Although it is usually clear from the context which is meant, this ambiguity can lead to confusion because there are times when that distinction plays a role. For that reason we want […]

Curves in Computer Graphics (Geometric Modeling) Part 2

Hermite Interpolation To avoid the polynomial oscillation problem with Lagrange interpolation, one could piece together polynomials of degree two, but one would in general get corners where they meet. However, if we try using cubic polynomials, then we have enough degrees of freedom to force the polynomials to have the same slope where they meet. […]

Curves in Computer Graphics (Geometric Modeling) Part 3

Spline Interpolation The interpolation problems described in the last section and the functions that solve them can be generalized. Definition. A spline of degree m and order m + 1 is a function.    for    which there exist real numbersso that (1) S is a polynomial of degreeand (2) S is a Cm-1 function. The xi […]

Curves in Computer Graphics (Geometric Modeling) Part 4

Bezier Curves This section and the next will deal with curves that are defined by control points but do not interpolate them in general. We shall return to the interpolation problem in Section 11.5.5. Although the geometric coefficients approach to defining curves is a big improvement over having to specify the algebraic coefficients, specifying tangent […]

Curves in Computer Graphics (Geometric Modeling) Part 5

B-Spline Curves The Standard B-Spline Curve Formulas One common problem with the curves discussed so far is that any change to a control point forces recomputation of the whole curve. This is very undesirable. B-spline curves solve this problem. Changes to control points will have only a local effect on the curve. B-splines can be […]

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