Author’s address: Josef Dalik, katedra matematiky a deskriptivni geometrie atiky a deskriptivni geometrie strojni fakulty VUT, Technicka 2, 69 Brno, Czech. Rozvrh předmětu Deskriptivní geometrie (design) v zimním semestru a. r. / 15 [en], Design – Descriptive geometry, Courses, Department of. Found it Kolomajzlik found Deskriptivni geometrie. Monday, July 25, Jihomoravský kraj, Czechia.

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We show that for a rational surface the Gram determinant of its tangent space is a perfect square if and only if the Gram determinant of its normal space is a perfect square. Consequently the polynomial surfaces of a given degree with polynomial area element can be constructed from the prescribed normal fields solving a system of linear equations. The degree of the constructed surface depending on the degree and the properties of the prescribed normal field is investigated.

We use the presented approach to interpolate a network of points and associated normals with piecewise polynomial surfaces with polynomial area element. The gauss map at the inflection point is not regular and in the neighborhood is typically not injective.

The support function is thus not regular and typically multivalued. We describe this function using an implicit algebraic equation and the rational Puiseux series of its branches. We show the correspondence between the degree of the approximation of the primary curve using Taylor series and the degree of the approximation of the support function using Puiseux series.

Based on this results we are able to approximate curve with inflections by curves with a simple support function which consequently possess rational offsets. We also analyze the approximation degree of this kind of dual approximation. Such a phenomenon was experienced e.

### Závěrečná práce: Mgr. Jindřich Červinka: Rotační plochy

We propose an alternative adaptive subdivision scheme, which ensures the approximation degree 4 both for the inflection—free segments and the segments with inflections. References [1] Gruber, P. Handbook of convex geometry. Curves and surfaces represented by polynomial support functions.

Theoretical Computer Science— Hermite interpolation by hypocycloids and epicycloids with rational offsets. Computer Aided Geometric Design 27, — There are examples showing that the multiplicity of the intersection can be easily composed out of the intersection multiplicity of the corresponding curve and there are others where it is not so direct.

The known algebraic description lacks geometric intuition or interpretation behind. We suggest certain explanations. Very often, in technical practice, we meet the concept of oval. This term denotes a curve composed of circular arcs, but many times also the ellipse itself.

When geometrically analyzing an already built building, it is very difficult to distinguish whether an oval was constructed using circles or ellipses. The quality of approximate constructions may be the reason. In the contribution, we will focus on some interesting constructions used by architects, theorists such as Sebastiano Serlio and Guarino Guarini.

A new viewpoint can give us new ideas and concepts that lead to a simplified solution of the mathematical problem. Hence, we deal with rational, elliptic or hyperelliptic curves that are birational to plane curves in the Weierstrass form and thus they are square-root parameterizable.

We design a simple algorithm for computing an approximate piecewise rational parametrization using topological graphs of the Weierstrass curves.

Predictable shapes reflecting a number of real roots of a univariate polynomial and a possibility to approximate easily the branches separately play a crucial role in the approximation algorithm. That means each point has a ball-like neighbourhood.

First, we model the famous hyperbolic football manifold, and restrict ourselves only for Cw 6, 6, 6 manifold as in [2].

### Deskriptivní geometrie II pro 2. ročník SPŠ stavebních. Autor: Musálková

Starting with a re-formulation of metrical quantities in n-dimensional affine space using linear algebraic tools, we can then talk about visualising geometric objects in n-dimensional space. Marrying these two concepts together with the concept of a symmetric bilinear form, we can make significant progress with regards to the understanding of trigonometry in higher dimensions over a general metrical framework.

We will pay specific attention to the three-dimensional space and the most important object in it: The technique we use applies to both types of surfaces, because they can be represented as curves within the afore mentioned quadrics.

These parameters are to be determined by solving a system of algebraic equations. The degrees of the equations admit a prediction of the number of possible solutions. Together with geometric criteria, useful drskriptivni, i.

Our main goal is the interpolation of Gk data at the boundaries deskruptivni ruled surfaces and canal surfaces. Nevertheless, we aim at low degree interpolants, and therefore, we choose the lowest possible n in any case.

