Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups.

Thus, in Euclidean geometry three non-collinear points determine a circle ( as the circumcircle of the triangle they define ), but four points in general do not ( they do so only for cyclic quadrilaterals ), so the notion of " general position with respect to circles ", namely " no four points lie on a circle " makes sense.

Thus through transformations students learn about Euclidean plane isometry.

Thus, non-Euclidean geometry did not contradict Euclidean geometry, but integrated it into a larger framework.

Thus ( x + y ) and ( x − y ) each contain proper factors of n. Computing the greatest common divisors of ( x + y, n ) and of ( x − y, n ) will give us these factors ; this can be done quickly using the Euclidean algorithm.

Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry.

Thus differential geometry may study differentiable manifolds equipped with a connection, a metric ( which may be Riemannian, pseudo-Riemannian, or Finsler ), a special sort of distribution ( such as a CR structure ), and so on.

Thus, the only logical possibilities are to accept non-Euclidean geometry as physically real, or to reject the entire notion of physical tests of the axioms of geometry, which can then be imagined as a formal system without any intrinsic real-world meaning.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Thus one can understand equations by a pure understanding of abstract topology or geometry — this idea is of importance in algebraic geometry.

Thus, maybe, causality lies in the foundation of the spacetime geometry.

Thus some prior information of the geometry is needed to define the correct normal direction.

Thus, in spherical geometry angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects ; for example, the sum of the interior angles of a triangle exceeds 180 degrees.

Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube.

Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry.

Thus, all of Vasquez ' work leads directly to the application of physical technology to directly access superspace geometry.

Thus, M is a space which has a point for every object that could appear in a family, and whose geometry reflects the ways objects can vary in families.

Thus, the value of a geometric analog to is 4 in this geometry.

Thus the geometry of the manifold is infinitesimally identical to that of the Klein geometry, but globally can be quite different.

Thus, generalizing the covariant derivative in bosonic geometry which constructs tensors from tensors, the superspace covariant derivative constructs superfields from superfields.

Thus inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students.

Thus, the initial step corresponds to k =

Thus, one would expect to find a factor that roughly corresponds to " left " and " right ", as this is the dominant framing for politics in our society, but the basis of Eysenck's " tough / tender-minded " thesis ( the second, " T "- factor ) may well represent nothing beyond an abstract mathematical construct.

Thus every analytic function corresponds to a steady incompressible, irrotational fluid flow in the plane.

Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation.

Thus, sensitive detection of specific optical frequencies necessitates optical heterodyne detection, in which two different ( close-by ) wavelengths of light illuminate the detector so that the oscillating electrical output corresponds to the difference between their frequencies.

Thus, a primary decomposition of ( n ) corresponds to representing ( n ) as the intersection of finitely many primary ideals.

Thus, the DCT-I corresponds to the boundary conditions: x < sub > n </ sub > is even around n = 0 and even around n = N-1 ; similarly for X < sub > k </ sub >.

Thus, the bow room corresponds exactly to the saloon in terms of its importance to the two smaller suites.

Thus, the DST-I corresponds to the boundary conditions: x < sub > n </ sub > is odd around n =- 1 and odd around n = N ; similarly for X < sub > k </ sub >.

Thus 1989 corresponds to Shōwa 64 up to the seventh day of the first month ( 7 January ) and to since the eighth day of the first month ( 8 January ).

Thus, swapping left for right in a sequent corresponds to negating all of the constituent formulae.

Thus D corresponds to 1 March, G to 1 April, B to 1 May, E to 1 June, G to 1 July, C to 1 August, F to 1 September, A to 1 October, D to 1 November, and F to 1 December — a result which Durandus recalled by the following distich:

Thus " kva ", corresponds to " 10 21 0 ".

Thus, the year 2000 CE corresponds to the year 1922 of the Salivahana Era.

Thus, Mandarin qì ( 氣, ' breath, air, spirit ') corresponds to Japanese ki.

Thus, the Mandarin reading of " Tokyo " ( 東京 ; Eastern ( 東 ) Capital ( 京 )) is Dōngjīng ; this corresponds to Japanese Tōkyō, with sound history for 京 being supposed approximately * kiæŋ -> kyau -> kyō ( for comparison: Southern Min 京 ( colloquial ) is kiaⁿ with a nasal diphthong ).

) Thus Mandarin Rìběn ( 日本, Japan ) corresponds to Japanese Nippon.

Thus, Mandarin Huángbò ( 黄檗 ) corresponds to Japanese Ōbaku, and Rúlái ( 如来 ) to Nyorai.

Thus, Mandarin hànzì ( 漢字 ) corresponds to Japanese kanji, hànwén ( 漢文, Chinese written language ) to kanbun, and zuìhòu ( 最後, last ) to saigo.

Thus J corresponds to multiplication by i on the ( 1, 0 )- vector fields of the complexified tangent bundle, and multiplication by − i on the ( 0, 1 )- vector fields.

Thus, each cell of the matrix corresponds to the strength of the association between the row item and the column item.

Thus at each step one can either go down ( append a (, 1 ) to the end ) or go right ( add 1 to the last number ) ( except the root, which is extra and can only go down ), which shows the correspondence between the infinite binary tree and the above numbering ; the sum of the entries ( minus 1 ) corresponds to the distance from the root, which agrees with the 2 < sup > n − 1 </ sup > nodes at depth n − 1 in the infinite binary tree ( 2 corresponds to binary ).