Lecture
In many fields of computer graphics there arises a need to represent three-dimensional shapes: in design, in reconstructing three-dimensional bodies from an image.
by their cross-sections, constructed using computer graphics, and many others. The need to describe spatial objects arises in two situations:
1) When we want to describe an object that already exists. In the general case, a real existing object cannot fully correspond to its representation. For this, in the worst case, an infinite number of triples (𝑥, 𝑦, 𝑧) would be required – one for each point of the object's surface. In the best case, the object can be represented by a combination of mathematical surfaces (planes, spheres, etc.). As a rule, we choose a certain set of points on the object and require that the representation of the object correspond to the object itself, with a given accuracy, only at these selected points. The representation of existing surfaces must therefore satisfy these constraints.
2) Computer-aided design. In this case there is no pre-existing model of the object being designed: the designer, guided perhaps only by an approximate sketch, creates the spatial shape interactively. The method used must be convenient for working in dialogue mode, so that the surface can easily be brought to the desired shape.
The most common ways of representing three-dimensional surfaces in space are: polygonal meshes and parametric bicubic patches.
Parametric bicubic patches describe the coordinates of points on a curved surface using three equations (one each for 𝑥, 𝑦, 𝑧).
Each equation has two variables (parameters), and the exponents on them are no higher than the third (hence the name bicubic). The boundaries of the patches are parametric bicubic curves. To represent surfaces with a given accuracy, a considerably smaller number of bicubic patches is required than when approximating with a polygonal mesh. However, algorithms for working with bicubic objects are substantially more complex than algorithms dealing with polygons.
When using either method, a three-dimensional body is represented as a closed surface.
In contrast, when modeling solid bodies, one deals with rigid objects. As a rule, many 3D editors and libraries use primitives such as a cube, cone, sphere, cylinder, torus, pipe, box, plane, and, for some reason, a teapot, which can be combined and subtracted to form various shapes.
To describe surfaces, let us first consider the representation of spatial curves, and then make a mathematical generalization from spatial curves to three-dimensional curved surfaces.
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