Except for the most simple loci such as lines, circles or possibly conics, this topic is not contained in most geometry texts. The reason might be difficulties when visualizing various objects with different movements. The use of dynamic geometry software Cabri, GeoGebra, Sketchpad, Whereas in the past the study of loci by DGS was based on numerical methods, now we are facing the introduction of symbolic methods based on the theory of automated theorem proving into DGS.

The result is the implicit equation of the locus. In the talk a few concrete examples, including drawbacks which can occur in some cases, are given. And the history of developing this idea is interesting either. Who is the original author of the concept? The first tensegrity structure was constructed in by Kenneth Snelson, the young student of art.

But this new idea was described, almost at the same geometrei, by deskritpivni patents prepared by three diferent authors – the scientist, the artist and the architect. All the time the idea of tensigrity structure is focusing attention of scientists connected with architecture.

Nowadays new groups of scientists are looking for the best and precisly definition for tensigrity and they are trying to defined true or false tensigrity structures. Authors want to present the basic concepts of tensigrity structures which are described in the original patents and on the base of this present the realization of tensigrity structures which were realized in architecture.

The normal or orthogonal quadratic cones geometdie circular sections being orthogonal to vertex generators. These cones can be generated by congruent pencils of planes with intersecting axes.

The corresponding conics are the spherical analogues of Thales circles. Equilateral quadratic cones are characterized by a vanishing trace. The associated equilateral spherical conics have the property that the three vertices of a regular right-angled spherical triangle can simultaneously move along.

Dualization yields cones which are the envelopes of triples of mutually orthogonal planes. If cones of this type are tangent to a regular quadric then their apices are located on a sphere. This reveals that in general ellipsoids are still movable within a fixed circumscribed box.

The Universe of Conics. Springer Spektrum, Heidelberg G. The Universe of Quadrics.

## Numerical solution of the Kiessl model

Springer Spektrum, in preparation. Bolek have submitted a new projection method.

The given method is based on imaging of Euclidean space on a plane. A projection plane and two projection directions, not parallel to the plane, are at first assumed. Each point of space has two points assigned, that come from projections of the geomehrie point in the defined directions. The method does not enable restitution of the given point basing on its imaging neither application of metrical constructions. Addition of possibility to save the directions of projection broadens way of using the imaging.

The satellite picture of the same object taken from two different directions may be treated as two-image parallel projection. Thank to the addition, basing on the image, it will be possible to figure out geometrical features of the photographed deskrkptivni.

An actual problem affected by the precision of positioning geodetic points using new GNSS technologies in coordinate systems is the precision of the map projection.

## Rotační plochy

Map projections are coming out of geometric expression of properties of reference surfaces of Earth using methods of differential geometry, as well as of relation between two linear manifolds – reference ellipsoid and map plane.

Choice of cartographic projection is determined by the geometrical characteristics of the territory and choice criteria for distortion of map elements. The aim of this paper is to show the role of geometry and mathematics in the cartography and different options for access to the distortions of the territory, such as optimization of extreme value of distortion, summing and integral criterion on area territory, in some case using criterion with the requirement of a minimum mean value of scale distortion in a given area.

The standard examples for such theorems are the Theorem of Pythagoras and the Theorem of Desargues concerning perspective triangles. But there are many others, too. This construction is based on a mechanical interpretation of the Apollonius definition an E. Tschirnhaus used this idea for constructing tangents of curves defined by a finite set of focal points and constant weighted sum of distances from these focal points.

Therefore, the question arises, what are the conditions which a figure should fulfil to visualise what an author aims at. As an answer it has to be stated that a proper visualisation is only one part, the other, more important, is the addressee.

Therefore, the lecture takes up the cudgels on behalf of a solid education in Geometry, including Descriptive Geometry. The concept of choosing an infinite hyperplane is used for making hypothesis in an affine space to solve projective problems and vice-versa. Their mixtures with the analytic use of homogenous coordinates is applied on projective theorems.