The three harmonic homologies theorem Heidari, Fahimeh; Honari, Bijan
Expositiones mathematicae,
March 2024, 2024-03-00, Letnik:
42, Številka:
2
Journal Article
Recenzirano
In this paper, we give a complete answer to the question: “Under what conditions the product of three harmonic homologies of the real projective space Pn(R) is a harmonic homology again?” Among other ...things, we prove the three harmonic homologies theorem in Pn(R) by which the product of three harmonic homologies with collinear centers is again a harmonic homology if and only if the hyperplanes are polars of the centers with respect to a quadric. It is shown that the three reflections theorem, the three inversions theorem, notably Pascal’s theorem and Miquel’s theorem in Laguerre geometry are special cases of this theorem.
Diffraction tomography of strain Lionheart, W R B; Withers, P J
Inverse problems,
04/2015, Letnik:
31, Številka:
4
Journal Article
Recenzirano
Odprti dostop
We consider whether it is possible to recover the three dimensional strain field tomographically from neutron and x-ray diffraction data for polycrystalline materials. We show that the distribution ...of strain transverse to a ray cannot be deduced from one diffraction pattern accumulated along that path, but that a certain moment of that data corresponds to the transverse ray transform of the strain tensor and so may be recovered by inverting that transform given sufficient data. We show that the whole strain tensor can be reconstructed from diffraction data measured using rotations about six directions that do not lie on a projective conic. In addition we give an inversion formula for complete data for the transverse ray transform. We also show that Bragg edge transmission data, which has been suggested for strain tomography with polychromatic data, cannot provide the strain distribution within the material but only the average along the ray path.
Degenerations of Pascal lines Chipalkatti, Jaydeep; Da Silva, Sergio
Beiträge zur Algebra und Geometrie,
09/2023, Letnik:
64, Številka:
3
Journal Article
Recenzirano
Odprti dostop
Let
K
denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points
A
,
B
,
C
,
D
,
E
,
F
on
K
, the three intersection points
A
E
∩
B
F
,
A
D
∩
C
F
,
...B
D
∩
C
E
are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s
hexagrammum mysticum
.
As one of the classical theories of projective geometry, Pascal’s theorem is closely related to the conic, which is the image of a circle. A circle is the geometric element typically used as a ...template for camera calibration methods. In this work, we propose a calibration method based on Pascal’s theorem and its inference, in which a circle is used as the calibration template. The proposed calibration method can be applied to solve the image of the circular points via the Newton iteration method. The intrinsic parameters of the camera can then be determined based on the constraints of the images of the circular points and the image of the absolute conic. The results of simulations and experiments conducted verify the validity and feasibility of the proposed calibration method.
The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues’s Theorem, harmonic conjugates, projectivities, ...involutions, conics, Pascal’s Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify the consistency of the axiom system; it is based on the usual model in three-dimensional Euclidean space, using only constructive properties of the real numbers. The methods of strict constructivism, following principles put forward by Errett Bishop, reveal the hidden constructive content of a portion of classical geometry. A number of open problems remain for future studies.
In this article we check, with the Mizar system
, Pascal’s theorem in the real projective plane (in projective geometry Pascal’s theorem is also known as the Hexagrammum Mysticum Theorem)
. Pappus’ ...theorem is a special case of a degenerate conic of two lines.
For proving Pascal’s theorem, we use the techniques developed in the section “Projective Proofs of Pappus’ Theorem” in the chapter “Pappus’ Theorem: Nine proofs and three variations”
. We also follow some ideas from Harrison’s work. With HOL Light, he has the proof of Pascal’s theorem
. For a lemma, we use PROVER9
and OTT2MIZ by Josef Urban
,
,
. We note, that we don’t use Skolem/Herbrand functions (see “Skolemization” in
).
Projective invariants are not only important objects in mathematics especially in geometry,but also widely used in many practical applications such as in computer vision and object recognition. In ...this work,we show a projective invariant named as characteristic number,from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property,two high-dimensional generalizations of Pascal＇s theorem are given,one establishing the connection of hypersurfaces of distinct degrees,and the other concerned with the intersections of a hypersurface and a simplex.
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