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A swept surface is generated from a profile curve and a sweep curve by employing the latter to define a continuous family of transformations of the former. By using polynomial or rational curves, and specifying the homogeneous coordinates of the swept surface as bilinear forms in the profile and sweep curve homogeneous coordinates, the outcome is guaranteed to be a rational surface compatible with the prevailing data types of CAD systems. However, this approach does not accommodate many geometrically intuitive sweep operations based on differential or integral properties of the sweep curve - such as the parametric speed, tangent, normal, curvature, arc length, and offset curves - since they do not ordinarily have a rational dependence on the curve parameter. The use of Pythagorean-hodograph (PH) sweep curves surmounts this limitation, and thus makes possible a much richer spectrum of rational swept surface types. A number of representative examples are used to illustrate the diversity of these novel swept surface forms - including the oriented-translation sweep, offset-translation sweep, generalized conical sweep, and oriented-involute sweep. In many cases of practical interest, these forms also have rational offset surfaces. Considerations related to the automated CNC machining of these surfaces, using only their high-level procedural definitions, are also briefly discussed.

A swept surface is generated from a profile curve and a sweep curve by employing the latter to define a continuous family of transformations of the former. By using polynomial or rational curves, and specifying the homogeneous coordinates of the swept surface as bilinear forms in the profile and sweep curve homogeneous coordinates, the outcome is guaranteed to be a rational surface compatible with the prevailing data types of CAD systems. However, this approach does not accommodate many geometrically intuitive sweep operations based on differential or integral properties of the sweep curve - such as the parametric speed, tangent, normal, curvature, arc length, and offset curves - since they do not ordinarily have a rational dependence on the curve parameter. The use of Pythagorean-hodograph (PH) sweep curves surmounts this limitation, and thus makes possible a much richer spectrum of rational swept surface types. A number of representative examples are used to illustrate the diversity of these novel swept surface forms - including the oriented-translation sweep, offset-translation sweep, generalized conical sweep, and oriented-involute sweep. In many cases of practical interest, these forms also have rational offset surfaces. Considerations related to the automated CNC machining of these surfaces, using only their high-level procedural definitions, are also briefly discussed.

Lawrence Edwards has shown that many flowering plants have buds with an outline in the form of a path curve, the curve that a point follows under repeated projective transformation of the plane into itself. Edwards, however, did not give a formula for these curves nor did he fit the curves by the standard method of least squares. This paper gives an elementary exposition of the method used by Edwards, shows its relation to projective geometry, and then uses homogeneous coordinates, linear differential equations and characteristic values and vectors of a matrix to derive the formula for path curves. This formula is then used to fit path curves by least squares to data provided by Edwards for the buds of 150 plants. Most buds are fit very closely.

Cílem této bakalářské práce je vytvoření metod pro získání 3D dat. Tyto data mají být použitelná jako vstupní data pro algoritmy, které je zpracují do formátů zprostředkující prostorový vjem člověka. Je řešen problém rektifikace, kdy srovnáváme jeden snímek podle druhého, a výpočet modelu scény snímané dvěma kamerami., The aim of this bachelor thesis is to create methods for obtaining 3D data from 2D images. This data will be used as an input data for algorithms which will process them into data formats creating 3D perception. In this work problem of rectification when we compare one image according to the other is solved as well as the problem of calculation of the model of a scene scanned with two cameras.

Cílem této bakalářské práce je vytvoření metod pro získání 3D dat. Tyto data mají být použitelná jako vstupní data pro algoritmy, které je zpracují do formátů zprostředkující prostorový vjem člověka. Je řešen problém rektifikace, kdy srovnáváme jeden snímek podle druhého, a výpočet modelu scény snímané dvěma kamerami., The aim of this bachelor thesis is to create methods for obtaining 3D data from 2D images. This data will be used as an input data for algorithms which will process them into data formats creating 3D perception. In this work problem of rectification when we compare one image according to the other is solved as well as the problem of calculation of the model of a scene scanned with two cameras.

Cílem této bakalářské práce je vytvoření metod pro získání 3D dat. Tyto data mají být použitelná jako vstupní data pro algoritmy, které je zpracují do formátů zprostředkující prostorový vjem člověka. Je řešen problém rektifikace, kdy srovnáváme jeden snímek podle druhého, a výpočet modelu scény snímané dvěma kamerami., The aim of this bachelor thesis is to create methods for obtaining 3D data from 2D images. This data will be used as an input data for algorithms which will process them into data formats creating 3D perception. In this work problem of rectification when we compare one image according to the other is solved as well as the problem of calculation of the model of a scene scanned with two cameras.